InP ; c ß f Ä X N Ë¶ ¥c Ü R GaP x Sb 1−x 8 ý O © X N Ëc Ç

(1)

InP ; c ß f Ä X N Ë¶ ¥c Ü R GaP ^x Sb 1−x 8 ý O © X N Ëc Ç

9 + ä ] 8 ; · »¦ P ^∗

â

l @ / < Æ § Ó ü t o < Æõ , Ã º" é ¶ 440-760 (2009¸ 3 Z 4 10{ 9 ~ Ã Î6 £ §)

:

r 7 Hë H \ " f H " é ¶ o½ + ËÓ ü t ì ø Í ¸^ GaP

x

Sb

1−x

_ D ¥ ½ + Ë0 l x ¸q \ É r \ -t ½ × ¼ ½ ¨ ¸ o\ ¦ # 3 6 x

&

h x 9 ] X ½ + Ë (UTB: universal tight binding) ~ ½ ÓZ O ` ¦ s 6 x # ½ ¨ % i . ^ 0 l x ¸\ É r(0 ≤ x ≤ 1) Å

Òכ ¹ @ /g A& h (Γ, L, X)_ ½ × ¼Ì s o H : r ¸ 300 K\ " f 0.73 ≤ E(Γ) ≤ 2.88 (eV), 1.17 ≤ E(L) ≤ 2.72 (eV), 1.72 ≤ E(X) ≤ 2.16 (eV) s % 3 . GaP

^x

Sb

1−x

H x < 0.56 t H E(Γ) \ _ ô Ç f ] X ½ × ¼Ì ss

%

3 x > 0.56\ " f H E(X) \ _ ô Ç ç ß ] X ½ × ¼Ì se ` ¦ · ú Ã º e % 3 . GaP

x

Sb

1−x

ü < InP H D ¥ ½ + Ë0 l x ¸ x = 0.35 \ " f ½ + ËU ´s 2.541 ˚ A(< Ê É r, © Ã º 5.868 ˚ A) Ü ¼ Ð & ñ ½ + Ër ~ ´ Ã º e . & ñ ½ + Ë ) a GaP

0.35

Sb

0.65

ü < InP_ â > \ " f_ { ¸á Ô! Ó õ ¸{ ¸á Ô! Ó > í ß õ H 0.22 eV ü < 0.12 eV s % 3 .

PACS numbers: 71.22.+i, 71.20.Nr

Keywords: GaP

x

Sb

1−x

, ½ × ¼& ñ § > =, { ¸á Ô! Ó , ¸{ ¸á Ô! Ó

I. " e Â ] Ø

III-V " é ¶( " é ¶ < Ê É r " é ¶) o½ + ËÓ ü t ì ø Í ¸^ H D ¥ ½ + Ë $ í ì r s

0 l x ¸\ 0 A& h Ü ¼ Ð & h : £ ¤$ í ` ¦ ¸] X ½ + É Ã º e

H @ /é ß y Ä »6 x ô Ç Ó ü t| 9 Ð" f, Y Us $ s ¸× ¼ F g Ã Ðt l

1 p x F g ¦î ß ¾ ¡ § _ l ó ø Í, x 4 ¤8 £ x, Ö ¸$ í 8 £ x1 p x Ü ¼ Ð æ ¼

#

t ¦ e . \ x Ã Ìr ° ú É r q ¨ î + þ A & ñ Ò q t © l Õ ü t µ 1 Ï² ú Ð í

H ¸ Z } É r III-V " é ¶ o½ + ËÓ ü t ì ø Í ¸^ \ ¦ Ò q t$ í r ~ ´ Ã º e > H

d \ , s \ @ /ô Ç s : r& h x 9 z ´+ « >& h ½ ¨ Ö ¸ µ 1 Ïy s

À Ò# Q4 R M ® o [1–4].

" é ¶ o½ + ËÓ ü t ì ø Í ¸^ GaP ^x Sb 1−x H 530 ^◦ C \ " f _

% ò

% i \ (0.01<x<0.99) 5 g D ¥ o$ í Ì s(miscibility gap)s + þ

A$ í ÷ &# Q [5] & ñ Ò q t © s _ Ô ¦ 0 p x ô Ç Ó ü t| 9 Ð · ú 94 R M

®

o Ü ¼ 9, s \ @ /ô Ç z ´+ « >& h ½ ¨ü < & h 6 £ x6 x s ] jô Ç÷ &

#

Q M ® o . Õ ª Q 1988¸ OMVPE(Organometallic vapor phase epitaxy) ~ ½ ÓZ O ` ¦ s 6 x # % 6 £ § Ü ¼ Ð GaP ^x Sb 1−x

& ñ Ò q t © \ $ í / B N # [6], GaAs l ó ø Í\ & ñ ½ + Ër GaP _0.68 Sb _0.32 _ \ -t Ì ss 1.43 eVs H z ´+ « >& h õ

\ ¦ % 3 % 3 . ¢ ¸ É r ½ ¨ õ \ " f H, GaP x Sb 1−x ` ¦ GaAs \ & ñ ½ + Ër ( ` ¦ M : ç ß ] X Ì s` ¦ ° ú > ÷ &# Q & h 6

£

x6 x u b # Qt InP\ & ñ ½ + Ër ( ` ¦ M : H Y

Us $ s ¸× ¼(tunable laser diodes) 1 p x Ü ¼ Ð" f_ 6 £ x6 x

0 p x$ í s ¸d Û ¼Y U ] jl ÷ &% 3 [7,8]. : £ ¤ y â > \ " f_

∗

E-mail: [email protected]

½ × ¼ & ñ § > = © I { ¸á Ô! Ó (VBO: valence band offset) õ ¸{ ¸á Ô! Ó (CBO: conduction band offset)` ¦

&

ñ S X y H כ É r s Ó ü t| 9 _ 6 £ x6 x 0 p x$ í \ @ /ô Ç & ñ & h

&

ñ Ð ) a [9–11]. Õ ª Q f s Ó ü t| 9 \ @ /ô Ç ½ × ¼& ñ

§ >

=(band alignment) © I \ @ /ô Ç s : r& h õ H & ñ S X y

· ú

9 \ O .

: r 7 Hë H \ " f H " é ¶ o½ + ËÓ ü t ì ø Í ¸^ GaP x Sb _1−x _ $ í ì

r q \ É r \ -t ½ × ¼ ½ ¨ ¸ o\ ¦ # 3 6 x& h x 9 ] X

½

+ Ë (UTB: universal tight binding) ~ ½ ÓZ O [12, 13]` ¦ s 6 x

# ½ ¨ô Ç . Å Òכ ¹ @ /g A& h Γ, L, X\ " f_ ¸{ _ þ j è

° ú

כ(CBM: conduction band minimum)õ { _ þ j

@

/° ú כ(VBM: valence band maximum) o\ ¦ > í ß # f

] X ½ × ¼Ì s(direct band gap)\ " f ç ß ] X ½ × ¼Ì s(indirect band gap) Ü ¼ Ð_ s \ ¦ ½ ©" î ô Ç . ¢ ¸ô Ç GaP x Sb 1−x \ ¦ InP \ & ñ ½ + Ër ( ` ¦ M :, â > \ " f_ { ü <

¸ { _ ½ × ¼& ñ § > =(band alignment) © I \ ¦ ½ ¨ô Ç .

II. T Â ] Ø

# 3

6 x& h x 9 ] X ½ + Ë ~ ½ ÓZ O É r III-V7 á ¤ " é ¶ o½ + ËÓ ü t ì ø Í ¸^ _

& h © I \ ¦ > í ß H X < e # Q B Ä º 4 ¤ ¸ ú Ù þ ¡~ Á º| 9 " f

´

òõ \ ¦ q §& h ç ß é ß ¦ & ñ S X > ½ ¨½ + É Ã º e H ´ òõ & h s

¦ ¼ # o ô Ç ~ ½ ÓZ O Ü ¼ Ð · ú 94 R e [12, 13]. UTB\ _ ô

Ç " é ¶ o½ + ËÓ ü t ì ø Í ¸^ GaP x Sb 1−x _ K x 9 Ðm î ß É r s " é ¶

-580-

(2)

o½ + ËÓ ü t ì ø Í ¸^ GaPü < GaSb_ Ä »´ òK x 9 Ðm î ß (effective Hamiltonian)` ¦ × æ¨ î ç H ô Ç ° ú כÜ ¼ Ð · p .

H ˆ _GaP

_x

_Sb

_1−x

= x ˆ H _GaP + (1 − x) ˆ H _GaSb (1)

#

l " f s " é ¶ o½ + ËÓ ü t ì ø Í ¸^ _ Ä »´ òK x 9 Ðm î ß É r | 9 " f K

x 9 Ðm î ß (ordered Hamiltonian)õ Á º| 9 " fK x 9 Ðm î

ß (disordered Hamiltonian)_ ½ + ËÜ ¼ Ð ³ ð & ³ ) a .

H ˆ GaP = ˆ H _GaP ⁰ + ∆ ˆ H GaP

H ˆ _GaSb = ˆ H _GaSb ⁰ + ∆ ˆ H _GaSb (2)

| 9

" fK x 9 Ðm î ß ˆ H _GaP ⁰ ( ˆ H _GaSb ⁰ ) H ¢ - a & ñ é # Qo © I GaP (GaSb) _ K x 9 Ðm î ß s 9, Á º| 9 " f K x 9 Ðm î ß

∆ ˆ H _GaP (∆ ˆ H _GaSb ) H GaP (GaSb) _ 6 £ § s : r u ¨ 8 \ _ ô

Ç q + þ A´ òõ \ ¦ Ä » ¸ H ª ∆ ˆ H GaP :Sb (∆ ˆ H GaSb:P ) Ü ¼

Ð" f u ¨ 8 $ í ì r q _ Y L Ü ¼ Ð · p .

∆ ˆ H GaP = (1 − x)∆ ˆ H GaP :Sb

∆ ˆ H _GaSb = x∆ ˆ H _GaSb:P (3) (2)−(3)d ` ¦ (1)d \ @ /{ 9 , " é ¶ o½ + ËÓ ü t GaP x Sb _1−x _

D ¥ ½ + Ë 0 l x ¸\ É r 8 ú x K x 9 Ðm î ß ` ¦ ½ ¨½ + É Ã º e .

H GaP

_x

Sb

_1−x

= xH _GaP ⁰ + (1 − x)H _GaSb ⁰

+x(1 − x)(∆ ˆ H _{GaP :Sb} + ∆ ˆ H _GaSb:P )(4)

#

l " f 8 ú x Á º| 9 " fK x 9 Ðm î ß ∆ ˆ H _{GaP :Sb} + ∆ ˆ H _GaSb:P H

¿

º s " é ¶ o½ + ËÓ ü t _ | 9 " fK x 9 Ðm î ß _ ˆ H _GaP ⁰ − ˆ H _GaSb ⁰ Ð é

H .

ì

ø Í ¸^ _ \ -t © I H & ñ ? /_ Schrodinger ~ ½ Ó& ñ d _

K \ ¦ ½ ¨K Í Ç rÜ ¼ Ð+ · ú Ã º e .

HΨ ˆ n (~ k, ~ r) = ε n (k)Ψ n (~ k, ~ r) (5)

& ñ _ ¦Ä » © I < ÊÃ º Ψ ⁿ (~ k, ~ r) H Bloch & ñ o \ ¦ ë ß 7 á ¤ r v

H " é ¶ C ¸ < ÊÃ º[ þ t φ ^ν _αj (~ r) _ + þ A ½ + ËÜ ¼ Ð · p .

Ψ n (~ k, ~ r) = 1

√ N ⁰ X

α,j,ν

A ⁿ _αν (~ k)e ^i~ ^{k·( ~} ^R

^j

^+~ ^τ

^ν

⁾ φ ^ν _αj (~ r − ~ τ ν ) (6)

#

l " f, N ⁰ É r & ñ _ é ß 0 A[ j í(unit cell)_ Ã º, ~ R j H j P

: Ú Ô Z s Û ¼ 0 Au , é ß 0 A[ j í î ß _ ν P : s : r _

© @ /0 Au H ~ τ ν Ð ? / 9 φ ^ν αj H ν P : s : r _ þ jü @y

"

é

¶ C ¸ < ÊÃ ºs . o½ + ËÓ ü t ½ ¨ ¸\ ¦ ° ú H III-V ì ø Í ¸

^

H sp ³ s Ú Ôo × ¼ Ð ½ + Ë © I \ ¦ s À Ò# Q ª s : r õ 6 £ § s

: r \ y 4> hm (α = s, p _x , p y , p z ) 8 ú x 8 > h_ þ jü @y " é ¶

Table 1. The TB parameters and bond length for the binary compounds GaP, GaSb, and InP.

TB parameters

and bondlength GaP GaSb InP

P

1

= E

ss

(000)0 −6.285 −6.093 −6.294 P

2

= E

ss

(000)1 −2.789 −3.886 −3.429 P

3

= E

xx

(000)0 1.094 0.810 1.843 P

4

= E

xx

(000)1 2.382 2.348 2.609 P

5

= 4E

ss

(0.5, 0.5, 0.5) −7.750 −6.218 −6.220 P

6

= 4E

sx

(0.5, 0.5, 0.5)01 5.260 5.100 4.700 P

7

= 4E

sx

(0.5, 0.5, 0.5)10 4.950 4.110 4.380 P

8

= 4E

xx

(0.5, 0.5, 0.5) 2.440 1.600 2.280 P

9

= 4E

xy

(0.5, 0.5, 0.5) 5.560 5.500 5.300 P

10

= 4E

xx

(0, 1, 1)0 −1.138 −0.920 −0.720 P

11

= 4E

xx

(0, 1, 1)1 −1.186 −2.012 −1.338 P

12

= 4E

xy

(1, 1, 0)0 0.760 0.760 0.520 P

13

= 4E

xy

(1, 1, 0)1 1.330 1.330 0.910 P

14

= 4E

xx

(1, 1, 0)0 0.854 0.599 0.384 P

15

= 4E

xx

(1, 1, 0)1 1.189 1.008 0.739 P

16

= 4E

sx

(1, 1, 0)0 0.024 0.042 0.087 P

17

= 4E

sx

(1, 1, 0)1 0.083 0.060 0.102 P

18

= 4E

ss

(1, 1, 0)0 −0.167 −0.163 −0.032 P

19

= 4E

ss

(1, 1, 0)1 −0.181 −0.224 −0.034 P

20

= 4E

sx

(0, 1, 1)0 1.180 0.750 0.900 P

21

= 4E

sx

(0, 1, 1)1 −0.080 −0.240 −0.006 P

22

= 4E

xy

(0, 1, 1)0 0.0 0.0 0.0 P

23

= 4E

xy

(0, 1, 1)1 0.0 0.0 0.0

d

⁰

2.358 2.639 2.541

C

¸ < ÊÃ º K x 9 Ðm î ß _ l $ < ÊÃ º ÷ &# Q, y K x 9 Ð m

î ß ' § > =כ ¹ è H

H mn =< Ψ m |H|Ψ n > (7) s

. s Ð" f 8 × 8 K x 9 Ðm î ß ' § > =d ` ¦ % 3 > ÷ & 9, s

\

¦ @ /y o(diagonalize) r ( Ü ¼ Ð" f, 4> h_ { ü <

4 > h_ ¸{ 8 ú x 8 > h_ \ -t ¦Ä »° ú כ` ¦ % 3 > ) a . s M

:, K x 9 Ðm î ß _ " é ¶ C ¸ < ÊÃ º\ @ /ô Ç & h ì r É r × æd " é ¶

( ª s : r õ 6 £ § s : r) \ TB B > h Ã º(TB parame- ter) E αβ (E αβ = <φ ^ν _αj |H|φ ^ν _βj

⁰

>) Ð é H . o½ + ËÓ ü t ½ ¨

¸\ ¦ ° ú H III-V o½ + ËÓ ü t ì ø Í ¸^ \ @ /ô Ç TB B > h Ã º H ¿ º

P : î r " é ¶ \ @ /ô Ç © ñ 6 x t ¦ 9 23> h

9 כ ¹ [14].

(4)d −(7)d ` ¦ : x K " é ¶ o½ + ËÓ ü t ì ø Í ¸^ GaP x Sb _1−x _

\

-t ¦Ä »° ú כ É r s " é ¶ o½ + ËÓ ü t ì ø Í ¸^ GaPü < GaSb_ TB B

> h Ã º(E ^αβ (x)) ü < D ¥ ½ + Ëq _ ½ ÓÜ ¼ Ð ? / H כ

`

¦ · ú Ã º e H X <,

E αβ (x) = xE _{GaP ,αβ} + (1 − x)E _GaSb,αβ

+x(1 − x)∆E αβ (GaPSb) (8)

(3)

Table 2. The energy eigen values for the alloy GaP _x Sb _1−x (with x = 0.0 − 1.0) and InP at the symmetrical points Γ, L, and X.

Material Γ L X

GaSb −11.87,0.0,0.0,0.0 −10.25,−7.13,−1.08,−1.08 −9.33,−6.76,−2.61,−2.61 0.73,3.44,3.44,3.44 1.17,5.21,6.33,6.33 1.72,1.79,8.7,8.7 GaP

0.1

Sb

0.9

−11.91,0.09,0.09,0.09 −10.31,−7.05,−1.07,−1.07 −9.33,−6.73,−2.61,−2.61

0.97,3.70,3.70,3.70 1.39,5.20,6.38,6.38 1.78,1.87,8.64,8.64 GaP

0.2

Sb

0.8

−11.97,0.15,0.15,0.15 −10.37,−6.98,−1.08,−1.08 −9.34,−6.71,−2.61,−2.61

1.20,3.94,3.94,3.94 1.60,5.19,6.44,6.44 1.84,1.96,8.60,8.60 GaP

0.3

Sb

0.7

−12.05,0.20,0.20,0.20 −10.43,−6.93,−1.08,−1.08 −9.35,−6.72,−2.62,−2.62

1.43,4.16,4.16,4.16 1.79,5.19,6.49,6.49 1.89,2.06,8.56,8.56 GaP

0.4

Sb

0.6

−12.14,0.23,0.23,0.23 −10.48,−6.90,−1.1,−1.1 −9.36,−6.74,−2.63,−2.63

1.65,4.36,4.36,4.36 1.97,5.2,6.55,6.55 1.95,2.16,8.53,8.53 GaP

0.5

Sb

0.5

−12.24,0.24,0.24,0.24 −10.53,−6.89,−1.12,−1.12 −9.37,−6.77,−2.64,−2.64

1.87,4.55,4.55,4.55 2.13,5.21,6.61,6.61 1.99,2.26,8.51,8.51 GaP

0.6

Sb

0.4

−12.36,0.23,0.23,0.23 −10.58,−6.89,−1.15,−1.15 −9.38,−6.82,−2.65,−2.65

2.08,4.72,4.72,4.72 2.27,5.23,6.66,6.66 2.03,2.36,8.49,8.49 GaP

0.7

Sb

0.3

−12.50,0.20,0.20,0.20 −10.63,−6.92,−1.18,−1.18 −9.40,−6.88,−2.66,−2.66

2.29,4.87,4.87,4.87 2.40,5.26,6.72,6.72 2.07,2.47,8.49,8.49 GaP

0.8

Sb

0.2

−12.65,0.16,0.16,0.16 −10.67,−6.96,−1.22,−1.22 −9.42,−6.96,−2.67,−2.67

2.49,5.01,5.01,5.01 2.52,5.28,6.77,6.77 2.11,2.57,8.48,8.48 GaP

0.9

Sb

0.1

−12.82,0.09,0.09,0.09 −10.71,−7.02,−1.27,−1.27 −9.44,−7.05,−2.69,−2.69

2.69,5.13,5.13,5.13 2.62,5.31,6.83,6.83 2.13,2.68,8.49,8.49 GaP −13.00,0.0,0.0,0.0 −10.75,−7.09,−1.32,−1.32 −9.46,−7.15,−2.7,−2.7

2.88,5.24,5.24,5.24 2.72,5.35,6.89,6.89 2.16,2.79,8.5,8.5 Inp −11.34,0.0,0.0,0.0 −9.68,−6.41,−0.89,−0.89 −8.85,−6.28,−2.09,−2.09

1.42,4.64,4.64,4.64 2.13,5.83,6.77,6.77 2.36,3.24,8.6,8.6 GaP

0.35

Sb

0.65

−12.09,0.22,0.22,0.22 −10.46,−6.92,−1.09,−1.09 −9.35,−6.73,−2.62,−2.62

1.54,4.26,4.26,4.26 1.88,5.19,6.52,6.52 1.92,2.11,8.55,8.55

#

l " f ∆E ^αβ (GaPSb) H ¿ º s " é ¶ o½ + ËÓ ü t TB B > h Ã º_

E GaP ,αβ − E _GaSb,αβ ) a . TB B > h Ã º × æ \ " f : £ ¤ y

' Í P : î r " é ¶ [ þ t ç ß _ © ñ 6 x` ¦ ? / H TB B

> h Ã º(P ⁵ − P 9 ) H s [ þ t _ ½ + ËU ´s ] jY L \ ì ø Íq Y V < Ê

`

¦ Harrison s µ 1 ß) ? /% 3 Ü ¼ 9 [15], s \ ¦ ¦ 9 d ¯ ² (x)E αβ (x) = x(d ⁰ _GaP ) ² E GaP,αβ

+(1 − x)(d ⁰ _GaSb ) ² E GaSb,αβ

+x(1 − x)∆d ² (GaPSb)∆E αβ (GaPSb) (9) GaP _x Sb _1−x _ ¨ î ç H ½ + ËU ´s o d(x) ¯ H GaP ü <

GaSb _ ½ + ËU ´s \ ¦ D ¥ ½ + Ë0 l x ¸\ × æ¨ î ç H ô Ç ° ú כs 9, ∆d(GaPSb) H ¿ º s " é ¶ o½ + ËÓ ü t _ ½ + ËU ´s s

∆d(GaPSb) = d ⁰ _GaP − d ⁰ _GaSb Ð é H . GaP x Sb _1−x \ @ /ô Ç

¨ î

ç H ½ + ËU ´s H ¯ d GaP

_x

Sb

_1−x

= (1 − x)d ⁰ _GaSb + xd ⁰ _GaP s

¦ InP_ ½ + ËU ´s H 2.541 ˚ A Ð" f [16], GaP x Sb 1−x \

@

/ô Ç ¨ î ç H ½ + ËU ´s 2.541 ˚ A(7 £ ¤, © Ã º H 5.868 ˚ A:

© Ã º = ^√ ⁴ ₃ X ½ + ËU ´s ){ 9 M : & ñ ½ + Ës ) a .

7 £ ¤, 2.541 = 2.639(1 − x) + 2.358x` ¦ ë ß 7 á ¤ r v H D ¥ ½ + Ë0 l x

¸ x = 0.35 Ð GaP 0.35 Sb _0.65 ü < InPs & ñ ½ + Ë ) a .

III. 4 m õ m Í + s ÇÊ Ý

GaP _x Sb _1−x _ \ -t © I \ ¦ ½ ¨ l 0 AK 6 x ô Ç s

"

é

¶ o½ + ËÓ ü t GaP, GaSb, InP _ TB B > h Ã º x 9 ½ + ËU ´s

H Table 1 \ e . : r 7 Hë H \ " f 6 x ô Ç TB B > h

Ã º H s " é ¶ o½ + ËÓ ü t ì ø Í ¸^ _ { þ j@ /° ú כ(VBM)s 0 s ÷ &> " f, 300 K\ " f Å Òכ ¹@ /g A& h \ " f_ \ -t Ì

s[ þ t` ¦ ¦ 9 # [17] Talwarü < Tings ½ ¨ô Ç TB B > h Ã

º [14]\ ç ß ¸& ñ ) a כ s . Table 1` ¦ s 6 x # D ¥ ½ + Ë 0

l

x ¸ x\ É r GaP x Sb 1−x _ Å Òכ ¹ @ /g A& h Γ, L, X\ " f _

\ -t ° ú כs Table 2\ e . Table 2\ " f Ð

(4)

Table 3. The principal energy band gaps (E(Γ), E(L), E(X)) for the alloy GaP _x Sb _1−x (with x = 0.0 − 1.0) and InP.

Material E(Γ) (eV) E(L) (eV) E(X) (eV)

GaSb 0.73 1.17 1.72

GaP

0.1

Sb

0.9

0.88 1.30 1.69

GaP

0.2

Sb

0.8

1.05 1.45 1.69

GaP

0.3

Sb

0.7

1.23 1.59 1.69

GaP

0.4

Sb

0.6

1.42 1.74 1.72

GaP

0.5

Sb

0.5

1.63 1.89 1.75

GaP

0.6

Sb

0.4

1.85 2.04 1.80

GaP

0.7

Sb

0.3

2.09 2.20 1.87

GaP

0.8

Sb

0.2

2.33 2.36 1.95

GaP

0.9

Sb

0.1

2.60 2.53 2.04

GaP 2.88 2.72 2.16

InP 1.42 2.13 2.36

GaP

0.35

Sb

0.65

1.32 1.66 1.70

Fig. 1. Composition dependent band gap variations for the alloy GaP _x Sb _1−x .

H ü < ° ú s Γ\ " f ] j Ð% i ~ VBM H x 7 £ x ½ + ÉÃ º2 ¤ x = 0.5 H % \ " f 0.24 eV 7 £ x Ù þ ¡ r y è % i .

s

ï r 0 A o\ ¦ · ú Ã º e .

GaP _x Sb _1−x _ Å Òכ ¹ ½ × ¼Ì s E(Γ), E(L), E(X) É r y

@ /g A& h _ ¸{ þ j è° ú כ(CBM)\ " f { þ j@ /

° ú

כ(VBM)` ¦ É ° ú כÜ ¼ Ð ½ ¨½ + É Ã º e .

E(Γ) = CBM (Γ) − V BM (Γ) E(L) = CBM (L) − V BM (Γ)

Fig. 2. Band alignment of the alloy GaP 0.35 Sb 0.65 lattice- matched to InP.

E(X) = CBM (X) − V BM (Γ) (10) Table 2 _ õ ü < (10)d ` ¦ s 6 x # , D ¥ ½ + Ë0 l x ¸\ É r Å

Òכ ¹ ½ × ¼Ì s o H Table 3 õ Fig. 1\ e . Å Ò כ

¹ @ /g A& h (Γ, L, X)\ " f_ ^ 0 l x ¸\ É r(0 ≤ x ≤ 1) ½ × ¼Ì s o H ; 0.73 ≤ E(Γ) ≤ 2.88 (eV), 1.17 ≤ E(L) ≤ 2.72 (eV), 1.72 ≤ E(X) ≤ 2.16 (eV) s % 3 . Õ ªa Ë >

\

" f Ð H ü < GaP ^x Sb 1−x H x < 0.56 t H E(Γ) \ _ ô

Ç f ] X ½ × ¼Ì s` ¦ ? / x > 0.56{ 9 M : H E(X) \ _

ô Ç ç ß ] X ½ × ¼Ì s` ¦ ° ú H . GaP ^x Sb 1−x \ ¦ InP \

&

ñ ½ + Ër GaP _0.35 Sb _0.65 /InP \ @ /K " f H f ] X Ì ss ¦ s M

:_ CBM É r 1.54 eV s ¦ VBM É r 0.22 eV Ü ¼ Ð" f ½ × ¼ Ì

s ° ú כ É r 1.54 eV − 0.22 eV = 1.32 eV s . > á ¤ ° ú É r r Û ¼

% 7

GaP x Sb 1−x ` ¦ GaAs (d ⁰ _GaAs = 2.448 ˚ A) \ & ñ ½ + Ë r

~ ´ â Ä º D ¥ ½ + Ë0 l x ¸ ¸| É r x = 0.68 s ÷ & ¦ X ½ × ¼\ _

ô Ç ç ß ] X ½ × ¼Ì s` ¦ ° ú H . s ü < ° ú s GaP x Sb _1−x ` ¦ # Q

"

² ú t H X <, GaAs\ & ñ ½ + Ër ~ ´ M : H ç ß ] X Ì ss InP \ & ñ ½ + Ër ( ` ¦ M : H f ] X Ì ss ÷ &Ù ¼ Ð, InP\

r v H כ s ´ òÖ ¦& h F g < Æ ´ òõ \ ¦ 4 R : r .

InP \ & ñ ½ + Ër GaP _0.35 Sb _0.65 _ â > \ " f_

½

+ É Ã º e . InP_ VBMü < CBM É r 0 eV ü < 1.42 eV s 9, GaP ^0.35 Sb 0.65 _ VBMü < CBM É r 0.22 eV ü <

1.54 eV Ð" f s \ @ /ô Ç ¸³ ð H Fig. 2 \ " fü < ° ú .

s

\ ¦ : x K GaP 0.35 Sb _0.65 ü < InP_ â > \ " f_ CBO

(=∆E _c ) H 0.12 eV s ¦ VBO(=∆E v ) H 0.22 eV Ð" f ] j

2+ þ A ì ø Í ¸^ (type-II semiconductor) H d` ¦ · ú Ã º e .

(5)

GaP x Sb 1−x ü < InP_ q & ñ ½ + Ëq 0.5 % â >

\

" f Ã º' ô Ç z ´+ « > õ [7]\ " f H ∆E c = 0.05 eV ü <

∆E _v = 0.5 eV ] j2+ þ A ì ø Í ¸^ H d` ¦ µ 1 ß+ À I . ¢ - a

& ñ ½ + Ë ) a â > \ @ /ô Ç : r 7 Hë H _ s : r& h õ ü < q

& ñ ½ + Ëq 0.5 %\ @ /ô Ç z ´+ « >& h õ ü < q §½ + É M :, ] j 2+ þ A ì

ø Í ¸^ H & ñ $ í & h õ H { 9 u % i Ü ¼ VBO ç ß s z

´+ « >u Ð s ` É r ° ú כ` ¦ % 3 6 £ § Ü ¼ Ð" f & ñ | ¾ Ó& h s

\

¦ ? /% 3 . s H & ñ ½ + Ëq ü < z ´+ « > ¸| ( : r ¸

. s > _ ½ × ¼& ñ § > =\ @ /ô Ç É r z ´+ « >& h õ [ þ t É r Ð

¦ ) a \ O . Ð: x â > \ " f_ ½ × ¼ ¸á Ô! Ó \ @ /ô Ç z ´ +

«

: ´ ú § .

~

&

h o\ ¦ · ú Ã º e # Q " é ¶/ s " é ¶ o½ + ËÓ ü t ì ø Í ¸^ _ â >

\ " f & h 6 x ½ + É Ã º e H ~ ½ ÓZ O s . s ~ ½ ÓZ O É r s " é ¶/ s " é ¶

.

Y

c p w Ã U Ø ô

[1] K. Iga and S. Kinoshita, Process Technology for Semiconductor Lasers (Springer, Berlin, 1996).

[2] I. Vurgaftman, J. R. Meyer and L. R. Ram-Mohan, J. Appl. Phys. 89, 5815 (2001).

[3] M. Quillec, Materials for Optoelectronics (Kluwer Academic Pub., Boston, 1996).

[4] A. B. Chen and A. Sher, Semiconductor Alloys (Plenum Press, New York, 1995).

[5] G. B. Stringfellow and M. J. Cherng, J. Cryst.

Growth 64, 413 (1983).

[6] M. J. Jou, Y. T. Cherng, H. R. Jen and G. B.

Stringfellow, Appl. Phys. Lett. 52, 549 (1988).

[7] S. Loualiche, A. Le Corre, S. Salaun, J. Caulet, B.

Lambert, M. Gauneau, D. Lecrosnier and B. De- veaud, Appl. Phys. Lett. 59, 423 (1991).

[8] T. Y. Seong, G. R. Booker, A. G. Norman, F. Glas, and G. B. Stringfellow, J. Korean Phys. Soc. 52, 471 (2008).

[9] K. Shim and H. Rabitz, Appl. Phys. Lett. 80, 4543 (2002).

[10] F. K¨ ohler, G. B¨ ohm, R. Meyer and M. C. Amann, Appl. Phys. Lett. 87, 32102 (2005).

[11] J. Park, J. Leem and S. Noh, J. Korean Phys. Soc.

28, 499 (1995).

[12] K. Shim and H. Rabitz, Phys. Rev. B 57, 12874 (1998).

[13] K. Shim and H. Rabitz, J. Appl. Phys. 85, 7715 (1999).

[14] D. N. Talwar and C. S. Ting, Phys. Rev. B 25, 2660 (1982).

[15] Sverre Foyen and W. A. Harrison, Phys. Rev. B 20, 2420 (1979).

[16] J. M. Baranowski, J. Phys. C 17, 6287 (1984).

[17] O. Madelung, M. Schulz and H. Weiss, Numerical

Data and Functional Relationships in Sience and

Technology; Semiconductors (1982), Vol. 17.

(6)

Band Alignment of GaP _x Sb _1−x Lattice-matched to InP

Jung Wook Bae and Kyurhee Shim ^∗

Department of Electro-Physics, Kyonggi University, Suwon 443-760 (Received 10 March 2009)

The composition-dependent energy-band structure of the ternary alloy semiconductor GaP

x

Sb

1−x

was obtained using the universal tight binding (UTB) method. The principal band gaps at 300 K were calculated to be 0.73 ≤ E(Γ) ≤ 2.88 (eV), 1.17 ≤ E(L) ≤ 2.72 (eV), and 1.72 ≤ E(X) ≤ 2.16 (eV). The alloy GaP

x

Sb

1−x

was found to have a direct gap induced by E(Γ) in the composition range x < 0.56 while an indirect gap induced by E(X) was found in the composition range x > 0.56.

The alloy GaP

x

Sb

1−x

and InP can be lattice-matched at x = 0.35 with the lattice parameter 5.868

˚ A. For the band alignment of the alloy GaP

0.35

Sb

0.65

lattice-matched to InP, the calculated values of the valence band offset (VBO) and the conduction band offset(CBO) were 0.22 eV and 0.12 eV, respectively.

PACS numbers: 71.22.+i, 71.20 Nr

Keywords: GaP

x

Sb

1−x

, Band alignment, Valence band offset, Conduction band offset

∗

InP ; c ß f Ä X N Ë¶  ¥c Ü R GaP x Sb 1−x 8 ý  O © X N Ëc  Ç

InP ; c ß f Ä X N Ë¶  ¥c Ü R GaP x Sb 1−x 8 ý  O © X N Ëc  Ç

9

+ ä ] 8 ; ·   »¦ P  ∗

 â

l @ / < Æ §    Ó ü t o  < Æõ , Ã º" é ¶ 440-760 (2009¸  3 Z 4 10{ 9  ~ Ã Î6 £ §)

 :

r  7 Hë  H \ " f  H   " é ¶ o½ + ËÓ ü t ì ø Í ¸^  GaP

Sb

_   D ¥ ½ + Ë0 l x  ¸q \    É r \  -t   ½ × ¼ ½ ¨ ¸   o\  ¦ # 3 6 x

&

h x 9 ] X   ½ + Ë (UTB: universal tight binding) ~ ½ ÓZ O `  ¦ s 6 x #  ½ ¨ % i  .   ^  0 l x  ¸\    É r(0 ≤ x ≤ 1) Å

Òכ ¹ @ /g A& h (Γ, L, X)_   ½ × ¼Ì  s   o  H  : r  ¸ 300 K\ " f 0.73 ≤ E(Γ) ≤ 2.88 (eV), 1.17 ≤ E(L) ≤ 2.72 (eV), 1.72 ≤ E(X) ≤ 2.16 (eV) s % 3  . GaP

Sb

  H x < 0.56  t   H E(Γ) \  _ ô  Ç f  ] X  ½ × ¼Ì  ss 

%

3   x > 0.56\ " f  H E(X) \  _ ô  Ç ç ß ] X   ½ × ¼Ì  se  `  ¦ · ú  Ã º e  % 3  . GaP

Sb

ü < InP  H  D ¥ ½ + Ë0 l x  ¸ x = 0.35 \ " f   ½ + ËU ´s  2.541 ˚ A(< Ê É r,     © Ã º 5.868 ˚ A) Ü ¼ Ð     & ñ ½ + Ër ~ ´ Ã º e   .     & ñ ½ + Ë ) a GaP

Sb

ü < InP_   â >   \ " f_     {   ¸á Ô! Ó õ     ¸{   ¸á Ô! Ó > í ß    õ   H 0.22 eV ü < 0.12 eV s % 3  .

PACS numbers: 71.22.+i, 71.20.Nr

Keywords: GaP

Sb

,  ½ × ¼& ñ § > =,    {   ¸á Ô! Ó ,    ¸{   ¸á Ô! Ó

I. " e Â ] Ø

III-V " é ¶(   " é ¶ < Ê É r  " é ¶) o½ + ËÓ ü t ì ø Í ¸^   H  D ¥ ½ + Ë $ í ì  r s

  0 l x  ¸\       0 A& h Ü ¼ Ð    & h : £ ¤$ í `  ¦  ¸] X ½ + É Ã º e 

  H @ /é ß y  Ä »6 x ô  Ç Ó ü t| 9  Ð" f, Y Us $  s  ¸× ¼  F g  Ã Ðt  l

 1 p x F g     ¦î ß ¾ ¡ § _  l ó ø Í, x 4  ¤8 £ x,  Ö ¸$ í 8 £ x1 p x Ü ¼ Ð æ ¼

#

t  ¦ e   . \ x  Ã Ìr ° ú  É r q ¨ î + þ A  & ñ Ò q t © l Õ ü t µ 1 Ï² ú  Ð í

 H  ¸ Z  } É r III-V " é ¶ o½ + ËÓ ü t ì ø Í ¸^ \  ¦ Ò q t$ í r ~ ´ Ã º e  >  H



d \    , s \  @ /ô  Ç s  : r& h x 9 z ´+ « >& h   ½ ¨  Ö ¸ µ 1 Ïy  s

À Ò# Q4 R M ® o  [1–4].





" é ¶ o½ + ËÓ ü t ì ø Í ¸^  GaP x Sb 1−x   H 530 ◦ C \ " f  _   

% ò

% i \ (0.01<x<0.99)   5 g  D ¥ o$ í Ì  s(miscibility gap)s  + þ

A$ í ÷ &# Q [5]   & ñ Ò q t © s   _  Ô  ¦ 0 p x ô  Ç Ó ü t| 9  Ð · ú  94 R M

®

o Ü ¼ 9, s \  @ /ô  Ç z ´+ « >& h   ½ ¨ü <    & h 6 £ x6 x s  ] jô  Ç÷ &

#

Q M ® o . Õ ª Q  1988¸  OMVPE(Organometallic vapor phase epitaxy) ~ ½ ÓZ O `  ¦ s 6 x #  % 6 £ § Ü ¼ Ð GaP x Sb 1−x

 

& ñ Ò q t © \  $ í / B N #  [6], GaAs l ó ø Í\      & ñ ½ + Ër   GaP 0.68 Sb 0.32 _  \  -t  Ì  ss  1.43 eVs    H z ´+ « >& h    õ

\  ¦ % 3 % 3  . ¢ ¸  É r   ½ ¨   õ \ " f  H, GaP x Sb 1−x `  ¦ GaAs \     & ñ ½ + Ër (  `  ¦ M : ç ß ] X Ì  s`  ¦ ° ú >  ÷ &# Q    & h 6

£

x6 x u  b  # Qt   InP\     & ñ ½ + Ër (  `  ¦ M :  H   Y

Us $  s  ¸× ¼(tunable laser diodes) 1 p x Ü ¼ Ð" f_  6 £ x6 x

0 p x$ í s   ¸d  Û ¼Y U ] jl ÷ &% 3   [7,8]. : £ ¤ y   â >   \ " f_ 

E-mail: [email protected]



½ × ¼ & ñ § > = © I      {   ¸á Ô! Ó (VBO: valence band offset) õ     ¸{   ¸á Ô! Ó (CBO: conduction band offset)`  ¦

&

ñ S X  y     H  כ  É r s  Ó ü t| 9 _  6 £ x6 x 0 p x$ í \  @ /ô  Ç   & ñ & h

&

ñ  Ð  ) a  [9–11]. Õ ª Q   f  s  Ó ü t| 9 \  @ /ô  Ç  ½ × ¼& ñ

§ >

=(band alignment)  © I \  @ /ô  Ç s  : r& h   õ   H & ñ S X  y 

· ú

 9    \ O  .



: r  7 Hë  H \ " f  H   " é ¶ o½ + ËÓ ü t ì ø Í ¸^  GaP x Sb 1−x _  $ í ì

 r q \    É r \  -t   ½ × ¼ ½ ¨ ¸   o\  ¦ # 3 6 x& h x 9 ] X   

½

+ Ë (UTB: universal tight binding) ~ ½ ÓZ O  [12, 13]`  ¦ s 6 x

#  ½ ¨ô  Ç . Å Òכ ¹ @ /g A& h  Γ, L, X\ " f_     ¸{ _  þ j è

° ú

כ(CBM: conduction band minimum)õ     { _  þ j

@

/° ú כ(VBM: valence band maximum)   o\  ¦ > í ß  #  f 

] X  ½ × ¼Ì  s(direct band gap)\ " f ç ß ] X  ½ × ¼Ì  s(indirect band gap) Ü ¼ Ð_    s \  ¦ ½ ©" î ô  Ç . ¢ ¸ô  Ç GaP x Sb 1−x \  ¦ InP \      & ñ ½ + Ër (  `  ¦ M :,  â >   \ " f_      { ü <



  ¸ { _   ½ × ¼& ñ § > =(band alignment)  © I \  ¦ ½ ¨ô  Ç .

II. T  Â ] Ø

# 3

6 x& h x 9 ] X   ½ + Ë ~ ½ ÓZ O  É r III-V7 á ¤ " é ¶ o½ + ËÓ ü t ì ø Í ¸^ _ 

InP ; c ß f Ä X N Ë¶ ¥c Ü R GaP x Sb 1−x 8 ý O © X N Ëc Ç

InP ; c ß f Ä X N Ë¶ ¥c Ü R GaP ^x Sb 1−x 8 ý O © X N Ëc Ç

+ ä ] 8 ; · »¦ P ^∗

â

l @ / < Æ § Ó ü t o < Æõ , Ã º" é ¶ 440-760 (2009¸ 3 Z 4 10{ 9 ~ Ã Î6 £ §)

:

r 7 Hë H \ " f H " é ¶ o½ + ËÓ ü t ì ø Í ¸^ GaP

_ D ¥ ½ + Ë0 l x ¸q \ É r \ -t ½ × ¼ ½ ¨ ¸ o\ ¦ # 3 6 x

h x 9 ] X ½ + Ë (UTB: universal tight binding) ~ ½ ÓZ O ` ¦ s 6 x # ½ ¨ % i . ^ 0 l x ¸\ É r(0 ≤ x ≤ 1) Å

Òכ ¹ @ /g A& h (Γ, L, X)_ ½ × ¼Ì s o H : r ¸ 300 K\ " f 0.73 ≤ E(Γ) ≤ 2.88 (eV), 1.17 ≤ E(L) ≤ 2.72 (eV), 1.72 ≤ E(X) ≤ 2.16 (eV) s % 3 . GaP

H x < 0.56 t H E(Γ) \ _ ô Ç f ] X ½ × ¼Ì ss

3 x > 0.56\ " f H E(X) \ _ ô Ç ç ß ] X ½ × ¼Ì se ` ¦ · ú Ã º e % 3 . GaP

ü < InP H D ¥ ½ + Ë0 l x ¸ x = 0.35 \ " f ½ + ËU ´s 2.541 ˚ A(< Ê É r, © Ã º 5.868 ˚ A) Ü ¼ Ð & ñ ½ + Ër ~ ´ Ã º e . & ñ ½ + Ë ) a GaP

ü < InP_ â > \ " f_ { ¸á Ô! Ó õ ¸{ ¸á Ô! Ó > í ß õ H 0.22 eV ü < 0.12 eV s % 3 .

, ½ × ¼& ñ § > =, { ¸á Ô! Ó , ¸{ ¸á Ô! Ó

III-V " é ¶( " é ¶ < Ê É r " é ¶) o½ + ËÓ ü t ì ø Í ¸^ H D ¥ ½ + Ë $ í ì r s

0 l x ¸\ 0 A& h Ü ¼ Ð & h : £ ¤$ í ` ¦ ¸] X ½ + É Ã º e

H @ /é ß y Ä »6 x ô Ç Ó ü t| 9 Ð" f, Y Us $ s ¸× ¼ F g Ã Ðt l

1 p x F g ¦î ß ¾ ¡ § _ l ó ø Í, x 4 ¤8 £ x, Ö ¸$ í 8 £ x1 p x Ü ¼ Ð æ ¼

t ¦ e . \ x Ã Ìr ° ú É r q ¨ î + þ A & ñ Ò q t © l Õ ü t µ 1 Ï² ú Ð í

H ¸ Z } É r III-V " é ¶ o½ + ËÓ ü t ì ø Í ¸^ \ ¦ Ò q t$ í r ~ ´ Ã º e > H

d \ , s \ @ /ô Ç s : r& h x 9 z ´+ « >& h ½ ¨ Ö ¸ µ 1 Ïy s

À Ò# Q4 R M ® o [1–4].

" é ¶ o½ + ËÓ ü t ì ø Í ¸^ GaP ^x Sb 1−x H 530 ^◦ C \ " f _

% i \ (0.01<x<0.99) 5 g D ¥ o$ í Ì s(miscibility gap)s + þ

A$ í ÷ &# Q [5] & ñ Ò q t © s _ Ô ¦ 0 p x ô Ç Ó ü t| 9 Ð · ú 94 R M

o Ü ¼ 9, s \ @ /ô Ç z ´+ « >& h ½ ¨ü < & h 6 £ x6 x s ] jô Ç÷ &

Q M ® o . Õ ª Q 1988¸ OMVPE(Organometallic vapor phase epitaxy) ~ ½ ÓZ O ` ¦ s 6 x # % 6 £ § Ü ¼ Ð GaP ^x Sb 1−x

& ñ Ò q t © \ $ í / B N # [6], GaAs l ó ø Í\ & ñ ½ + Ër GaP _0.68 Sb _0.32 _ \ -t Ì ss 1.43 eVs H z ´+ « >& h õ

\ ¦ % 3 % 3 . ¢ ¸ É r ½ ¨ õ \ " f H, GaP x Sb 1−x ` ¦ GaAs \ & ñ ½ + Ër ( ` ¦ M : ç ß ] X Ì s` ¦ ° ú > ÷ &# Q & h 6

x6 x u b # Qt InP\ & ñ ½ + Ër ( ` ¦ M : H Y

Us $ s ¸× ¼(tunable laser diodes) 1 p x Ü ¼ Ð" f_ 6 £ x6 x

0 p x$ í s ¸d Û ¼Y U ] jl ÷ &% 3 [7,8]. : £ ¤ y â > \ " f_

½ × ¼ & ñ § > = © I { ¸á Ô! Ó (VBO: valence band offset) õ ¸{ ¸á Ô! Ó (CBO: conduction band offset)` ¦

ñ S X y H כ É r s Ó ü t| 9 _ 6 £ x6 x 0 p x$ í \ @ /ô Ç & ñ & h

ñ Ð ) a [9–11]. Õ ª Q f s Ó ü t| 9 \ @ /ô Ç ½ × ¼& ñ

=(band alignment) © I \ @ /ô Ç s : r& h õ H & ñ S X y

9 \ O .

: r 7 Hë H \ " f H " é ¶ o½ + ËÓ ü t ì ø Í ¸^ GaP x Sb _1−x _ $ í ì

r q \ É r \ -t ½ × ¼ ½ ¨ ¸ o\ ¦ # 3 6 x& h x 9 ] X

+ Ë (UTB: universal tight binding) ~ ½ ÓZ O [12, 13]` ¦ s 6 x

# ½ ¨ô Ç . Å Òכ ¹ @ /g A& h Γ, L, X\ " f_ ¸{ _ þ j è

כ(CBM: conduction band minimum)õ { _ þ j

/° ú כ(VBM: valence band maximum) o\ ¦ > í ß # f

] X ½ × ¼Ì s(direct band gap)\ " f ç ß ] X ½ × ¼Ì s(indirect band gap) Ü ¼ Ð_ s \ ¦ ½ ©" î ô Ç . ¢ ¸ô Ç GaP x Sb 1−x \ ¦ InP \ & ñ ½ + Ër ( ` ¦ M :, â > \ " f_ { ü <

¸ { _ ½ × ¼& ñ § > =(band alignment) © I \ ¦ ½ ¨ô Ç .

II. T Â ] Ø

6 x& h x 9 ] X ½ + Ë ~ ½ ÓZ O É r III-V7 á ¤ " é ¶ o½ + ËÓ ü t ì ø Í ¸^ _

& h © I \ ¦ > í ß H X < e # Q B Ä º 4 ¤ ¸ ú Ù þ ¡~ Á º| 9 " f

òõ \ ¦ q §& h ç ß é ß ¦ & ñ S X > ½ ¨½ + É Ã º e H ´ òõ & h s

¦ ¼ # o ô Ç ~ ½ ÓZ O Ü ¼ Ð · ú 94 R e [12, 13]. UTB\ _ ô

Ç " é ¶ o½ + ËÓ ü t ì ø Í ¸^ GaP x Sb 1−x _ K x 9 Ðm î ß É r s " é ¶

o½ + ËÓ ü t ì ø Í ¸^ GaPü < GaSb_ Ä »´ òK x 9 Ðm î ß (effective Hamiltonian)` ¦ × æ¨ î ç H ô Ç ° ú כÜ ¼ Ð · p .

H ˆ _GaP

_Sb

= x ˆ H _GaP + (1 − x) ˆ H _GaSb (1)

l " f s " é ¶ o½ + ËÓ ü t ì ø Í ¸^ _ Ä »´ òK x 9 Ðm î ß É r | 9 " f K

x 9 Ðm î ß (ordered Hamiltonian)õ Á º| 9 " fK x 9 Ðm î

ß (disordered Hamiltonian)_ ½ + ËÜ ¼ Ð ³ ð & ³ ) a .

H ˆ GaP = ˆ H _GaP ⁰ + ∆ ˆ H GaP

H ˆ _GaSb = ˆ H _GaSb ⁰ + ∆ ˆ H _GaSb (2)

" fK x 9 Ðm î ß ˆ H _GaP ⁰ ( ˆ H _GaSb ⁰ ) H ¢ - a & ñ é # Qo © I GaP (GaSb) _ K x 9 Ðm î ß s 9, Á º| 9 " f K x 9 Ðm î ß

∆ ˆ H _GaP (∆ ˆ H _GaSb ) H GaP (GaSb) _ 6 £ § s : r u ¨ 8 \ _ ô

Ç q + þ A´ òõ \ ¦ Ä » ¸ H ª ∆ ˆ H GaP :Sb (∆ ˆ H GaSb:P ) Ü ¼

Ð" f u ¨ 8 $ í ì r q _ Y L Ü ¼ Ð · p .

∆ ˆ H _GaSb = x∆ ˆ H _GaSb:P (3) (2)−(3)d ` ¦ (1)d \ @ /{ 9 , " é ¶ o½ + ËÓ ü t GaP x Sb _1−x _

D ¥ ½ + Ë 0 l x ¸\ É r 8 ú x K x 9 Ðm î ß ` ¦ ½ ¨½ + É Ã º e .

= xH _GaP ⁰ + (1 − x)H _GaSb ⁰

+x(1 − x)(∆ ˆ H _{GaP :Sb} + ∆ ˆ H _GaSb:P )(4)

l " f 8 ú x Á º| 9 " fK x 9 Ðm î ß ∆ ˆ H _{GaP :Sb} + ∆ ˆ H _GaSb:P H

º s " é ¶ o½ + ËÓ ü t _ | 9 " fK x 9 Ðm î ß _ ˆ H _GaP ⁰ − ˆ H _GaSb ⁰ Ð é

H .

ø Í ¸^ _ \ -t © I H & ñ ? /_ Schrodinger ~ ½ Ó& ñ d _

K \ ¦ ½ ¨K Í Ç rÜ ¼ Ð+ · ú Ã º e .

& ñ _ ¦Ä » © I < ÊÃ º Ψ ⁿ (~ k, ~ r) H Bloch & ñ o \ ¦ ë ß 7 á ¤ r v

H " é ¶ C ¸ < ÊÃ º[ þ t φ ^ν _αj (~ r) _ + þ A ½ + ËÜ ¼ Ð · p .

√ N ⁰ X

A ⁿ _αν (~ k)e ^i~ ^{k·( ~} ^R

^+~ ^τ

⁾ φ ^ν _αj (~ r − ~ τ ν ) (6)