3 464-8602 1 [1] 2 3 (epitaxy) (homoepitaxy) (heteroepitaxy) 1 Makio Uwaha. E-mail:uwaha@nagoya-u.jp; http://slab.phys.nagoya-u.ac.jp/uwaha/ 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 3 3 3 3=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) 1 3 3 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
3.1: ( ) ( ) ( ) [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 + 2 3 μ (3.8) (bulk modulus: ) (3.2) ɛ ξζ (3.2) ɛ ξζ σ αγ 4 σ αβ σ Poisson
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 = 0 5 1 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
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)
(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
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
1 3.4 2 2 V (r) a 1 4 4 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 < 0 7 3.4 σ 0 σ 0 / 2 3.4 ( 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
d 2 3.4 ( λ μ σ 0 ) 5 1 2 8 2 - - [5] 1/d 2 σ 0 3.3 2.4 ( ) 3 ( 3.5) ( 3.6(c)) (Stranski-Krastanov: SK) SK 2.4 (Frank-van der Merwe: FM) (Volmer-Weber: VW) 2 3 FM VW 8 3.4 [2]
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 0 3.7 [6] f FM f SK VW σ 0 FM SK VW SK VW
3.6: ( )[8] (a) (FM) (b) (VW) (c) (SK) 0.3 σ 0 0.2 VW SK FM SK VW 0.1 0 0.1 0 0.1 f 3.7: f ( )σ 0 [6] σ 0 (misfit dislocation) b a 3.8 3.8(a)
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 ) 9 1 1 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
0.1 FM SK transient islands SK+D FM+D J/Ka 2 0.01 VW+D (01) unstable 0.001 VW 0 0.025 0.05 0.075 0.1 0.125 3.9: f J [12] +D f (3.23) 3 L h 10 2 L h L [9] PbSe(001) PbTe [10] 3.7 3.9 f J( 2 V (a) ) FM SK VW 3.7 f J VW J FM SK VW SK FM [12] 10 m xx = Eah
[1] Crystal Letters No.40 (2009) 3; No.41 (2009) 3. [2] A. Pimpinelli and J. Villain, Physics of Crystal Growth, Cambridge University, Cambridge, 1998. [3] Y. Saito, H. Uemura and M. Uwaha, Phys. Rev. B 63 (2001) 045422. [4] : ( 2007 3 ). [5] Y. Saito, J. Phys. Soc. Jpn.73 (2004) 1816. [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) 1296. [8], 2 ( 2008). [9] H. Katsuno, M. Uwaha and Y. Saito, J. Phys. Soc. Jpn. 76 (2007) 044605. [10] G. Springholz and K. Wiesauer, Phys. Rev. Lett. 88 (2002) 015507. [11] H. Katsuno, M. Uwaha and Y. Saito, J. Cryst. Growth, 310 (2008) 1380. [12] H. Katsuno, M. Uwaha and Y. Saito, Surf. Sci., 602 (2008) 3459.