QMAS SiC 7661 24 2 28
SiC SiC QMAS(Quantum MAterials Simulator) VASP(Vienna Ab-initio Simulation Package) SiC 3C, 4H, 6H-SiC EV VASP VASP 3C, 4H, 6H-SiC (0001) (11 20) (1 100) MedeA SiC QMAS - C Si (0001) VASP (0001) QMAS VASP
1 2 2 3 2.1... 3 2.2 SiC... 3 2.2.1 (Lely-method)... 3 2.2.2 (MSE:Metastable Solvent Epitaxy) 4 2.3... 7 2.3.1... 7 2.3.2... 10 3 12 3.1 (first principles calculations)... 12 3.1.1 Schrödinger... 12 3.2 QMAS(Quantum MAterials Simulator)... 13 3.2.1 qmas-atom... 14 3.2.2 qmas-band... 16 3.3 QMAS... 22 4 24 4.1 SiC... 24 4.1.1... 24 4.1.2... 24 4.1.3... 25 4.2... 27 4.2.1... 30 4.2.2... 30 4.2.3... 32 4.2.4... 34 4.2.5... 36 4.2.6... 38 5 41 1
1 SiC VASP(Vienna Ab-initio Simulation Package) SiC 3C, 4H, 6H SiC (0001) (11 20) (1 100) VASP SiC (0001) SiC VASP QMAS(Quantum MAterials Simulator) QMAS SiC VASP 2 SiC SiC 3 QMAS 4 5 2
2 2.1 (SiC) Si C 19 SiC 1955 (Lely ) SiC Lely 10mm 1981 Tairov Lely SiC Lely SiC (2000 ) 1980 SiC SiC (Metastable Solvent Epitaxy) SiC MSE [1] VASP 2 SiC 2.2 SiC 2.2.1 (Lely-method) SiC Lely Lely 3
Si C 2.1 Lely (2200 2400) 2500 [2] 2.1: Lely SiC [2] 2.2.2 (MSE:Metastable Solvent Epitaxy) MSE SiC 2.2 MSE MSE TaC 2 SiC Si MSE (feed) 3C-SiC 4H-SiC 4
(seed) SiC 1800 SiC Si C µm Lely MSE 2.3 MSE Si C C 1800 4S-SiC 2.3(a) 4H 3C 3C 4H 2.3(b) 3C-SiC 4H-SiC SiC µ 2.3(b) 3C µ c 4H µ c 4H 4H-SiC 4H-SiC C C C 2.3(c) Si C 4H-SiC 3C-SiC MSE [2] 2.2: MSE SiC [2] 5
2.3: MSE C C (a) 4H 3C (b) Si-C (c) [2] 6
2.3 2.3.1 2 SiC SiC 2.4 2.5 2.6 3C-SiC 4H-SiC 6H-SiC (111) (0001) (1 10) (11 20) (11 2) (1 100) 4H 6H-SiC (0001) 3C-SiC (111) 2.4 2.6 3C-SiC (1 10) (11 2) 4H 6H-SiC (11 20) (1 100) Si C 3C-SiC (111) 4H 6H-SiC (0001) Si (111) (0001) Si C Si C Si C [2] 7
図 2.4: 3C-SiC の表面 青色で示したのが Si 原子 黒色で示したのが C 原子 図 2.5: 4H-SiC の表面 青色で示したのが Si 原子 黒色で示したのが C 原子 8
図 2.6: 6H-SiC の表面 青色で示したのが Si 原子 黒色で示したのが C 原子 この構造では (0001) 面と (0001 ) 面の終端面の構成元素の組み合わせにより 表 面エネルギーが変わってくる この様子を模式的に表すと図 2.7 のようになり 表 面を覆う Si 原子の割合 (被覆率 coverage ratio)θsi によって 全く違った構造を 取れる事が分かる 本研究では 図 2.7 のようなスラブモデルを作成し それぞれ の表面エネルギーを計算した θsi = 0.5 でないモデルでは モデル中の Si 原子数 と C 原子数の比が 1:1 でなくなる この状態での表面エネルギー計算のためには 化学ポテンシャルに依存した計算法を利用する必要がある そこで 西谷研究室では VASP を用いて化学ポテンシャルに依存した計算手法 を用いて表面エネルギー計算してきた しかし この計算手法だけでは電場勾配 による影響を考慮するという面で不安を残している そこで新たな計算手法での 再計算が必要である 9
2.7: Si θ Si Si-rich C-rich θ Si =0.5 a-type b-type [2] 2.3.2 SiC 2.8 SiC C (a) 3C-SiC (b) 6H-SiC 2.8 (a) (b) 3C 6H SiC 3C 6H Si 1414 (a) 1404 5 10
2.8: SiC [3] (a) Olenski Abbaschian 1996 (b) Fromm Gebhardt 1976 SiC VASP SiC 6H > 4H > 3C VASP 11
3 3.1 (first principles calculations) Schrödinger (first principles calculations) 3.1.1 Schrö dinger Schrödinger Hψ = ɛψ (3.1) ( d2 + V )ψ = ɛψ (3.2) dx2 (Hamiltonian:H) (wave function:ψ) (energy Eigen value:ɛ) (Kinetic Energy) (d 2 ψ/dx 2 ) (Vψ) (potential:v ) (nuclearpotential) ( exchange-correlation interaction) 3.1 3.2 ψ self consistent loop [4] Full Potential pseudo potential 2 PseudoP 12
ultra soft norm [4] (Atomic Orbital) (Plane Wave) 2 AO (Linear Combination) LCAO PW (augument) APW (Linearlized) FP All Electron AE Projector Augmented Wave Pseudo P [4] (Local Density Approximation:LDA) (Generalized Gradient Approximation:GGA) version Perdew-Wang 91 [4] PW (K-space) (Fast Fourier Transformation) (real space) PW (periodic boundary condition) [4] 3.2 QMAS(Quantum MAterials Simulator) QMAS(Quantum MAterials Simulator) PAW(Project Augmented-Wave) 13
QMAS PAW FLAPW(Full-potential Linear Augmented Plane Wave) 3.1 3 FLAPW PAW [5] 3.1: FLAPW PAW FLAPW PAW PAW 3.1 QMAS 3 (qmas-atom qmas-band qmas-gamma) qmas-atom qmas-band 3.2.1 qmas-atom qmas-atom 3.1 3.2 V qmas-band (total number of electrons) (principal quantum number) (azimuthal quantum number) LDA GGA GGA 3.1 qmas-atom 14
3.1: qmas-atom cdir z 3.1 C 6.0 nc C 1 K 2 L 3.1 1 nv 3.1 C 2 1 2p 2 15
iexc LDA GGA 1 LDA 2 GGA nn 1,2,3.. K L M.. ll 0,1,2.. s p d f.. 3.1 C 3 5 occ C 2 3.1 2.0 icv 1 2 3.1 C 3 1 2 2 3.2.2 qmas-band qmas-band Schrödinger 3.1 3.2 ɛ qmas-atom self consistent loop QMAS (rydberg:ry) 16
(hartree:e h ) 3.2 3.4 qmas-atom 4H-SiC 3.2 3.3 3.4 3.2: qmas-band title 8 unit 1 2 Bohr number atom Si 4 C 4 3.3 17
number element 4H-SiC Si C 2 lattice v angle v (Primitive Vector) a,b,c 4H-SiC a : b : c = 3.09286:3.09286:10.1222[ ] α : β : γ = 90:90:120[ ] element list qmas-atom qmas-band 3.3: qmas-band number space 230 1 4H-SiC P6 3 mc 3.3 186 18
number wycoff pos number atom number space 4H-SiC 4 wycoff position 2 (1/3,2/3,1/4) a 1/3 b 2/3 c 1/4 (1/3,2/3,1/4) (0,0,0) (1.309,0.755,2.531) 3 Primitive Vector P 0,P 1,P 2 (x, y, z) (X, Y, Z) P 0 = p 0,0 p 0,1 p 0,2,P 1 = p 1,0 p 1,1 p 1,2,P 2 = p 2,0 p 2,1 p 2,2 (3.3) x B = y (3.4) z X Y = P 0 x + P 1 y + P 2 z (3.5) Z 2 2 19
3.4: qmas-band 20
cutoff wf QMAS Ry VASP number max iteration 100 100 10 number max relax 1 2 th charge 1 cal stress 1 2 21
3.5 (force) cell [4] 3.5: [4] 3.3 QMAS VASP QMAS 2 Si C E SiC(surface) E H SiC(slab) 22
E SiC(bulk) E H(bulk) E SiC(surface) = E H SiC(slab) E SiC(bulk) E H(bulk) (3.6) QMAS QMAS SiC 2.8 QMAS SiC VASP 23
4 4.1 SiC QMAS 3C 4H 6H-SiC 2.8 QMAS 4.1.1 E-V E-V x y y 2 E-V x y [4] 4.1.2 4.1 4.2 Qmas SiC 24
4.1: SiC lattice constant polytype length[ ] angle[ ] symmetry c/a a b c 3C 4.36 4.36 4.36 1 90 90 90 F-43m 4H 3.08 3.08 10.05 3.263 90 90 120 P6 3 mc 6H 3.08 3.08 15.12 4.909 90 90 120 P6 3 mc polytype 4.2: atomic coordinate Si C x y z x y z 3C 0 0 0 0.25 0.25 0.25 4H 0 0 0 0 0 0.17 0.33333 0.66667 0.25 0.33333 0.66667 0.43750 0 0 0 0 0 0.125 6H 0.33333 0.66667 0.66667 0.33333 0.66667 0.45833 0.33333 0.66667 0.33333 0.33333 0.66667 0.79167 4.1.3 4.1 (a) 3C 4H 6H-SIC E-V (B) 4.3 3C > 6H > 4H 3C E-V 0K 4.1 2.8 25
4.1: 3C,4H,6H-SiC (a) (E-V) (b) E-V 4.3: lattice constant polytype length[ ] V[ ] E[eV/SiC] c/a a b c 3C 4.3502 4.3502 4.3502 1 20.5815-885.3260 4H 3.0718 3.0718 10.0533 3.2505 20.9318-884.9040 6H 3.1077 3.1077 15.2440 4.9274 20.6852-885.1308 VASP SiC 3C 4H SiC 4.2 3C 4H 4.1 4.2 4.2 Qmas SiC Qmas 26
4.2: SiC 4.2 SiC QMAS QMAS 1 2.8 SiC 4.4 4.5 VASP VASP 2 SiC Lely C C-rich SiC (0001) 4.3(a) MSE TaC 2 3C-SiC Si 1800 4H-SiC 4H-SiC Si Si-rich 4.3(b) 2 SiC 4.4 4.5 4.4 Si-rich (0001) 4.5 C-rich 27
(0001) Si-rich C-rich Si-rich C-rich Si-rich MSE SiC C-rich Lely SiC 4.3: SiC [2] 28
4.4: Si-rich SiC [2] 4.5: C-rich SiC [2] 29
4.2.1 1. MedeA SiC 2. QMAS 1 3. QMAS Maple 4. 5. 1 4 Maple 4.2.2 2 4.6 4.8 VESTA(Visualization for Electronic and STructural Analysis) 4H-SiC (11 20) Si C 4.6 (1 100) 4.7 (0001) 4.8 (11 20) - - Lely - MSE - 30
4.6: VESTA 4H-SiC (11 20) (1 100) Si C 4.7: VESTA 4H-SiC (11 20) (0001) Si C 31
4.8: VESTA 4H-SiC (11 20) (11 20) Si C 4.2.3 Si C SiC 3C, 4H, 6H-SiC 4H, 6H-SiC (0001) (11 20) (1 100) 3C-SiC (111) (1 10) (11 2) QMAS E Si(bulk) E C(bulk) E SiC(bulk) E SiC(slab) E Si(bulk) E C(bulk) 4.1 QMAS E SiC(bulk) E SiC(slab) E = E SiC(slab) E SiC(bulk) (4.1) 4.2 ( E) (S [m 2 ]) (E SiC(surface) ) E SiC(surface) = E S (4.2) 4.4 3 3 4.9 4.10 10 2 [2] 32
4.9: (0001) (11 20) (1 100) (111) (1 10) (11 2) [2] 4.10: [2] 33
4.4: SiC lattice constant polytype length[ ] angle[ ] symmetry c/a a b c 3C 4.36 4.36 4.36 1 90 90 90 F-43m 4H 3.08 3.08 10.05 3.263 90 90 120 P6 3 mc 6H 3.08 3.08 15.12 4.909 90 90 120 P6 3 mc 4.2.4 SiC (0001) (000-1) 2 2.7 4 θ Si =0.5 Si C 1:1 4.2 µ E i(bulk) n i 4.3 µ i = E i(bulk) n i (4.3) SiC µ SiC(bulk) Si µ Si C µ C A B AB AB=A+B+ H f SiC 4.4 H f µ Si + µ C = µ SiC(bulk) = µ Si(bulk) + µ C(bulk) + H f (4.4) 4.4 SiC (µ Si µ C ) H f 0 (4.5) µ Si(bulk) + H f µ Si µ Si(bulk) (4.6) µ C(bulk) + H f µ C µ C(bulk) (4.7) 34
4.6 4.7 µ Si µ C 4.4 µ Si = µ Si(bulk) µ C = µ C(bulk) + H f C-rich Si-rich E SiC(suface) (S [m 2 ]) E SiC(suface) = E SiC(slab) n Si (µ Si(bulk) + H f ) n C µ C(bulk) S (C rich) (4.8) E SiC(suface) = E SiC(slab) n Si µ Si(bulk) n C (µ C(bulk) + H f ) (Si rich) (4.9) S 4.8 4.9 4.11 Si-C Si100 % C100 % SiC 1:1 Si C SiC SiC Si SiC Si µ Si(bulk) SiC µ Si SiC µ C SiC Si SiC µ Si = µ Si(bulk) µ C = µ C(bulk) + H f C-rich µ C = µ C(bulk) µ Si = µ Si(bulk) + H f [2] 4.11: Si-C Si100 % C100 % SiC 1:1 [2] 35
4.2.5 QMAS Maple E-V 4.12 4.13 4.12 3C-SiC(1 10) Maple 36 4.13 3C-SiC(111) θ Si =1 96 4.12 4H, 6H-SiC 4.13 4H-SiC error Maple 36
4.12: 3C-SiC(1 10) Maple 4.13: 3H-SiC(111) Si-rich Maple 37
4.2.6 SiC 4.5 4.5: SiC [n i ] Cutoff-E k-mesh polytype Energy [ev] n i Cutoff-E [ev] k-mesh 3C-SiC -.88531813E+03 n Si :n C =1:1 600 10 10 10 4H-SiC -.17698033E+04 n Si :n C =2:2 600 884 6H-SiC -.26553633E+04 n Si :n C =3:3 600 882 3C-Si -.63181719E+03 n Si :n C =1:0 408 444 3C-C -.25015897E+03 n Si :n C =0:1 408 888 (0001) (11 20) (1 100) 4.6 4.6: (0001) (11 20) (1 100) [n i ] Cutoff-E k-mesh polytype Surface Energy [ev] n i Cutoff-E [ev] k-mesh 3C-SiC (1 10) -.14152012E+05 n Si :n C =16:16 600 513 (11 2) -.21228032E+05 n Si :n C =24:24 600 251 4H-SiC (11 20) -.28298307E+05 n Si :n C =32:32 600 312 (1 100) -.28308125E+05 n Si :n C =32:32 600 512 6H-SiC (11 20) -.42447364E+05 n Si :n C =48:48 600 331 (1 100) -.42453879E+05 n Si :n C =48:48 600 551 (0001) 4.7 polytype a-type b-type 2.7 θ Si = 1/2 38
4.7: (0001) (θ Si ) [n i ] Cutoff-E k-mesh a-type b-type 2.7 θ Si = 1/2 polytype coverage (θ Si ) Energy [ev] n i Cutoff-E [ev] k-mesh 3C-SiC θ Si = 0 -.43445514E+05 n Si :n C =48:52 600 331 a-type θ Si = 1/2 -.42448991E+05 n Si :n C =48:48 600 331 b-type θ Si = 1/2 -.42408042E+05 n Si :n C =48:48 600 331 θ Si = 1 -.45003561E+05 n Si :n C =52:48 600 251 4H-SiC θ Si =0 error n Si :n C =64:68 600 331 a-type θ Si = 1/2 error n Si :n C =64:64 600 331 b-type θ Si = 1/2 error n Si :n C =64:64 600 331 θ Si =1 error n Si :n C =68:64 600 251 6H-SiC θ Si =0 error n Si :n C =48:52 600 331 a-type θ Si = 1/2 -.42449090E+05 n Si :n C =48:48 600 331 b-type θ Si = 1/2 -.42451077E+05 n Si :n C =48:48 600 331 θ Si = 1 -.45003537E+05 n Si :n C =52:48 600 251 4.8 4.9 4.8: [J/m 2 ]. 1[eV]=1.60218 10 19 [J] Surface 3C-SiC 4H-SiC 6H-SiC (11 20) 3.90 2.77 3.82 (1 100) 13.98 2.26 5.49 39
4.9: [J/m 2 ]. 1[eV]=1.60218 10 19 [J] environment coverage (θ Si ) 3C-SiC 4H-SiC 6H-SiC Si-rich θ Si = 0 30.62 error error a-type θ Si = 1/2 20.23 error 16.05 b-type θ Si = 1/2 38.13 error 15.18 θ Si =1 8.30 error 9.30 C-rich θ Si = 0 22.03 error error a-type θ Si = 1/2 20.23 error 16.05 b-type θ Si = 1/2 38.13 error 15.18 θ Si = 1 12.48 error 15.67 40
5 4 3C-SiC (1 10), (11 2) VASP QMAS 5.1 3C-SiC (111) Si-rich C-rich 5.2 QMAS VASP (1 10) VASP (11 2) 5.1: 3C-SiC (1 10), (11 2) VASP QMAS 41
5.2: 3C-SiC (111) Si-rich C-rich VASP QMAS QMAS VASP (111) 4 4.13 VASP Maple QMAS SiC QMAS QMAS 42
[1] SiC (2009) [2] (2011) [3] R.W.Olensinski, G.J.Abbaschian The C-Si System (1984) [4] VASP (2011) [5] ProjectorAugmented Wabe http://www.cmp.sanken.osakau.ac.jp/miniws0607/abstracts/ishibashi.pdf. 43
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