薄膜結晶成長の基礎3.dvi
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1 [1] 2 3 (epitaxy) (homoepitaxy) (heteroepitaxy) 1 Makio Uwaha. [email protected]; 2
2 3.1 [2] (strain) r u(r) ɛ αγ (r) = 1 ( uα + u ) γ (3.1) 2 x γ x α 3 u(r) r u(r) u α / x γ u α / x γ u γ x α (3.1) ɛ αγ 3 3=9 6 ( 3 3 ) (stress) σ αγ (r) ɛ αγ 3 3 σ αγ (r) = Ωαγ ξζ ɛ ξζ (r) (3.2) ξ=1 ζ=1 σ αγ ˆn γ σ αγ ˆn γ α (σ αγ ) Ωαγ ξζ =81 Ω ξζ αγ = μ(δ αξδ γζ + δ αζ δ γξ )+λδ αγ δ ξζ (3.3) μ λ (Lamé s constants) (3.2) (3.3) Ωαγ ξζ 3 σ αγ (r) = 2μɛ αγ (r)+λδ αγ ɛ ξξ (r) (3.4) ξ=1 = 2μ ɛ αγ (r) δ αγ ɛ ξξ (r) + ξ=1 (λ + 23 μ ) δ αγ 3 x α α =1, 2, 3 x y z 3 ɛ ξξ (r) (3.5) ξ=1
3 3.1: ( ) ( ) ( ) [3] (3.5) 2 1 ( ) μ λ ( Young s modulus: )E (Poisson ratio: )σ 3λ +2μ E = μ, λ + μ (3.6) λ σ = 2(λ + μ) (3.7) 4 (3.5) 1 μ ( modulus of rigidity, shear modulus) 2 K = λ μ (3.8) (bulk modulus: ) (3.2) ɛ ξζ (3.2) ɛ ξζ σ αγ 4 σ αβ σ Poisson
4 3.2: ( ) 3.1 [3] ( ) (adsorbed atom, adatom) (force dipole, elastic dipole) m αγ r r α f γ r (3.9) r fr fr = fr( 3.1) m αγ 2 2 d 4 3 ( ) 2 ( 3.2) 3 f ext ɛ αγ : σ αγ (r) = fα ext x. (3.10) γ γ γ σ αγ ˆn γ f ext 5
5 σ αγ (r) = 0 (3.11) x γ γ σ αγ ɛ ξζ u (3.11) u(r) 2 u =0 6 u 1/r u(r) 1/r r 2 : u(r) u(r +Δr) 1 r 1 r +Δr 1 Δr. (3.12) r2 3 : ɛ αγ u/ r 1/r 3 r 2 3 : W f u(r) f u(r +Δr) f ( ) 1 r 2 1 (r +Δr) 2 f Δr r 3. (3.13) d 2 1/d 3 : U adatom adatom 1 d 3. (3.14) 2 d 2 ( 3.3(b)) 3.3 x y U adatom line adatom line 1 (d 2 + y 2 dy ) 3/2 6 : (λ + μ)r(r u)+μr 2 u =0 1 d 2 (3.15)
6 (a) 0 d (b) 0 d (c) 0 d (d) 0 d 3.3: (a) - : U 1/d 3 (a) - : U 1/d 2 (a) - : U 1/d (a) - : U ln d d 1 U layer adatom dx dy d (x 2 + y 2 ) 3/2 1 d (3.16) ( 3.3(c)) d (y ) U layer layer 0 1 dx 1 dx 2 d(y 1 y 2 ) d [(x 1 x 2 ) 2 +(y 1 y 2 ) 2 ] 3/2 ln d (3.17) ( 3.3(d)) 1 1
7 3.4: [4] U step step 1 d 2. (3.18) ( ) 1 (3.17) U step step ln d. (3.19) Si(001) 1 90 [2] 3.2
8 V (r) a E ( N =2 V (a)+v( ) 2a) (3.20) E (3.20) a V (a)+ 2V ( 2a) = 0 (3.21) f 1 = V (a)( σ 0 ) > 0 f 2 = V ( 2a) = σ 0 / 2 < σ 0 σ 0 / ( σ 0 ) σ 0 σ 0 / 2 σ 0 V (r) k 1 = V (a)( 3.4 ) k 2 = V ( 2a)( 3.4 ) k 1 /k 2 =2 ( ) 7 σ 0 ( ) σ αγ σ
9 d ( λ μ σ 0 ) [5] 1/d 2 σ ( ) 3 ( 3.5) ( 3.6(c)) (Stranski-Krastanov: SK) SK 2.4 (Frank-van der Merwe: FM) (Volmer-Weber: VW) 2 3 FM VW [2]
10 σ 1y σ 1x σ 2 adsorbate interface a substrate (a) a (b) (c) 3.5: (a) [6] (b)(c) [7] : : a b f = b a a (3.22) σ [6] f FM f SK VW σ 0 FM SK VW SK VW
11 3.6: ( )[8] (a) (FM) (b) (VW) (c) (SK) 0.3 σ VW SK FM SK VW f 3.7: f ( )σ 0 [6] σ 0 (misfit dislocation) b a (a)
12 y h z L (a) x (b) 3.8: (a) L [4] (b) [11] h 1 (y x L ) (L h a) U D (f, h, L) = (α Ea2 8π ln h ) Eafh + 2ζ(2) a πe ( ) Eah 2 (3.23) (E ) a h U dis = (α Ea2 8π ln h ) (3.24) a (α 1 ) 2 L h 1 f f (a/l) U relax = 1 2 E ( f a L) 2 Lh 1 2 Ef2 Lh Eafh (3.25) (3.24) (3.25) f = 1 ( ln α h ) c (3.26) 8πh c a h c 9 ζ(2) ζ(2) = π 2 /6 L
13 0.1 FM SK transient islands SK+D FM+D J/Ka VW+D (01) unstable VW : f J [12] +D f (3.23) 3 L h 10 2 L h L [9] PbSe(001) PbTe [10] f J( 2 V (a) ) FM SK VW 3.7 f J VW J FM SK VW SK FM [12] 10 m xx = Eah
14 [1] Crystal Letters No.40 (2009) 3; No.41 (2009) 3. [2] A. Pimpinelli and J. Villain, Physics of Crystal Growth, Cambridge University, Cambridge, [3] Y. Saito, H. Uemura and M. Uwaha, Phys. Rev. B 63 (2001) [4] : ( ). [5] Y. Saito, J. Phys. Soc. Jpn.73 (2004) [6] H. Katsuno, H. Uemura, M. Uwaha and Y. Saito, J. Cryst. Growth 275 (2005) e283. [7] H. Uemura, M. Uwaha and Y. Saito, J. Phys. Soc. Jpn. 71 (2002) [8], 2 ( 2008). [9] H. Katsuno, M. Uwaha and Y. Saito, J. Phys. Soc. Jpn. 76 (2007) [10] G. Springholz and K. Wiesauer, Phys. Rev. Lett. 88 (2002) [11] H. Katsuno, M. Uwaha and Y. Saito, J. Cryst. Growth, 310 (2008) [12] H. Katsuno, M. Uwaha and Y. Saito, Surf. Sci., 602 (2008) 3459.
第86回日本感染症学会総会学術集会後抄録(I)
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薄膜結晶成長の基礎2.dvi
2 464-8602 1 2 2 2 N ΔμN ( N 2/3 ) N - (seed) (nucleation) 1.4 2 2.1 1 Makio Uwaha. E-mail:[email protected]; http://slab.phys.nagoya-u.ac.jp/uwaha/ 2 [1] [2] [3](e) 3 2.1: [1] 2.1 ( ) 1 (cluster) ( N
k m m d2 x i dt 2 = f i = kx i (i = 1, 2, 3 or x, y, z) f i σ ij x i e ij = 2.1 Hooke s law and elastic constants (a) x i (2.1) k m σ A σ σ σ σ f i x
k m m d2 x i dt 2 = f i = kx i (i = 1, 2, 3 or x, y, z) f i ij x i e ij = 2.1 Hooke s law and elastic constants (a) x i (2.1) k m A f i x i B e e e e 0 e* e e (2.1) e (b) A e = 0 B = 0 (c) (2.1) (d) e
19 σ = P/A o σ B Maximum tensile strength σ % 0.2% proof stress σ EL Elastic limit Work hardening coefficient failure necking σ PL Proportional
19 σ = P/A o σ B Maximum tensile strength σ 0. 0.% 0.% proof stress σ EL Elastic limit Work hardening coefficient failure necking σ PL Proportional limit ε p = 0.% ε e = σ 0. /E plastic strain ε = ε e
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1 9 v.0.1 c (2016/10/07) Minoru Suzuki T µ 1 (7.108) f(e ) = 1 e β(e µ) 1 E 1 f(e ) (Bose-Einstein distribution function) *1 (8.1) (9.1)
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72 9 Hooke,,,. Hooke. 9.1 Hooke 1 Hooke. 1, 1 Hooke. σ, ε, Young. σ ε (9.1), Young. τ γ G τ Gγ (9.2) X 1, X 2. Poisson, Poisson ν. ν ε 22 (9.) ε 11 F F X 2 X 1 9.1: Poisson 9.1. Hooke 7 Young Poisson G
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No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
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量子力学 問題
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Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e
7 -a 7 -a February 4, 2007 1. 2. 3. 4. 1. 2. 3. 1 Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e z
