::::::::::::::::::::::::::::::: Si/SiO 2 ::::::::::::::::::: Si/SiO 2 ::::::::::::::::::::::::: Si/SiO 2 :::::::::::::

Transcription

1 Si/SiO 2 1p ο 78p

2 ::::::::::::::::::::::::::::::: Si/SiO 2 ::::::::::::::::::: Si/SiO 2 ::::::::::::::::::::::::: Si/SiO 2 ::::::::::::::::::::: Si/SiO 2 : : : : : ::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::: book-keeping :::::::::::::::::::::::::: Ewald ::::::::::::::::::::::::::: :::::::::::::::::::::: :::::::::::::::::::::::::: Born-Oppenheimer :::::::::::::::::::::: :::::::::::::::::: ::::::::::::::::::::: : : : VASP ::::::::::::::::: ::::::::::::::::::: :::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::: ::::::::::: :::::::::::::::::::::::::::::: 33 1

3 3.3.1 ::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::: SiO 2 ::::::::::::::::::::::::::::: Si :::::::::::::::::::::::::::: SiO 2 ::::::::::::::::::::::::::: c-si/c-sio 2 :::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::: ::::::::::::::::: ::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::: Ewald :::::::::::::::: :::::::::::::::::::::::::: ::::::::::::::::::::::::::: ,, ::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::: ::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::: c-si/c-sio 2 :::::::::::::::::::::: :::::::::::::::::::::: a-sio 2 /c-si ::::::::::::::::::::::::::

4 1.1 A schematic illustlation of MOS-DRAM and various types of silicon dioxide films.([1]) ::::::::::::::::::::::::::::: A schematic illustlation of Si/SiO 2 interface.([1]) ::::::::::::: A schematic illustlation of pseudotridymite Si/SiO 2 interface.([2]) :::: Intermediate-oxidation states at Si/SiO 2 (100) interface, identified by their Si 2p core-label shifts([5]). :::::::::::::::::::::::: Periodic boundary condition :::::::::::::::::::::::: book-keeping method for MD calculation ::::::::::::::::: Ewald sum ::::::::::::::::::::::::::::::::: rectangle approximated band :::::::::::::::::::::::: Binding energies as a function of volume. ::::::::::::::::: Binding energies as a function of volume in Ref.[25]. : : : : : : : : : : : ff-quartz structure obtained by DFT caluculation : : : : : : : : : : : : : fi-quartz structure obtained by DFT caluculation : : : : : : : : : : : : : ff-cristobalite structure obtained by DFT caluculation : : : : : : : : : : : Ideal fi-cristobalite structure obtained by DFT caluculation :::::::: Ideal-fi-tridymite structure obtained by DFT caluculation. : : : : : : : : : Stishovite structure obtained by DFT caluculation. :::::::::::: I 1 O structure obtained by DFT caluculation. :::::::::::::::: V 1 O structure obtained by DFT caluculation. :::::::::::::::: c-si/c-sio 2 interface structure obtailned by my DFT calculation. Left panel is Q model. Center panel is C model. Right panel is T model. :::: Floachart of GA. :::::::::::::::::::::::::::::: ff-quartz structre before annealing(upper panel) and after annealing(lower panel). ::::::::::::::::::::::::::::::::::: ff-cristobalite structre before annealing(left panel) and after annealing(right panel). ::::::::::::::::::::::::::::::::::: 62 3

5 6.3 Q model strucure before annealing(upper panel) and after annealing(lower panel). ::::::::::::::::::::::::::::::::::: C model structre before annealing(left panel) and after annealing(right panel). ::::::::::::::::::::::::::::::::::: T model structre before annealing(left panel) and after annealing(right panel). ::::::::::::::::::::::::::::::::::: Amorphous SiO 2 structure ::::::::::::::::::::::::: pair correlation function of amorphous SiO 2 :::::::::::::::: Si-O-Si bond angle distribution in amorphous SiO 2 obtained by my MD calculation. ::::::::::::::::::::::::::::::::: Si-O-Si bond angle distribution of amorphous SiO 2 by experimental data (curve 2 is silica glass). :::::::::::::::::::::::::: Structure of a-sio 2 /c-si and suboxide atoms at interface. From left panel, suboxide 1 + atoms, suboxide 2 + atoms, suboxide 3 + atoms and all atoms : 72 4

6 2.1 Calculation Conditions ::::::::::::::::::::::::::: Calculations for equation of states ::::::::::::::::::::: Calculation conditions of DFT calculations. ::::::::::::::: Structural prameters for ff-quartz obtained by our DFT calculation. : : : Structural prameters for fi-quartz obtained by our DFT calculation. : : : Structural prameters for ff-cristobalite obtained by our DFT calculation. : Structural prameters for ideal-fi-cristobalite obtained by our DFT calculation. ::::::::::::::::::::::::::::::::::: Structural prameters for ideal-fi-tridymite obtained by our DFT calculation. :::::::::::::::::::::::::::::::::::: Structural prameters for stishovite obtained by our DFT calculation. : : : Defect formation energies of an oxygen interstitials in silicon crystal obtained by my DFT calculation. (I N O means silicon crystal in which N oxygen atoms are added around one Si atom. ). :::::::::::::::: Defect formation energies for oxygen vacancies in ff-quartz obtained by my DFT calculation (V N O means ff-quartz in which N oxygen atoms are removed around one Si atom.). :::::::::::::::::::::: Interface enegies obtained by my DFTcalculation. Reference states are silicon crystal and suitably strained SiO 2 crystal. Energies are in ev. : : : body potential parameters :::::::::::::::::::::::: body potential parameters :::::::::::::::::::::::: body potential parameters :::::::::::::::::::::::: Structural prameters for ff-quartz obtained by our MD calculation. :::: Structural prameters for fi-quartz obtained by our MD calculation. :::: Structural prameters for ff-cristobalite obtained by our DFT calculation. : Structural prameters for ideal-fi-cristobalite obtained by our DFT calculation. ::::::::::::::::::::::::::::::::::: 63 5

7 6.8 Defect formation energies of an oxygen interstitials in silicon crystal by my MD calculation. (I N O means silicon crystal in which N oxygen atoms are added around one Si atom. ). ::::::::::::::::::::: Defect formation energies for oxygen vacancies in ff-quartz obtained by my MD calculation (V N O means ff-quartz in which N oxygen atoms are removed around one Si atom.). :::::::::::::::::::::: Interface enegies obtained by my MD calculation. Reference states are silicon crystal and suitably strained SiO 2 crystal. Energies are in ev. : : : O coordination number of Si atom in amorphous SiO 2 structure. : : : : : Si coordination number of O atom in amorphous SiO 2 structure. : : : : : Density of amorphous SiO 2. ::::::::::::::::::::::: 70-6

8 Si/SiO 2 Si, (SiO 2 ),., Si,,,,, (Fig. 1.1 ), Si Si/SiO 2. Fig. 1.1: A schematic illustlation of MOS-DRAM and various types of silicon dioxide films.([1]),, (CVD).,. 7

9 ,,,, Si/SiO 2 CVD. Si/SiO 2,,,,., 10nm,, Volmer-Weber VW Capillary-induced growth stress VW Si/SiO Si SiO 2. Si, a- SiO 2, c-si., Fig. 1.2 SiO 2 Si SiO x. a-sio 2 Si/SiO 2,, Si-Si, Si/SiO 2 Si,, Si., Si/SiO 2 SiO 2., Si 1, SiO 2, SiO 2,. A. Ourmazd et al.[2] (HR-TEM) Si/SiO 2 (001), Fig.1.3 8

10 Fig. 1.2: A schematic illustlation of Si/SiO 2 interface.([1]) Si., Si/SiO 2 (001) T. Fig. 1.3: A schematic illustlation of pseudotridymite Si/SiO 2 interface.([2]) Y. Iida et al.[3], Si/SiO 2 (001), X CTR,., Si/SiO 2 (001) C. Kageshima and Shiraishi[4] Si(100),., Si/SiO 2 (001) Q. 9

11 1.1.3 Si/SiO Si/SiO 2 SiO 2. Si-O x Si 4 x (x=1,2,3,4)( Si x+ ). Fig. 1.4 Himpsel et al.[5] Si/SiO 2 (100) Si2p. Si/SiO 2 (100) Si 1+,Si 2+,Si 3+,Si 4+. Fig. 1.4: Intermediate-oxidation states at Si/SiO 2 (100) interface, identified by their Si 2p core-label shifts([5]). Si SiO 2, 10

12 [6][7], [7]. Si sp3. SiO 2 Si O, O, 50% sp3, 50%., Si/SiO 2 Si,,. Capron et al.[8] Si/SiO 2. [9] Si/SiO 2 Si/SiO 2. R. Beczko et al.[10] T, Q, C. T.Yamasakiel al.[11] T, Q, C, SiO 2. A. Bongiorno and A. Pasquarello[7] Si/SiO 2,. Kageshima and Shiraishi[4] Si(001),., 10,. Si/SiO 2. J. Tersoff[12] Keating, a-si/c-sio 2 (001),. T. Watanebe and I. Ohdomari SW Si/SiO 2 MO [13], Si(001). T. Umeno et al.[14] Tersoff Si/SiO 2., Si/SiO 2. 11

13 1.2 c-si/a-sio 2.,,, c-si/a-sio 2., c-si/a-sio 2, Si/SiO 2., Si/SiO ,,. 2,. 3,. 4,. 5,. 6,. 7 12

14 2 13

15 2.1,,. 2.2,,,.. (1) (2) (book-keeping ) (3) (4) Verlet, t (5) (6) (2) ,.., L 1 (unit cell), (image unit) ( 2.1),,, book-keeping,. 14

16 boundary boundary boundary unit cell boundary image cell Fig. 2.1: Periodic boundary condition r c, r c,., r c, r c R c (>r c ). book-keeping R c, N up. T K v T ave. 3 ( ), v T max ' 3vT ave, ( )N up t r c = v T N t, r max c r c N up, R c (= r c + r c ) r c Ewald,. Ewald., 15

17 r c Rc r c Fig. 2.2: book-keeping method for MD calculation,., Fig. 2.3(a) i Fig. 2.3(b) Fig. 2.3(c) i ( ), Fig. 2.3(c) , i, i. 2,,. 3 i, i.,. Φ = e2 Φ 1 = 1 2 Φ 2 = 1 4ß 2 Ω 4ß" (Φ 1 +Φ 2 +Φ 3 ) (2.1) X X i j q i q j X X 0X R erfc (ff jr ij + Rj) 0 X q i q j exp g! 4ff 2 i j 0 ψ g2 X 4ß = exp Ω g jr ij + Rj ψ g2 4ff 2! (2.2) cos (g r ij ) 1 g 2 (2.3) 16

18 Fig. 2.3: Ewald sum 17

19 2( X 2 ( X 4 z i cos (g r i )) + i i z i sin (g r i ) ) g 2 (2.4) Φ 3 = p ff X z 2 ß i (2.5) i, Φ, Φ 1 Fig. 2.3(b), Φ 2 Fig. 2.3(c), Φ 3 Fig. 2.3(c). q i i, r i i, r ij i j, R, ff, g, Ω., (2.2) 0 i=j R = 0, (2.3)(2.4) 0 g = 0.,. F 1;i = 1 4ß" = e2 4ß" q i (2.6) X j 0 X q j (r ij + R) jr ij + Rj R e 2 8 < : 2ff p ß exp ff 2 jr ij + Rj 2 + erfc ff jr ij Rj = ; jr i j + Rj 2 (2.7) F 2;i = 2 (2.8) i = e2 4ß 4ß" q i Ω = e2 X j 0 X q j 0X g 4ß 4ß" q i exp Ω g 28 < X 4 : g g 2 j! exp ψ g2 sin (g r 4ff 2 ij ) g (2.9) g 2! 4ff 2 ψ g2 9 = z j cos (g r i ) ; sin (g r i) 8 < X : j 9 = z j sin (g r i ) ; cos (g r i) 3 5 (2.10) ff, ff, Φ 1, Φ 2,,., L, ff = 5 L. 18

20 2.2.4 MD.,. ES+ [15]., E es. E es = X i E i (q i )+ 1 2 X i6=j V ij (r ij ; q i ;q j ) (2.11) E i i, Taler. E i (q i )=E i (0) + χ 0 i q i J 0 i q 2 i (2.12), χ i J i atomic hardness., V ij,. ZZ ρi (r 1 ; q i ) ρ j (r 2 ; q j ) V ij (r ij ; q i ;q j )= dr 1 dr 2 (2.13) r 12, ρ i (r; q i ) q i i. ρ i (r; q i ). ρ i (r;q i )=Z i effi (r r i )+(q i Z i ) ef i (r r i ) (2.14) Z i,f i. (2.14) (2.13), (2.11),. E es = E 0 + X i q i χ i X i;j q i q j V ij (2.15) f i Slater 1s, f i (r r i )= ο3 i ß exp ( 2ο i jr r i j).. E 0, χ i, V ij. E 0 = X i E i (0) X i6=j Z i Z j (2.16)

21 ψ! [f i jf j ] [ijf j ] [jjf i ]+ 1 r i j χ i = χ 0 i + X j, [ρ a jρ b ] [ajρ b ]. [ρ a jρ b ]= (2.17) Z j ([jjf i ] [f i jf j ]) (2.18) V ij = J 0 i ffi ij +[f i jf j ] (2.19) ZZ ρa (r 1 ) ρ b (r 2 ), f R i, (2.18) (2.19). χ i = χ 0 i + X j dr 1 dr 2 (2.20) r 12 Z [ajρ b ]= dr ρ b (r) (2.21) jr r a j 8 < X Z j : R» jjf R i V ij = J 0 i ffi ij +» f 0 i jfr j X» f 0 i jfr j R f i (r i R) 9 = ; (2.22) (2.23), P i q i =0 = χ i i L = E es μ X i X j X j q i (2.24) V ij q j μ =0 (2.25) V ij q j = μ χ i (2.26). μ = χ P i + j V ij q μ = μ 0 P i q i = 0. q i i., P i q i =0 (exact penalty method)[16] 20

22 , ffi = E tot (q 1 q n )+W ψ X i q i! 2 (2.27). E tot, W. Lagrange, MD.,. ffie tot (q 1 q n ) ffiq i +2Wq i (2.28) 2.3 (FPMD) (Density Functional Theory : DFT).,,,., Car-Parrinello,, [17] Born-Oppenheimer, Born-Oppenheimer ( )., Born-Oppenhaimer.,,. 21

23 ρ (r), E tot [ρ]. E tot [ρ] = Z V (r) ρ (r) dr 3 + T [ρ] + e2 2 ZZ ρ (r0) ρ (r) r 0 r dr 03 dr 3 + E xc [ρ] (2.29) 2 ρ, 3, 4. E tot [ρ] ρ (r), Kohn-Sham. " ψ! # μh 2 r 2 + V ef f (r) ψ i (r) =E i ψ i (r) (2.30) 2m V ef f (r) =V (r) +V H (r) +μ xc (r) (2.31) V H (r) = Z e2 ρ (r 0 ) jr 0 rj dr0 (2.32) μ xc (r) = ffie xc [ρ] ffiρ (2.33) Kohn-Sham Schrödinger,. Kohn-Sham,., ρ. ρ (r) = Xocc i jψ i (r)j 2 (2.34),., (LDA), ffl xc, E xc [ρ (r)] = Z ffl xc (r) ρ (r) dr 3 (2.35) μ xc (r) = ffl xc (r) +ρ (r) ffiffl xc ffiρ (2.36) 22

24 ., (GGA), f xc (ρ; rρ), E x c [ρ (r)] = Z f xc (ρ; rρ) dr 3 (2.37) μ xc (r) (2.38) kohn-sham,., screen,... j k + G >= Ω 1 2 exp [i (k + G)] (2.39), Ω, G, k Brillouin.,. Ω 1 Z exp [ig r] dr 3 = ffi G;0 (2.40),. hffi j ψi =, (2.39), (2.40), (2.41) hk + G j k + G 0 i = Z ffi Λ (r) ψ (r) dr 3 (2.41) Z Ω 1 Λ 2 exp [i (k + G)] Ω 1 2 exp [i (k + G)] dr 3 Z = Ω 1 exp [ i (k + G)] exp [i (k + G)] dr 3 Z = Ω 1 exp [i (G 0 G)] dr 3 = ffi G 0 G ;0 = ffi G;G 0 (2.42) 23

25 ., kohn-sham. (2.39), (2.40) = * k + G fi ψ! fi fi μh 2 fi + fififi fi r 2 k + G 0 2m " Λ Z Ω 1 2 exp [i (k + G) r] = Ω 1 Z = Ω 1 Z = Ω 1 ψ μh 2 exp [ i (k + G) r] exp [ i (k + G) r] 2m = Ω 1 ψ μh 2 = = = ψ μh 2 2m ψ μh 2 2m ψ μh 2 2m! 2m!! " ψ μh 2 2m ψ μh 2 2m ψ μh 2 ( 2m! r 2 #! r 2 # D k + G j μh E 2 r 2 j k + G 0 2m Ω 1 2 exp [i (k + G 0 ) r] dr 3 exp [i (k + G 0 ) r] dr 3!) ni 2 j k + G 0 j 2o exp [i (k + G 0 ) r] dr 3 Z j k + G 0 j 2 exp [ i (k + G) r]exp[i (k + G 0 ) r] dr 3 Z j k + G 0 j 2 exp [i (G 0 G) r] dr 3 j k + G 0 j 2 ffi G 0 G ;0!! j k + G 0 j 2 ffi G;G 0 j k + G j 2 ffi G;G 0 (2.43). (2.31 V (r), V PS (r). a V L a (r) ( ) P l P + l V NL a;l (r)p l ( P l a l, )., V L (r) V NL (r). V (r) =V L (r) +V NL (r) (2.44) V P L (r) = R Pt a V L a (r R t a) t a a, R ), u (r)( R, u (r + R) =u (r)) G, u (r) = X u (G) exp [ig r] (2.45) G 24

26 , u (G) =Ω 1 Z u (r)exp[ ig r] dr 3 (2.46), V L (r). X V L (r) = V L (G) exp (ig r) (2.47) G Z V L (G) = Ω 1 V L (r) exp ( ig r)dr (2.48) = Ω 1 X R = Ω 1 X R = Ω 1 c V L a (G) = Z X t a X Z t a X Z t a V L a (r R t a)exp( ig r)dr (2.49) V L a (r R t a)exp( ig (r R t a ))dr (2.50) V L a (G)exp( ig r) (2.51) V L a (r)exp ( ig r) dr (2.52), Ω c., V L (r) < k+gj jk + G 0 >, E Dk + G j V L (r) j k + G 0 (2.53) = = * fi fi k + Gfi X Z G Ω 1 2 exp [i (k + G) r] = Ω 1 Z = X V L (G) exp [ig r] exp [ i (k + G) r] X G 00 G 00 V L (G 00 )Ω 1 X = V L (G 00 ) ffi G 0 +G 00 G G X 00 = V L (G 00 ) ffi G 00 ; G G 0 G 00 = V (G G 0 ) Z fi + fi fi fi k + G0 Λ X V L (G 00 )exp[ig 00 r]ω 1 2 exp [i (k + G 0 ) r] dr 3 G 00 V L (G 00 ) exp [ig 00 r]exp[i (k + G 0 ) r] dr 3 exp [i (G 0 + G 00 G) r] dr 3 ;0., Kohn-Sham. hk + G jhjk + G 0 i (2.55) 25 (2.54)

27 =! ψ μh 2 j k + G 0 j 2 ffi 2m G;G + V 0 L (G G 0 ) +V NL (k + G; k + G 0 )+V H (G G 0 )+μ xc (G G 0 ) (2.56), V H, V H (G) = 8ßρ (G) jgj 2 (2.57). n E k n C n k+g, ψ kn (r). ψ kn (r) = X G C k+g n jk + G > (2.58) E kn, 2.,. ρ (r) = = Xocc n B:Z: X k jψ kn (r)j 2 Xocc B:Z: X X X n k C k+g n C k+g n 0Ω 1 exp [i (G G 0 r)] (2.59) G G 0,.,, ρ new =(1 ff) ρ in + ffρ out,., N , Kohn-Sham,N 3.,, NlogN., Car-Parrrinello. Car-Parrrinello,,.,fψ i g Lagrange. ~E tot = E tot X ij ij (hψ i jψ j i ffi ij ) (2.60) 26

28 ~ E tot ψ i Λ tot ψ i Λ ψ i ψ i =[H ] ψ i (2.61) ( = hψ i jhjψ i i) [H ] ψ i 2 ( ) Car-Parrrinello. μ,. μ ψ i = [H ] ψ i (2.62) [H ] ψ i 1 ( ).. μ ψ _ i = [H ] ψ i (2.63) [H ] ψ i.. ψ Etot ~ ψi Λ = [H ] ψ i =0 ψ i. Hψ i = ψ i, Kohn- Sham VASP VASP(Vienna Ab-initio Simulation Package)[18].. ff quartz. K Monkhorst-Pack ,., Murnaghan-Fitting,. Table. 2.1 PAW,,,. PW91 Perdew Wang GGA. Table.?? Table. 2.1 ff quartz [25]. LDA GGA. GGA LDA, LDA,GGA. LDA-PAW, LDA-high, LDA-mid, LDA-low, 27

29 LDA- PAW Table 2.1: Calculation Conditions LDAhigh LDAmid LDAlow GGA- PAW GGAhigh GGAmid GGAlow PAW E cutof f [ev] xc type LDA LDA LDA LDA PW91 PW91 PW91 PW91 Table 2.2: Calculations for equation of states exp. LDA-PAW LDA-high LDA-mid LDA-low E[eV] a[å] c[å] B[GPa] GGA-PAW GGA-high GGA-mid GGA-low other theory E[eV] a[å] c[å]

30 ., LDA-PAW 1000eV., Si-O LDA-high. 2.4, ( ),,,,,,,. 2,,. 1, [19]. 2., ,. 2,,,,, ,.. Linear Crossover Linear Crossover χ (1;t) i 1 t i, χ (2;t) i 2 t (1;t) i, 0:5 χ i + χ (2;t) i, 1:5χ (1;t) i 0:5χ (2;t) i, 0:5χ (1;t) i +1:5χ (2;t) i 29

31 . A Native Crossover 2,,. 1 3 (1;t) Parent1: χ 1 ;χ (1;t) 2 ;χ (1;t) 3 ;χ (1;t) 4 ; ;χ (1;t) n (2;t) Parent2: χ 1 ;χ (2;t) 2 ;χ (2;t) 3 ;χ (2;t) 4 ; ;χ (2;t) n (1;t) Offspring1 : χ 1 ;χ (1;t) 2 ;χ (1;t) 3 ;χ (2;t) 4 ; ;χ (2;t) n Offspring2 : χ (2;t) 1 ;χ (2;t) 2 ;χ (2;t) 3 ;χ (1;t) 4 ; ;χ (1;t) n. Blend Crossover Blend Crossover (BLX-ff) μ i 0, i χ (1;t+1) i χ (1;t+1) i =(1 fl i )χ (1;t) i., fl i =(1+2ff) μ i ff. + fl i χ (1;t) i (2.64) Random Mutation, r i 0 1, i y (1;t+1) i i χ (U ) i i χ (L) i, y (1;t+1) (U ) i = r i χ i y (1;t+1) i χ (L) i (2.65) = χ (1;t) i +(r i 0:5) i (2.66) 30

32 . i i. Normally Distributed Mutation N (0;ff), i y (1;t+1) i y t+1 i = χ 1;t i + N (0;ff) (2.67). ff. 31

33 3 32

34 3.1,.,,.,. 3.2 Tersoff [20] Brenner [21], E self E ion, f q.. Ω = X i ffi ij = E rep ij E self i E cov ij X i6=j ffi ij (3.1) + E ion ij (3.2) E self i = E 0 + χ i q i J iq 2 i (3.3) E rep ij = f c (r ij )(q i ) A ij exp( 1;ij r ij ) (3.4) E cov ij = f c (r ij ) g cov (q i ) B ij b ij exp( 2;ij r ij ) (3.5) E ion ij = q iq j r ij (3.6) r ij i, j,q i i, χ i, J i i, f c i,j, A ij, B ij i, j,, 1;ij, 2;ij i,j,, b ij i, j. g cov,. 3.3 [22], i ( LDOS) n i (E), Fermi E F, E band;i E band;i = Z E F 1 En i (E)dE (3.7) 33

35 Fig. 3.1: rectangle approximated band., W, N 0, E c,, (LDOS ) N 0 =W, E band;i = Z E F " E N 0 E 2 W 2 +Ec W de N = 0 2 W # E F W 2 +Ec (3.8). N, Z E» F N 0 N = W 2 +Ec W de = E N E F 0 (3.9) W W 2 +Ec., (3.8), (3.9),. E band;i = W 2 N (N N 0 ) N 0 + NE c (3.10), i1 p μ p;i1, i,j H ij, μ p;i1 = = Z +1 X 1 i1;i2; ;ip E p n(e)de H i1;i2 H i2;i3 H ip;i1 (3.11)., 2 W 2 2 ( ), 2 H ij = H ji = h, W 2 = μ 2;i μ 0;i = 1 X H ij H Z ih 2 ji = (3.12) N 0 N 0 j 34

36 . Z i i., E bond;ij = E band;i Z i = 1 2 N (N N 0 ) W + N E c (3.13) N 0 Z i Z i. E c s, E rep, E cov, E rep = N E c Z i = N Z i A X j S ij = ANs (3.14) E cov = 1 2 (A ), N (N N 0 ) W (3.15) N 0 Z i E rep = N E c Z i = Ns / N (3.16) E cov / Z 1 2 i (3.17) E cov / N (N N 0 ) (3.18). (3.17) Abell-Tersoff [20][23],, (3.4) g rep (3.5) g cov g rep (q i ) = N i (3.19) g cov (q i ) = N i (N i N 0;i ) (3.20) N i = N neutral;i q i (3.21). N 0;i i,n neutral;i i.,,,, Φ = X ffi ij = E rep ij i E self i E cov ij X i6=j ffi ij (3.22) + E ion ij (3.23) E self i = E 0 i + χq i Jq2 i (3.24) 35

37 E rep ij = f c (r ij ) ψ1 + Qrij! E cov ij = f c (r ij ) f q (q i ) b ij X A exp ( A r ij ) (3.25) m=1;3 B m exp ( Bm r ij ) (3.26) E q iq ion j ij = (3.27) r ij b ij = ij = f c (r ij ) = 1+ ij ffi (3.28) f c (r ik ) nc + d (h cos 2o ijk ) X k6=i;j exp 8 >< >: 1 2» n o fi ff r ij R e ij (r ik R eik ) 1 r ij» R 1 h1 +cos ß(rij R 1 ) R 2 R 1 i R1 <r ij <R 2 (3.29) 0 r ij R 2 (3.30) f q (q i ) = N i (q i )(N 0 i N i (q i )) N i (0) (N 0 i N i (0)) (3.31) N i (q i ) = N neutral i q i (3.32) r ij i; j, q i i. f q, N i i, N 0 i., Si sp3, N 0 Si =8;Nneutral Si =4, O 2s 2p,2p, N 0 O =6;Nneutral O =4. χ, J, i, A; B m ; A ; Bm ;Q;R 1 ;R 2 ;R e i; j, a; c; d; h; ff; fi i; j; k. 3.4, Tersoff. 36

38 4 37

39 4.1.,,. VASP(Vienna Ab-initio Simulation Package)[18] LSDA. Vanderbilt [24], RMM-DIIS,., 520eV K Monkhorst- Pack Table 4.1: Calculation conditions of DFT calculations. speices number of Si atoms number of O atoms K point mesh by MP ff-quartz fi-quartz ff-cristobalite ideal-fi-cristobalite ideal-fi-tridymite stishovite I 1 O I 2 O I 3 O I 1 I I 2 I I 3 I

40 4.3 alpha-quartz. ff-quartz (4.1). x y 2 Si + y 2 SiO 2 (ffq) =Si x O y (4.1) E 4.2. E = x y E Si + y E SiO 2 (ffq) E SixOy (4.2) SiO 2,, c-si/c-sio SiO 2 Fig. 4.1 SiO 2. ideal-ficristobalite ideal-fi-tridymite. [25] Fig. 4.2, stishovite alpha-quartz beta-quartz alpha-cristobalite ideal-beta-cristobalite ideal-beta-tridymite stishovite energy[ev] volume[angstrom 3 /molecular unit] Fig. 4.1: Binding energies as a function of volume. 39

41 Fig. 4.2: Binding energies as a function of volume in Ref.[25]. 40

42 DFT Fig. 4.3 ff-quartz, Fig. 4.4 fi-quartz, Fig. 4.5 ff-cristobalite, Fig. 4.6 ideal-fi-cristobalite, Fig. 4.7 ideal-fi-tridymite, Fig. 4.8 stishovite. Table. 4.2 ff-quartz, Table. 4.3 fi-quartz, Table. 4.4 ff-cristobalite, Table. 4.5 ideal-fi-cristobalite, Table. 4.6 ideal-fi-tridymite, Table. 4.7 stishovite. ideal-ficristobalite ideal-fi-tridymite. idealfi-cristobalite ideal-fi-tridymite,,.,.,,lda, LDA. ff-quartz ff-cristobalite,2 Si-O. Fig. 4.3: ff-quartz structure obtained by DFT caluculation Si Table. 4.8 Si DFT., I N O Si Si N O Si. Fig. 4.9 DFT I 1. O 41

43 Table 4.2: Structural prameters for ff-quartz obtained by our DFT calculation. Expt. a this work (LDA) other calc. Λb 6 a axis (Å) c axis (Å) d SiO1 (Å) d SiO2 (Å) SiOSi (degree) K (GPa) E (ev/sio 2 ) a Ref. [26], b Ref. [25] Fig. 4.4: fi-quartz structure obtained by DFT caluculation Table 4.3: Structural prameters for fi-quartz obtained by our DFT calculation. Expt. a this work (LDA) other calc. Λb 6 a axis (Å) c axis (Å) d SiO (Å) SiOSi (degree) E (ev/sio 2 ) a Ref. [27], b Ref. [25] 42

44 Fig. 4.5: ff-cristobalite structure obtained by DFT caluculation Table 4.4: Structural prameters for ff-cristobalite obtained by our DFT calculation. Expt. a this work (LDA) other calc. Λb 6 a axis (Å) c axis (Å) d SiO1 (Å) d SiO2 (Å) SiOSi (degree) K (GPa) E (ev/sio 2 ) a Ref. [28], b Ref. [25] 6 Table 4.5: Structural prameters for ideal-fi-cristobalite obtained by our DFT calculation. this work (LDA) a axis (Å) d SiO (Å) SiOSi (degree) E (ev/sio 2 )

45 Fig. 4.6: Ideal fi-cristobalite structure obtained by DFT caluculation Fig. 4.7: Ideal-fi-tridymite structure obtained by DFT caluculation. 44

46 6 Table 4.6: Structural prameters for ideal-fi-tridymite obtained by our DFT calculation. this work (LDA) a axis (Å) c axis (Å) d SiO1 (Å) d SiO2 (Å) SiOSi (degree) E (ev/sio 2 ) Fig. 4.8: Stishovite structure obtained by DFT caluculation. Table 4.7: Structural prameters for stishovite obtained by our DFT calculation. Expt. this work (LDA) other calc. Λc 6 a axis (Å) a c axis (Å) a d SiO1 (Å) a d SiO2 (Å) a SiOSi (degree) a K (GPa) b 2.82 E (ev/sio 2 ) b a Ref. [29], b Ref. [30], Ref. [31], Ref. [32], c Ref. [25] 45

47 Fig. 4.9: I 1 O structure obtained by DFT caluculation. Table 4.8: Defect formation energies of an oxygen interstitials in silicon crystal obtained by my DFT calculation. (I N O means silicon crystal in which N oxygen atoms are added around one Si atom. ). Defect Formation Energy (ev) species this work (LDA) I 1 O 1.59 I 2 O 3.03 I 3 O

48 4.4.3 SiO 2 Table. 4.9 ff-quartz DFT., V N O Si N O ff-quartz. Fig DFT V 1 O ( Si-Si O ). Fig. 4.10: V 1 O structure obtained by DFT caluculation. Table 4.9: Defect formation energies for oxygen vacancies in ff-quartz obtained by my DFT calculation (V N O means ff-quartz in which N oxygen atoms are removed around one Si atom.). Defect Formation Energy (ev) species this work (LDA) V 1 O 0.95 V 2 O 2.08 V 3 O

49 4.4.4 c-si/c-sio 2 c-si/c-sio 2, LDA Si (5.39),. Fig Fig. 4.11: c-si/c-sio 2 interface structure obtailned by my DFT calculation. Left panel is Q model. Center panel is C model. Right panel is T model. Si, c-sio 2.,, Si, DFT SiO 2 SiO 2., c-si c-sio 2., Si-O, Si-Si, [7],[10] E interf = E tot N SiSi ffl SiSi N SiO ffl SiO (4.3). E interf, E tot, N SiSi, N SiO Si-Si,Si-O, ffl SiSi Si DFT Si-Si, ffl SiO DFT SiO 2 SiO 2 DFT Si-O., SiO 2, SiO 2,. Table c-si/c-sio

50 Table 4.10: Interface enegies obtained by my DFTcalculation. Reference states are silicon crystal and suitably strained SiO 2 crystal. Energies are in ev. speices this work (LDA) other calc. (GGA) a Q model C model T model a Reference[10] 49

51 5 50

52 5.1,.,,.,.,,,., c-si/a-sio 2,, Si/SiO , c-si/a-sio 2 a-sio 2 Si/SiO 2,, Si-Si, Si/SiO 2 Si,, Si, SiO 2, SiO 2, c-si/c-sio 2, Si, Si., Si-O SiO 2,. ff-quartz, fi-quartz, ff-cristobalite, ideal-fi-cristobalite, ideal-fi-tridymite.,, DFT, 80%, 85%, 90%, 95%, 98%, 99%, 101%, 102%, 105%, 110%, 115%, 120% DFT...,.,,., Si O 4, O Si 2 SiO 2. Si-O Si-O (Si-O, 2 ), Si-Si O Si (Si-O, 3 ) Si-Si O Si (Si-O, 6 ), Si-Si O bcc Si (Si-O, 8 )., O-Si Si-O (O-Si, 1 ), O-O Si O (O-Si, 3 ), O-O Si O (O-Si, 4 )., DFT, 80%, 85%, 90%, 95%, 98%, 99%, 101%, 102%, 105%, 110%, 115%, 120% 51

53 DFT,,.,1, Si-O2, Si-O-O3, O-Si-Si3. ff-quartz O, Si O +, c-si/c-sio 2, Q model, C model, T model. DFT,, MD CG, CG,. DFT,., Si-O-O3, O-Si-Si3, O-O-O3 3. O-O O 2, O, O. O-O2, O-O-O3. stishovite, Si O Si2 1, ff-quartz O, O, Si, Si O, O. suboxide, ideal-fi-cristobalite O Si +,Si 2+,Si 3+,., LDA, Si ff-quartz. Si ff-quartz. 5.3,,, f. f = E fit + F fit + P fit (5.1) E fit = F fit = P fit = vu u t P i W E DFT EMD E;i N vu u t P i W F;i max X i E;i 2 F DF T;i;1 FMD;i;1 F N P mn=xx;yy;zz jp DF T;i;mn P M D;i;mn j W P;i 3 P 52 2 ; ; F DF T;i;ni F MD;i;ni F 2 (5.2) (5.3) (5.4)

54 E fit, W E;i i, E MD;i i Si ff-quartz, E DF T;i i Si ff-quartz DFT, E;i i. F fit, W F;i i, F M D;i;j i j, F DF T;i;j i j DFT, n i i, F. P fit, W P;i i, P M D;i;mn i mn, P DF T;i;mn DFT i mn, F. DFT MD. DFT MD, DFT MD. DFT MD. 1, try&error., Si x O y i 4.3 ff-quartz E;i = x y EDF 2 T;Si + y E 2 DF T;SiO 2 (ffq), E;i = E DF T. F, 1.0eV=Å,, DFT 0.1eV=Å.,,. 5.4 Fig

55 Fig. 5.1: Floachart of GA. 54

56 5.4.1 Ewald Ewald,. Φ s1;ij = 1 2 Φ s2;ij = 1 2 0X R 4ß Ω erfc (ff jr ij + Rj) 0X g jr ij + Rj exp ψ g2 4ff 2! (5.5) cos (g r ij ) 1 g 2 (5.6) Φ s3 = ff p ß (5.7) (5.8) Ewald,. F s1;ij = 0 X R F s2;ij = 4ß Ω 8 < e 2 p : 2ff exp ff 2 jr ij + 2 Rj + ß (r ij + R) jr ij + Rj 0X g exp ψ g2 4ff 2! erfc ff jr ij + Rj 2 9 = ; jr i j + Rj 2 (5.9) sin (g r ij ) g g 2 (5.10) Ewald, ,,., Tersoff [20], Brenner [21],,, SiO 2 A ij, B ij. 55

57 5.4.3 tot (q 1 q n ) +2Wq i i, E cov;s;ij = 1 2 f c (r ij ) b ij X m=1;3 B m exp ( Bm r ij ) tot (q 1 q n i +2Wq i (5.13).,. i Si, i cov (q 1 q n q (q i q (q i i = X q (q i i E cov;s;ij (5.14) = q i 8 = 1 q i 4 (5.15) (5.16). Ewald (q 1 q n ) X = e2 q j (Φ s1;ij +Φ s2;ij i 4ß" j + e2 4ß" X j 56 q j (Φ s1;ji +Φ s2;ji )+ e2 4ß" 2q iφ s3 (5.17)

58 self (q 1 q n i = χ i + J i q i tot (q 1 q n i +2Wq i (5.19) i i i +2Wq i (5.20).,,,,,,. Ewald ,,.,.,..,,,.. Blend Crossover (BLX-ff)., ff =0:5. Random Mutation (2.66). 57

59 6 58

60 Table Table 6.1: 1-body potential parameters parameter Si O χ J Table Si-Si Tersoff3 Table 6.2: 2-body potential parameters parameter Si-Si Si-O O-O A B B B ; ; ; ffi Si Q Re R R Re SiSi Si, Re SiO SiO 2 59

61 . R 1;SiO,R 1;OO SiO 2, O 2. Table Table 6.3: 3-body potential parameters Si-Si-Si Si-Si-O Si-O-Si Si-O-O O-Si-Si O-Si-O O-O-Si O-O-O p c d h LDA,, CG, [GPa]. Table. 6.4 ff-quartz, Table. 6.5 fi-quartz, Table. 6.6 ff-cristobalite, Table. 6.7 ideal-fi-cristobalite,. Si-O 1.57Å, LDA. 2.,,., ff-quartz 3. ideal-fi-cristobalite., ff-quartz ff-cristobalite 500K 10psec,. 324 ff-quartz Fig. 6.1, 192 ff-cristobalite Fig

62 Fig. 6.1: panel). ff-quartz structre before annealing(upper panel) and after annealing(lower 61

63 Fig. 6.2: ff-cristobalite structre before annealing(left panel) and after annealing(right panel). Table 6.4: Structural prameters for ff-quartz obtained by our MD calculation. 6 Expt. a this work (LDA) this work (MD) BKS Λb Tsuneyuki Λc a axis (Å) c axis (Å) d SiO1 (Å) d SiO2 (Å) SiOSi (degree) K (GPa) C 11 (GPa) C 33 (GPa) E (ev/sio 2 ) a Ref. [26], b Ref. [33], c Ref. [34] 6 Table 6.5: Structural prameters for fi-quartz obtained by our MD calculation. Expt. a this work (LDA) this work (MD) a axis (Å) c axis (Å) d SiO (Å) SiOSi (degree) E (ev/sio 2 ) b Ref. [27] 62

64 Table 6.6: Structural prameters for ff-cristobalite obtained by our DFT calculation. 6 Expt. a this work (LDA) this work (MD) BKS Λb Tsuneyuki Λc a axis (Å) c axis (Å) d SiO1 (Å) d SiO2 (Å) SiOSi (degree) K (GPa) C 11 (GPa) C 33 (GPa) E (ev/sio 2 ) a Ref. [28], b Ref. [33], c Ref. [34] 6 Table 6.7: Structural prameters for ideal-fi-cristobalite obtained by our DFT calculation. this work (LDA) this work (MD) a axis (Å) d SiO (Å) SiOSi (degree) E (ev/sio 2 )

65 6.4 Table. 6.8 Si MD. (4.2). Si Si-Si O CG. DFT. Table 6.8: Defect formation energies of an oxygen interstitials in silicon crystal by my MD calculation. (I N O means silicon crystal in which N oxygen atoms are added around one Si atom. ). Defect Formation Energy (ev) species this work (LDA) this work (MD) I 1 O I 2 O Table. 6.9 ff-quartz MD. ff-quartz Si O CG. DFT. Table 6.9: Defect formation energies for oxygen vacancies in ff-quartz obtained by my MD calculation (V N O means ff-quartz in which N oxygen atoms are removed around one Si atom.). Defect Formation Energy (ev) species this work (LDA) this work (MD) V 1 O V 2 O c-si/c-sio 2 Table MD c-si/c-sio 2. c-si/c- SiO 2 (4.3). C model, T model DFT, Q model DFT., Q model,c model, T model 500K 10psec,. 64

66 Table 6.10: Interface enegies obtained by my MD calculation. Reference states are silicon crystal and suitably strained SiO 2 crystal. Energies are in ev. speices this work (LDA) this work (MD) other calc. (GGA) a Q model C model T model a Reference[10] 68 Q model Fig. 6.3, 80 C model Fig T model Fig Fig. 6.3: Q model strucure before annealing(upper panel) and after annealing(lower panel). 65

67 Fig. 6.4: C model structre before annealing(left panel) and after annealing(right panel). 66

68 Fig. 6.5: T model structre before annealing(left panel) and after annealing(right panel). 67

69 6.6 SiO kg/m ff-quartz 1psec 10000K, 6000K 0K 1K/fsec. 300K, 0Pa 10psec NPT,. 3. Fig Fig. 6.6: Amorphous SiO 2 structure Table Si O. 4, 70%, 3 25%, 4 95%. Table 6.11: O coordination number of Si atom in amorphous SiO 2 structure. O coord. of Si ratio

70 Table O Si. 2, 83%, 1 14%, 2 95%. Table 6.12: Si coordination number of O atom in amorphous SiO 2 structure. O coord. of Si ratio Fig , 3%., 4.0Å Si O 2 3.5Å., 3.0Å Si-Si., [36]. Pair Correlation Function [barns/å 2 ] this work exp distance [Å] Fig. 6.7: pair correlation function of amorphous SiO 2 Table SiO %, Si-O, ff-quartz, [37]. Table MD Si-O-Si. [37] Table

71 Table 6.13: Density of amorphous SiO 2. Density (kg/m 3 ) exp this work density of a-sio density of ff-qartz a-sio 2 /ff-q Si. 0.1 this work ratio Si-O-Si angle [degree] Fig. 6.8: Si-O-Si bond angle distribution in amorphous SiO 2 obtained by my MD calculation. 6.7 a-sio 2 /c-si xy Si (5.43 2) 144 ff-quartz 1psec 10000K, 6000K 0K 1K/fsec. 2000K, 0Pa 10psec xy, z,xy Si., z, xy 70

72 Fig. 6.9: Si-O-Si bond angle distribution of amorphous SiO 2 by experimental data (curve 2 is silica glass) Si Si, 1000K 20psec a-sio 2 /c-si. Fig Fig suboxide1 +, suboxide2 +, suboxide3 +,., suboxide3 + a-sio 2, suboxide1 + c-si, [38]., suboxide1 + 5, suboxide2 + 2, suboxide3 + 4, suboxide3 +, suboxide2 +, suboxide1 + [38],,., Si 1 a-sio 2, [38]. 71

73 Fig. 6.10: Structure of a-sio 2 /c-si and suboxide atoms at interface. From left panel, suboxide 1 + atoms, suboxide 2 + atoms, suboxide 3 + atoms and all atoms 72

74 7 73

75 ffl Si/SiO 2, Si/SiO 2 Tersoff. ffl Si, SiO 2,Si SiO 2,c-Si/c-SiO 2, DFT,,,. ffl MD, DFT SiO 2, c-si/c-sio 2. ffl MD 3, 2,,. ffl MD SiO 2 DFT,. ffl MD DFT. ffl MD C model, T model DFT. Q-model DFT,. 74

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78 ,..,.,. 4 GA... 2,..,. 2,,..,,,. Ewald, DFT.., DFT.,,,. 77

79 1ο