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z@/<Ƨ Óüto<Æõ, âíß 712-749 (2007¸ 10Z4 16{9 ~ÃÎ6£§)
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Q q&ñ|9 ìøÍ¸^ Óüt|9[þt\ &h6 x½+É Ãº eH K$3&h Fg<Æ ìríß 'a>d`¦ Ä»¸ %i. {çß כ¹ 1
l
x ¸+þA`¦ {çß s ü< s©_ Fg \-t\ @/ôÇ 5Åq&h f¨Ãº >ú\¦ >íß ¦ Kramer- Kronig ìríß 'a>d`¦ 6 x # ÏãJ]XÒ¦¸ >íß %i. #Q ëH³\ Ц)a a-Si, a-TiO2, a-Si3N4ü<
a-GaAs B|9[þt\ @/ # Fg<Æ ìríß 'a>d`¦ &h6 x # z´+«> õü< Bĺ ¸ú {9u H õ\¦ %3%3
.
PACS numbers: 42.25.B, 78.20.C, 78.66.J
Keywords: q&ñ|9 ìøÍ¸^, Fg<Æ&h ìríß, Kramer-Kronig ìríß 'a>d, a-Si3N4, a-GaAs, a-Si, a-TiO2
I. "e Â]Ø
q
&ñ|9 ìøÍ¸^H <ɪpeH Óüto&h :£¤$í M:ëH\ 6£x6 x
0px$ís ´ú§Ér Óüt|9Ð @/¿º÷&¦ eܼ 9, þjH\H #Q Õ
ªÒ¨\"f Fg è\ @/ôÇ ½¨ Ö¸µ1Ïy '÷&¦ e
[1]. Fg èÐ_ 6£x6 x`¦ 0AK q&ñ|9 ìøÍ¸^ {½¨
¸_ sK 9כ¹ . çß]X ;s+þA ìøÍ¸^ Óüt|9\ q K
´òÖ¦s Z}Ér f]X ;s+þA ìøÍ¸^ Óüt|9_ {½¨¸H Fg
l o<ÆZO (photoelectrochemical method) [2, 3], ¦:r
\
"f_ $í ¸ 8£¤&ñZO (intrinsic conduction measure- ment method), Fg ~½ÓئZO (photoemission method) [4, 5], ü@ÂÒ [O1lx`¦ # Fg<Æ&h Û¼&7àÔ!3_ o\¦ 8£¤
&
ñ
H ¸ ìrFgZO (modulation spectroscopy) [6]õ
¸Fg ìøÍZO (photoreflectance method) [7] 1pxܼР8£¤&ñ
½ +
É Ãº eܼ, ÅÒÐ çßéß "f¸ &ñSX > ìøÍ¸^ {½¨
¸\¦ ½+É Ãº eH Fg<Æ&h f¨Ãº ~½ÓZO (optical absorp- tion method)`¦ 6 xôÇ. q&ñ|9 ìøÍ¸^ Óüt|9_ Fg \
-t\ @/ôÇ Fg<Æ&h f¨ÃºH ß¼> ¿º t %ò%iܼРÐüt Ã
º e. 'Í P:H Fg \-t E {çß \-t Eg s
© Z}Ér f¨Ãº %ò%i (α ≥ 103cm−1)ܼÐ, Tauc [8] x9 Mottü< Davis [9] yy 6£§õ °ú Ér Fg \-t\ @/ ô
Ç f¨Ãº >ú 'a>d`¦ Ä»¸ %i.
α (E) = A(E − Eg)n
E (1)
#
l"f AH "é¶ Ð&ñ qYV ©Ãºs. nÉr {_ ©I x
9
¸ <Êú +þAI\ _>r H ©ÃºÐ íÓüt +þAI\¦ t
∗E-mail: [email protected]
H f]X ;s âĺ n = 2 [8] °úכ`¦ t¦ çß]X ;s
â
ĺ n = 3 [9] °úכ`¦ . ¿ºP:H Fg \-t { ç
ß
\-t Eg s ±úÉr f¨Ãº %ò%iܼÐ, Urbach [10, 11] µ1Ï|ôÇ Aü< °ú Ér â+«> dܼР³ð&³)a.
α (E) = α0exp [σ0(E − E0)/kT ] (2)
#
l"f α0, σ0, E0H yy q&ñ|9 ìøÍ¸^ Óüt|9_ :£¤
$ í
õ 'aº)a ©Ãº[þts. Forouhiü< Bloomer\ ØÔ [12], ¸@/Ð ;sôÇ Ä»ôÇôÇ Ãº"î`¦ |9 M:
_ ;s SXÒ¦ qÖ¦Ér 8s© ©ÃºÐ ÅÒ#Qtt ·ú§Ü¼ 9, ;¤ <Êúü< d (1)_ YLܼРÅÒ#Q. Alterovitz 1px [13]õ McGahanõ Woollam [14]Ér Forouhiü< Bloomer
Ä
»¸ôÇ dܼР#Q >h_ úè q&ñ|9 òøÍè ~ÃÌ}_ {çß
\-t 6£§_ °úכ`¦ tH âĺ\¦ µ1Ï| %i. Êê\ Forouhiü< Bloomerd`¦ +þA # McGahan 1px [15]s { _
©I x9¸ qíÓüt +þAI\¦ ¦ ¦9Ùþ¡tëß Taucd_ ôÇ> Z}Ér f¨Ãº %ò%i\ëß ²DGôÇ # K$3½+É Ã
º µ1Ú\ \O%3. q5pwôÇ rl\ Yamaguchi 1px [16]Ér {_
©
I x9¸ ©/BG <Êúü< tú <Êú_ YLܼРÅÒ#Q
¦ &ñ ¦ q&ñ|9 |9o ½©è (a-Si3N4)\¦ K$3Ùþ¡tëß Ó
ü
to&h Bjm7£§\ @/ôÇ /åLs \O#Q Ér q&ñ|9 ìøÍ¸
^
\ @/ôÇ &h6 xs #Q9°?. sü<H 1lq&hܼРAdachi [17] Taucd`¦ SX© # q&ñ|9 ìøÍ¸^ Óüt|9_ Fg \
-t\ @/ôÇ Fg<Æ&h f¨Ãº\¦ K$3Ùþ¡tëß :£¤&ñôÇ \-t
% ò
%i@/, 7£¤ e> \-tü< {çß \-t\"fëß ²DGôÇ÷&
#
Q K$3K H éß&hs e. sÊê Jellisonõ Modine [18]Ér Fg \-t\ @/ôÇ f¨Ãº >ú <Êú\¦ Tauc dõ Lorentz 1lx ¸+þA_ YLܼР³ð&³Ùþ¡. tëß Jellisonõ -268-
Fig. 1. The experimental data of refractive index and extinction coefficient against photon energy for a-Si ( [22]) with theoretic curves.
Modines Ä»¸ôÇ ¸+þAÉr tu> éßíHo)a ¸+þAܼÐ
:
r¸ x9 ·ú§4õ °ú Ér ü@ÂÒ o_ _>r$í`¦ \V8£¤½+É Ãº
\ O
`¦÷r ëß m {çß \-t s \"f f¨Ãº >úH
% ò
ܼР&ñ %i. s\¦ Т-a l 0AK Ferlauto 1px [19, 20]Ér Fg \-t e> \-t pëß %ò%i\"fH f¨Ãº
>
ú Urbach ZOgË:`¦ ØÔ 9, Õª s© %ò%i\"fH f¨Ãº
>
ú +þA)a Tauc dõ Lorentz 1lx ¸+þA_ YLܼÐ
Ér¦ &ñÙþ¡. ÕªQ Fg \-t e> \-t ü
< °ú `¦ âĺ 5Åq$í`¦ Щ l #Q§>H éß&hs e
.
:
r 7HëH\"fH @/\"f ¸@/Ð_ f¨ÃºH Taucd`¦ ØÔ 9 { â>\"f \-t SXÒ¦ ìrí t Ã
º&hܼРyè H Dunstan [21] ¸+þA\ ¸@/\"f_ Ä
»ôÇôÇ Ãº"î`¦ ¦9ôÇ Forouhiü< Bloomer d`¦ &h 6
x #, Fg \-t e> \-t s© x9 s ¿º %ò
% i
ܼРyy ¾º#Q Fg \-t\ @/ôÇ f¨Ãº >ú_ d
`
¦ K$3&hܼРϏ %i. ¢¸ôÇ ïr &hìrd (Cauchy integral formula) õ Ļú s:r (residue theorem) `¦ &h 6
x # Kramer-Kronig ìríß 'a>d`¦ :xK ÏãJ]XÒ¦\ @/ ô
Ç K$3&h d`¦ ¸Ø¦ %i¦ #Q q&ñ|9 ìøÍ¸^\¦ ê
r 7HëH[þt\"f /åL)a z´+«>&h õ\¦ :r 7HëH\"f Ä»¸ ô
Ç f¨ÃºÖ¦õ ÏãJ]XÒ¦\ @/ôÇ dܼРrÐ3x?/l %i.
II. T Â]Ø
Dunstan [21]s ]jîßôÇ f¨Ãº >ú\ @/ôÇ Fg \-t
H 6£§õ °ú s ÅÒ#Qt 9,
Fig. 2. The experimental data of refractive index and extinction coefficient against photon energy for a-Si ( [23]) with theoretic curves.
α (E) = Z ∞
0
P (ε) f (E − E0+ ε) dε (3) P (ε) = β exp [−βε]H { â>\"f tú&hܼРyè
H \-t SXÒ¦ ìrí\¦, α (E) = f (E − Eg)H ¸@/ ü
< ¸@/ s %ò%i\"f f¨Ãº >úРn = 2 f]X ; s
ìøÍ¸^ âĺ Tauc ]jîßôÇ d (1)õ °ú Ér íÓüt
<
Êú +þAIÐ ÅÒ#Qt¦, E0H e> \-ts. ¸@/\
"
f_ Ä»ôÇôÇ Ãº"î`¦ ¦9ôÇ Forouhiü< Bloomer [12]
d
_ ;¤ <Êú\¦ d (3)\ &h6 x ¦, f¨Ãº >ú αü< f¨ F
g >ú κ çß_ κ = αc/2ω 'a>\¦ s6 x # >íß f¨ F
g >ú κH
κ (E) =
D E2
©[(E0−E)+β1]2+β21ª
E2−BE+C , E ≥ E0 D
E2
exp[β(E−E0)]©
[2(E0−E)+β1]2+β21 ª
E2−BE+C , E0≥ E (4)
Ð jþt ú e. #l"f B = 2Epg, C = E2pg + Γ2, D = Ac¯hΓ2±
2π, Γ = ¯hγ/2, Epg = Eσ∗ − EσH Penn- gap \-t, γ−1H _ ú"î, Eσ∗H ¸@/_ ìøÍ
½ +
Ë \-t, EσH @/_ ½+Ë \-ts. Fg \
-t Eü< e> \-t E0 °ú `¦ âĺ f¨Fg >ú κH κ (E0) = 2D±
β2E02¡
E02− BE0+ C¢
Ð ÅÒ#Qt 9 f¨Fg
>
ú κ(E) Fg \-t\ @/K"f 5Åq&h <Êú e
`
¦ ·ú ú e. Kramer-Kronig ìríß 'a>d`¦ d (4)\
&
h
6 x ÏãJ]XÒ¦Ér
n (E)−n (∞) = 2 πP.V
Z ∞
0
E0
E02− E2[κ (E0) − κ (∞)]dE0 (5)
ü
< °ú s ÅÒ#Q. #l"f P.V H ÅÒu (principle vol- ume)\¦ ·p. Fg \-t E e> \-t E0 s
©
x9 s Ð &hìr ½¨çß`¦ ¾º¦ Fg \-t E e>
\
-t E0Ð Ér %ò%i\"f f¨Ãº >ú_ l# &h
¦ &ñ (5)dÉr
n (E) = n (∞) +2 πP.V
Z ∞
E0
E0κ (E0)
(E02− E2)dE0 (6)
Ð r jþt ú e. d (6)_ &hìr ½¨çß`¦ ¨8ôÇ Êê
ïr &hìrdõ Ļú s:r`¦ &h6 x ÏãJ]XÒ¦ nÉr
n (E) = n (∞) +ED2
£(E−E0)2+2β(E−E0)+2
β2
¤
E2−BE+C
+
D
½¡
3B24 −C¢£
(E0−1β)2+β21¤
−(B2+2C)(B2−8C)
16 −B24 (2C−E2)−2BC(E0−β1)
¾
(C−E2)2+B2¡
C−B24 ¢
(7)
Fig. 3. The experimental data of refractive index and extinction coefficient against photon energy for a-Si ( [24]) with theoretic curves.
ü
< °ú s ÅÒ#Q.
III. ÷mÇ]MöÊÝ8ý Rw õmÍ ÄZØV Ä
f
¨Fg >ú\ @/ôÇ d (4)ü< ÏãJ]XÒ¦\ @/ôÇ d (7)`¦ s 6
x # rÐ3x?/l ©Ãº[þt B, C, D, E0, Eg, β, n (∞)\¦
&
ñ
l 0AK Mathworks_ #36 x èáÔàÔJ?#Q Matlab á
ÔÐÕªÏþ_ Curve Fitting Tool`¦ 6 x %i. q&ñ|9 ìøÍ
¸^ B|9_ {çß \-t EgH Dunstans /åLôÇ E0− 1/β`¦ 6 x # ½¨ %i.
Fig. 1Ér pè ©u, IªFg t, Fgl2¤ B^
\
6 x H a-Si_ Fg \-t\ @/ôÇ f¨Fg >úü< ÏãJ ]
X
Ò¦`¦ ·p. z´+«> X<s'H Philipp [22]s z´oBH
&ñ`¦ c 7£xÃÌZOܼР]jôÇ q&ñ|9 z´oBH ~ÃÌ}
`
¦ ìøÍ Û¼&7àÔ!3 õ\¦ s6 x # %3Ér X<s's 9, z´
Fig. 4. The experimental data of refractive index and extinction coefficient against photon energy for a − T iO2
( [25]) with theoretic curves.
Ér Kramer-Kronig ìríß 'a>d`¦ 6 x # %3Ér s:r
õs.
Fig. 2H Aspnes 1px [23]s $:r &ñ z´oBH`¦ ëß[þtl 0
AK $·úo<Æl©7£xÃÌ (LPCVD : low pressure chemical vapor deposition) ZOܼР]jôÇ a-Si ìøÍ¸^ B|9Ð,
"
é
¶ ¼#FgZO`¦ s6 x # 1.5 eV\"f 6.0 eVt Fg \
-t %ò%i\"f f¨Fg >úü< ÏãJ]XÒ¦`¦ 8£¤&ñôÇ õü< d (4)ü< d (7) _ s:rõ_ q§\¦ Ð#ÅÒ¦ e.
Fig. 3Ér Piller [24] ]jôÇ a-Si ìøÍ¸^ Óüt|9_ f¨Fg
>
úü< ÏãJ]XÒ¦`¦ ·p ÕªaË>s.
Fig. 4H Joseph 1px [25]s ]jôÇ ¿ºa "fÐ Ér q
&
ñ
|9 TiO2 ìøÍ¸^ B|9_ f¨Fg x9 ÏãJ]XÒ¦`¦ "é¶ ¼#Fg Z
O
ܼР8£¤&ñ # ·p ÕªaË>s.
Fig. 5H Philipp [26]s SiH4ü< NH3 o½+ËÓüt`¦ 1000¸\"f 0lq#"f &ñ|9 z´oBH 0A\ 7£xÃÌ # ]jôÇ