χ (1) χ (3)

χ (1) χ (3)

L x, L y, L z ( ) ħ2 2 2m x + 2 2 y + 2 ψ (x, y, z) = E 2 z 2 x,y,z ψ (x, y, z) E x,y,z E x E y E z

ħ2 2m 2 x 2ψ (x) = E xψ (x) ħ2 2m 2 y 2ψ (y) = E yψ (y) ħ2 2m 2 z 2ψ (z) = E zψ (z) L x L y L z E nx,n y,n z = ħ2 π 2 2m ( n 2 x L 2 x + n2 y L 2 y + n2 z L 2 z ) (n x, n y, n z ) ħ2 2m 2 ψ(r, θ, ϕ) + V (r)ψ(r, θ, ϕ) = Eψ(r, θ, ϕ)

x = r ϕ θ, y = r ϕ θ, z = r ϕ 2 = 1 r 2 r 2r + 1 [ 1 r 2 ϕ ϕ ϕ ϕ + 1 2 ] 2 ϕ θ 2 ψ(r, θ, ϕ) = y nl (r)y m l (θ, ϕ) l = 0, 1, 2,..., m = l,...0,..., l Yl m (θ, ϕ) y nl (r) ( ) 1 d r 2dy nl(r) r 2 dr dr l(l + 1) y r 2 nl (r) + 2m ħ [E V (r)] y nl(r) = 0 2

SnO 2

(5 10 nm)

30 Zn II b 8 O 64 Zn(48.89 ), 66 Zn(27.81 ), 68 Zn(18.57 ) 16 (99.76 ). (1s) 2 (2s) 2 (2p) 6 (3s) 2 (3p) 6 (3d) 10 (4s) 2 (1s) 2 (2s) 2 (2p) 4. sp 3 sp 3

3.4eV 7.52eV T m = 2242K T mznse = 799K sp 3 1/4 Zn 2+ O 2 T d

(2 ZnO) 4 3m T 2 d F 4 3m 120 6 mm C 6 P 6 3 mc C 4 6 hkil a = b = 0.3249(6) nm c = 0.52042(20) nm d = 5.675 g cm 3

Zn 2+ O 2 Zn 2+ O 2 0.074 0.140 nm Z Z = 1.15 ± 0.15 (01 10) 2 1 10

30nm

k ψ Sk (x) S { ϕ i, i = 1,..., N} : ψ Sk (x) = N a i ϕ i (x) i=1 ϕ i (x) = x ϕ i ϕ i ϕ i (x) N

{ ϕ i, i = 1... } ψ Sk (x) ψ Sk (x) E k ϕ i (x) ϕ i (x)

a i a n = ϕ n(x)ψ Sk (x)dx M a n = w i ϕ n(x i )ϕ m (x i ) i=1 M w i ϕ i (x) δ nm = M w i ϕ n(x i )ϕ m (x i ) i=1 ϕ i (x) M M k ψ Sk (x) ϕ 1 (x), ϕ 2 (x),...ϕ N (x) ψ Sk (x) ϕ i (x) ψ Sk (x) ϕ i (x)

ψ Sk (x) S E i R ϕ Rk (x) R ψ Sk (x) ϕ Rk (x) ϕ Rk (x) ψ Sk (x) S E S ψ S ( ) } { ħ2 2m 2 + u S ( ) ψ S ( ) = E S ψ S ( ) = (x, y, z) m u S (r)

R u R ( ), ψ R ( ) E R } { ħ2 2m 2 + u R ( ) ψ R ( ) = E R ψ R ( ) u R ( ) u S ( ) R S u R ( ) u S ( ) ψ(, t) iħ t = } { ħ2 2m 2 + u t ( ) ψ(, t) u t ( ) = σ(t)u S ( ) + [1 σ(t)]u R ( ) σ(t) σ(t) = { 0 t ta 1 t t b t a < t < t b σ(t) σ(t) = a(t t a ) a = 1/(t b t a ) ψ(, t) t t a : ψ(, t) = ψ R (, t) = C R ( ie R t/ħ) ψ R ( ) t t b : ψ(, t) = ψ S (, t) = C S ( ie S t/ħ) ψ S ( ) t

t b t a N t t = (t b t a )/N N ψ(, t 1 ) ψ(, t a ) = 1 t 1 } { ħ2 iħ 2m 2 + u t ( ) ψ(, t)dt t a t 1 = t a + t. ψ(, t a ) ψ R (, t) ψ(, t a ) = ψ R (, t) δψ(, t) ψ R (, t) δψ(, t) = 1 t 1 } { ħ2 iħ 2m 2 + u t ( ) ψ(, t)dt t a t ψ R (, t a ) ψ(, t) u S ( ) u R ( ) N u t ( ) u R ( ) σ(t 1 ) t = t 1 t a 1/N N = 1000 u t ( ) = 10 3 u S ( ) + 0.999 u R ( )

ψ(, t 2 ) = ψ R (, t a ) + ψ(, t 1 ) + 1 iħ t 2 t 1 { } ħ2 2m 2 + u t ( ) ψ(, t)dt t 2 t a = 2 t ψ(, t 1 ) N ψ(, t N ) = ψ R (, t a ) + N δψ(, t k ) = ψ S (, t b ) k=1 t t b ψ S (, t) C S = (ie S t b /ħ) ψ S ( ) = ψ S (, t b ) E S S E S = } ψs( ) { ħ2 2m 2 + u S ( ) ψ S ( ) ψ(, t 1 ), ψ(, t 2 ) 10 o σ(t 10 ) = 10/N { ħ2 2m 2 + 10 N u S( ) + [ ] } 1 10 N ur ( ) ψ(, t 10 ) = E 10 ψ(, t 10 ) E 10 ψ(, t 10 )

S R R u R ( ) t S u S ( )

E 1, E 2 ω

ω 21 = (E 2 E 1 )/ħ ω ω 21 χ (1) (ω) χ (3) (ω) χ tot (ω) E(t) = ê ( E 0 e iωt + E 0e iωt) ψ(, t) iħ t = Hψ(, t) ψ(, t) ψ(, t) = c 1 (t)ψ 1 ( ) + c 2 (t)ψ 2 ( ) H = H 0 + H i H i Hψ 1 ( ) = H 0 ψ 1 ( ) + H i ψ 1 ( ) Hψ 1 ( ) = E 1 ψ 1 ( ) + H i ψ 1 ( ) Hψ 2 ( ) = E 2 ψ 1 ( ) + H i ψ 2 ( ) iħ c 1(t) t ψ 1 ( )+iħ c 2(t) ψ 2 ( ) = c 1 (t)e 1 ψ 1 ( ) + c 2 (t)e 2 ψ 2 ( ) t + c 1 (t)h i ψ 1 ( ) + c 2 (t)h i ψ 2 ( )

ψ1( ) iħċ 1 (t) ψ1( )ψ 1 ( )dv + iħċ 2 (t) ψ1( )ψ 2 ( )dv = c 1 (t)e 1 ψ 1( )ψ 1 ( )dv + c 2 (t)e 2 ψ 1( )ψ 2 ( )dv +c 1 (t) ψ1( )H i ψ 1 ( )dv + c 2 (t) ψ 1( )H i ψ 2 ( )dv ψ 1( )ψ 1 ( )dv = 1 ψ 1( )ψ 2 ( )dv = 0 iħċ 1 (t) = c 1 (t)e 1 + c 1 (t)(h i ) 11 + c 2 (t)(h i ) 12 (H i ) nm = ψn( )H i ψ m ( )dv ψ2( ) iħċ 2 (t) = c 2 (t)e 2 + c 1 (t)(h i ) 21 + c 2 (t)(h i ) 22 H i = d E(t)

d = e r iħċ 1 (t) = c 1 (t)e 1 c 1 (t) d 11 E(t) c 2 (t) d 12 E(t) iħċ 2 (t) = c 2 (t)e 2 c 1 (t) d 21 E(t) c 2 (t) d 22 E(t) d nm ψ n( ) dψ m ( )dv d 11 d 22 d 11 = d 22 = 0 iħċ 1 (t) = c 1 (t)e 1 c 2 (t) d 12 E(t) iħċ 2 (t) = c 2 (t)e 2 c 1 (t) d 21 E(t) iħċ 1 (t) = c 1 (t)e 1 c 2 (t) d 12 ê ( E 0 e iωt + E 0 e iωt) iħċ 2 (t) = c 2 (t)e 2 c 1 (t) d 21 ê ( E 0 e iωt + E 0 e iωt) d nm ê = d nm iħċ 1 (t) = c 1 (t)e 1 c 2 (t)d 12 ( E0 e iωt + E 0 e iωt) iħċ 2 (t) = c 2 (t)e 2 c 1 (t)d 21 ( E0 e iωt + E 0 e iωt) b 1 (t) = c 1 (t)e i(e 1/ħ)t, b 2 (t) = c 2 (t)e i(e 2/ħ)t

iħḃ1(t)e i(e 1/ħ)t + iħb 1 (t)e i(e 1/ħ)t ( i(e 1 /ħ)) = b 1 (t)e i(e 1/ħ)t E 1 b 2 (t)e i(e 2/ħ)t d 12 ( E0 e iωt + E 0 e iωt) iħḃ2(t)e i(e 2/ħ)t + iħb 2 (t)e i(e 2/ħ)t ( i(e 2 /ħ)) = b 2 (t)e i(e 2/ħ)t E 2 b 1 (t)e i(e 1/ħ)t d 21 ( E0 e iωt + E 0 e iωt) iħḃ1(t) = b 2 (t)d 12 ( E 0 e i (ω+ E 2 E 1 ħ )t + E 0 e i (ω E 2 E 1 ħ )t ) iħḃ2(t) = b 1 (t)d 21 ( E 0 e i (ω E 2 E 1 ħ )t + E 0 e i (ω+ E 2 E 1 ħ )t ) ω E 2 E 1 ħ ω 21 ω+ω 21 ω ω 21 iħḃ1(t) = b 2 (t)d 12 E 0 e i(ω ω 21)t iħḃ2(t) = b 1 (t)d 21 E 0 e i(ω ω 21)t a 1 (t) = b 1 (t), a 2 (t) = b 2 (t)e i(ω ω 21)t

iħȧ 1 (t) = d 12 E 0 a 2 (t) iħȧ 2 (t) = ħ a 2 (t) d 21 E 0 a 1 (t) ρ nm = c n c m P tot = N d P tot = ϵ 0 χ(ω)êe 0 iωt +ϵ 0 êχ (ω)e 0 iωt N ϵ 0 A = ψ (, t) ˆ Aψ(, t)dv d d = ψ (, t) dψ(, t)dv = [c 1(t)ψ 1( ) + c 2(t)ψ 2( )] d [c 1 (t)ψ 1 ( ) + c 2 (t)ψ 2 ( )] dv =

c 1(t)c 1 (t) ψ1( ) dψ 1 ( )dv + c 1(t)c 2 (t) 0 ψ 1( ) dψ 2 ( )dv +c 2(t)c 1 (t) ψ2( ) dψ 1 ( )dv + c 2(t)c 2 (t) ψ 2( ) dψ 2 ( )dv 0 d = ρ 21 (t) d 12 + ρ 12 (t) d 21 = σ 21 (t) d 12 e iωt + σ 12 (t) d 21 e iωt ρ nm σ nm = a n a m P tot = Nσ 21 (t) d 12 e iωt + Nσ 12 (t) d 21 e iωt P tot = êϵ 0 χ(ω)e 0 e iωt + êϵ 0 χ (ω)e 0 e iωt } χ(ω) = Nd 12σ 21 (t) ϵ 0 E 0 σ 21 (t) σ σ σ 21 (t) σ σ nm a n a m iħ σ t = [H, σ] = Hσ σh

iħ α t ] [ȧ1 = Hα iħ = H ȧ 2 [ a1 [ ] 0 d H = 12 E 0 d 21 E 0 ħ a 2 ] Hσ [ ] [ ] 0 d Hσ = 12 E 0 σ11 σ 12 d 21 E 0 ħ σ 21 σ 22 [ ] d Hσ = 12 E 0 σ 21 d 12 E 0 σ 22 d 21 E 0 σ 11 ħ σ 21 d 21 E 0 σ 12 ħ σ 22 σh [ ] [ ] σ11 σ σh = 12 0 d 12 E 0 σ 21 σ 22 d 21 E 0 ħ [ ] d21 E σh = 0 σ 12 d 12 E 0 σ 11 ħ σ 12 d 21 E 0 σ 22 d 12 E 0 σ 21 ħ σ 22 [ ] t = d 12 E 0 σ 21 + d 21 E 0 σ 12 d 12 E 0 (σ 22 σ 11 ) + ħ σ 12 d 21 E 0 (σ 11 σ 22 ) ħ σ 21 d 21 E 0 σ 12 + d 12 E 0 σ 21 iħ σ

σ 11 (t) = i ħ E 0(d 12 σ 21 d 21 σ 12 ) σ 22 (t) = i ħ E 0(d 21 σ 12 d 12 σ 21 ) σ 12 (t) = i ħ E 0d 12 (σ 22 σ 11 ) i σ 12 ) σ 21 (t) = i ħ E 0d 21 (σ 11 σ 22 ) i σ 21 ) σ 21 (t) σ 11 (t), σ 22 (t) σ 21 (t) σ 22 (t) = σ 11 (t) = σ 21 (t) = 0 σ 21 (t) σ 11 (t) = i ħ E 0(d 12 σ 21 d 21 σ 12 ) + σ 22 1 T 1 σ 22 (t) = i ħ E 0(d 21 σ 12 d 12 σ 21 ) σ 22 1 T 1

σ 21 (t) = i ħ E 0d 12 (σ 22 σ 11 ) + (i 1T2 ) σ 21 T 1 T 2 σ 22 (t) σ 11 (t) = 2i ħ E 0d 21 σ 12 2i ħ d 12σ 21 2σ 22 T 1 2σ 22 = σ 22 σ 11 + 1 σ 21 (t) = i ħ d 21E 0 (σ 22 σ 11 ) + (i 1 T 2 )σ 21 σ 22 (t) σ 11 (t) = 2i ħ E 0(d 21 σ 12 d 12 σ 21 ) σ 22 σ 11 + 1 T 1 σ 21 (t) = i ħ d 21 E 0 i 1 T 2 (σ 22 σ 11 ) σ 21(t) = i ħ d 21E 0 i 1 T 2 (σ 22 σ 11 ) 2i ħ E 0(d 21 σ 21 d 21σ 21 ) σ 22 σ 11 + 1 T 1 = 0 σ 22 σ 11 = 1 + 2 T 2 2 1 + 2 T 1 T 2

σ 21 σ 21 = i ħ d 21E 0 i T 2 2 + T 2 1 + 2 T 2 2 + Ω 2 T 1 T 2 Ω = d 21E 0 ħ σ 21 χ tot (ω) = N d 21 2 it 2 T2 2 ħϵ 0 1 + 2 T2 2 + Ω 2 T 1 T 2 χ tot (ω) = χ (ω) + iχ (ω) χ (ω) = N d 21 2 ħϵ 0 T 2 2 1 + 2 T 2 2 + Ω 2 T 1 T 2 χ (ω) = N d 21 2 ħϵ 0 T 2 1 + 2 T 2 2 + Ω 2 T 1 T 2 Ω 2 T 1 T 2

χ(ω) = N d 21 2 ħϵ 0 T 2 i + T 2 + N d 21 2 ħϵ 0 Ω 2 T 1 T 2 2 ( i + T 2 )(i + T 2 ) 2 χ(ω) = N d 21 2 T 2 ħϵ 0 i T 2 1 + 2 T 2 2 4N d 21 4 E 2 0T 1 T 2 2 ħ 3 ϵ 0 i T 2 (1 + 2 T 2 2 )2 χ(ω) χ (1) + 3 4 χ(3) E 2 0 χ (1) = N d 21 2 T 2 ħϵ 0 i T 2 1 + 2 T 2 2 χ (3) = 16 3 N d 21 4 T 1 T 2 2 ħ 3 ϵ 0 i T 2 (1 + 2 T 2 2 )2 Ω 2 = 8 d 21 2 I ħ 2 n ϵ 0 c χ(ω) = N d 21 2 ħϵ 0 it 2 T 2 2 1 + 2 T 2 2 + 8 d 21 2 I ħ 2 n ϵ 0 c T 1T 2

χ tot (ω) = χ (ω) + iχ (ω) = N d 21 2 ħϵ 0 it 2 T 2 2 1+ 2 T 2 2 +8 d 21 2 I ħ 2 n ϵ 0 c T 1T 2 χ (ω) = N d 21 2 ħϵ 0 T 2 2 1 + 2 T 2 2 + 8 d 21 2 I ħ 2 n ϵ 0 c T 1T 2 χ (ω) = N d 21 2 ħϵ 0 T 2 1 + 2 T 2 2 + 8 d 21 2 I ħ 2 n ϵ 0 c T 1T 2 χ (1) = N d 21 2 T 2 ħϵ 0 i T 2 1 + 2 T 2 2 χ (3) = 16 3 N d 21 4 T 1 T 2 2 ħ 3 ϵ 0 i T 2 (1 + 2 T 2 2 )2

1.5eV 3.3eV

ϵ CdT e = 7.1 ϵ ZnO = 8.66 ϵ in = ϵ CdT e ϵ ZnO ϵ out ϵ in = ϵ out ϵ out = 2.25 ϵ eff = 2ϵ out+ϵ in 3ϵ out χ(ω) = Nd 12σ 21 (t) ϵ 0 ϵ eff E 0 d 21 ϵ eff χ (1) = N d 21 2 T 2 ħϵ 0 ϵ 2 eff i T 2 1 + 2 T 2 2 χ (3) = 16 3 N d 21 4 T 1 T 2 2 ħ 3 ϵ 0 ϵ 4 eff i T 2 (1 + 2 T 2 2 )2 µ 21 = d 21 ϵ eff

χ tot (ω) = N µ 21 2 it 2 T2 2 ħϵ 0 1 + 2 T2 2 + 8 µ 21 2 I ħ 2 n ϵ 0 c T 1T 2 χ (ω) = N µ 21 2 ħϵ 0 T 2 2 1 + 2 T 2 2 + 8 µ 21 2 I ħ 2 n ϵ 0 c T 1T 2 χ (ω) = N µ 21 2 ħϵ 0 T 2 1 + 2 T 2 2 + 8 µ 21 2 I ħ 2 n ϵ 0 c T 1T 2 χ (1) = N µ 21 2 T 2 ħϵ 0 i T 2 1 + 2 T 2 2 χ (3) = 16 3 N µ 21 4 T 1 T 2 2 ħ 3 ϵ 0 i T 2 (1 + 2 T 2 2 )2 Ĥ = ˆp2 2m e + V con ( r) + Σ( r)

ˆp m e V con ( r) 0 r R core V con ( r) = V 0 R core r R shell r > R shell V 0 Σ( r)

Σ(r) = e 2 8πϵ 0 ϵ in R shell k=0 (k + 1)(ϵ in ϵ out ) kϵ in + (k + 1)ϵ out r 2k Rshell 2k ϵ 0 ϵ in = ϵ CdT e ϵ ZnO ϵ out ϵ in = ϵ out E gs (mev ) E fe (mev ) E 21 (mev ) µ 21 10 28 (C m) 3 225.5907 496.7228 271.1321 6.6698 5 209.2115 446.2394 237.0279 8.6380 7 198.1362 410.5901 212.4539 10.5644 9 193.4359 386.5507 193.1148 12.4778 (E gs ) (E fe )

ϵ in ϵ out E gs (mev ) E fe (mev ) E 21 (mev ) µ 21 10 28 (C m) 3 309.3013 591.3591 282.0578 3.3647 5 279.7913 537.0854 257.2941 4.3947 7 259.9441 484.5292 224.5851 5.9057 9 247.4170 454.5091 207.0921 6.7412 ϵ eff = 1 (χ tot, χ (1), χ (3) ) N N = 1.7 10 23 m 3 T 1 = 1 ps, T 2 = 0.14 ps χ (1)

χ (3) mev

χ (1) χ (3) µ 21 χ (3) χ (1) χ (1) µ 21 2 χ (3) µ 21 4 χ tot χ tot

ϵ eff χ (1) χ (3) χ (1) χ (3) µ 21 µ 21

µ 21

SnO 2 Li +

0.3 0.7