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A Study of the Optical Dispersion Characteristic of Amorphous Semiconductors

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x —¸+þA`¦  {ߖ s < s©œ_ F g \-t\ @Ç ƒ5Åq&h f¨Ãº >º\¦ >ߖ ¦ Kramer- Kronig ìߖ ›'a>d”`¦ 6 x #Œ ÏãJ]X¦•¸ >ߖ %i. #ŒQ ë³\ ˜Ð“¦)a a-Si, a-TiO2, a-Si3N4ü<

a-GaAs B|9[þt\ @/ #Œ F <Æ ìߖ ›'a>d”`¦ &h6 x #Œ z´+«> < Bº ¸ú˜ {9u H \¦ %3%3



.

PACS numbers: 42.25.B, 78.20.C, 78.66.J

Keywords: q&ñ|9 ìøÍ•¸^‰, F <Æ&h ìߖ, Kramer-Kronig ìߖ ›'a>d”, a-Si3N4, a-GaAs, a-Si, a-TiO2

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 [1]. F  ™èÐ_ 6£x6 x`¦ 0AK q&ñ|9 ìøÍ•¸^‰ {¨

›

¸_ sK €9¹ . çߖ]X …;s+þA ìøÍ•¸^‰ Óüt|9\ q K

 ´òÖ¦s ZÉr f”]X …;s+þA ìøÍ•¸^‰ Óüt|9_ {¨›¸H F g

„



l <ÆZO (photoelectrochemical method) [2, 3], “¦“:r

\

"f_ ”$í „¸ 8£¤&ñZO (intrinsic conduction measure- ment method), F g ~½ÓئZO (photoemission method) [4, 5], üÒ [O1lx`¦  #Œ F <Æ&h Û¼&7˜àÔ!3_ o\¦ 8£¤

&

ñ

H ¸ ìrF gZO (modulation spectroscopy) [6]õ 

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¸F g ìøÍZO (photoreflectance method) [7] 1p¼–Ð 8£¤&ñ

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É Ãº e”¼, ÅҖРçߖéߖ "¸ &ñSX‰ > ìøÍ•¸^‰ {¨

›

¸\¦ Œ•½+É Ãº e”H F <Æ&h f¨Ãº ~½ÓZO (optical absorp- tion method)`¦ 6 Ç. q&ñ|9 ìøÍ•¸^‰ Óüt|9_ F g \



-t\ @Ç F <Æ&h f¨ÃºH ß¼> ¿º t %ò%i¼–Ð üt Ã

º e”. 'Í PH F g \-t E {ߖ \-t Eg s

©œ“ ZÉr f¨Ãº %ò%i (α ≥ 103cm−1¼–Ð, Tauc [8] x9 Mottü< Davis [9] yŒ•yŒ• 6£§õ °ú “Ér F g \-t\ @/ ô



Ç f¨Ãº >º ›'a>d”`¦ Ä»•¸ %i.

α (E) = A(E − Eg)n

E (1)

#

Œl"f AH "é¶ ˜Ð&ñ qYV ©œÃºs. n“Ér {_ ©œI x

9

¸ †<Êú +þAI\ _>r H ©œÃº–Ð ŸíÓü +þAI\¦ t

E-mail: [email protected]





H f”]X …;s âĺ n = 2 [8] °úכ`¦ t¦ çߖ]X …;s

 â

ĺ n = 3 [9] °úכ`¦ ”. ¿ºPH F g \-t { ç

ß

– \-t Eg s  ±ú“Ér f¨Ãº %ò%i¼–Ð, Urbach [10, 11]  µ1Ï|Ç A< °ú “Ér â+«> d”¼–Ð ³ð‰&³)a.

α (E) = α0exp [σ0(E − E0)/kT ] (2)

#

Œl"f α0, σ0, E0H yŒ•yŒ• q&ñ|9 ìøÍ•¸^‰ Óüt|9_ :£¤

$ í

õ ›')a ©œÃº[þts. Forouhiü< Bloomer\ Ԁ [12], „¸@Ð …;sÇ „ Ä»ôÇôÇ Ãº"î`¦ |9 M: „



_ …;s SX‰Ò¦ q¦“Ér 8s©œ ©œÃº–Ð ÅÒ#Qtt ·ú§Ü¼ 9, ‚;Ÿ¤ †<Êúü< d” (1)_ Y¼–Ð ÅÒ#. Alterovitz 1px [13]õ McGahanõ Woollam [14]“Ér Forouhiü< Bloomer

Ä

»•¸ôÇ d”¼–Ð #ŒQ >h_ ú™è q&ñ|9 òøÍ™è ~ÃÌ}Œ•_ {ߖ





 \-t 6£§_ °úכ`¦ tH âĺ\¦ µ1Ï| %i. Êê\ Forouhiü< Bloomerd”`¦ +þA #Œ McGahan 1px [15]s { _

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º µ1Ú\ \O%3. q5pÇ rl\ Yamaguchi 1px [16]“Ér {_



©

œI x9¸ Š©œ/B †<Êúü< tº †<Êú_ Y¼–Ð ÅÒ#



¦ &ñ ¦ q&ñ|9 |9o ½©™è (a-Si3N4)\¦ K$3þ¡tߖ Ó

ü

to&h Bjm7£§\ @Ç ƒ/åLs \O#Q Ér q&ñ|9 ìøÍ•¸

^

‰\ @Ç &h6 xs #Q9°?. sH 1lq&h¼–Ð Adachi [17] Taucd”`¦ SX‰©œ #Œ q&ñ|9 ìøÍ•¸^‰ Óüt|9_ F g \



-t\ @Ç F <Æ&h f¨Ãº\¦ K$3þ¡tߖ :£¤&ñôÇ \-t

% ò

%i@/, 7£¤ e”> \-t< {ߖ \-t\"ߖ ²DÇ÷&

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Q K$3K H éߖ&hs e”. sê Jellisonõ Modine [18]“Ér F g \-t\ @Ç f¨Ãº >º †<Êú\¦ Tauc d” Lorentz ”1lx —¸+þA_ Y¼–Ð ³ð‰&³Ùþ¡. tߖ Jellisonõ -268-

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Fig. 1. The experimental data of refractive index and extinction coefficient against photon energy for a-Si ( [22]) with theoretic curves.

Modines Ä»•¸ôÇ —¸+þÉr tu> éߖíH)a —¸+þ¼–Ð

“ :

¸ x9 ·úš§4 °ú “Ér üÒ o_ _>r$í`¦ \V8£¤½+É Ãº

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% ò

ܼ–Ð &ñ %i. s\¦ ˜Ð¢-a l 0AK Ferlauto 1px [19, 20]“Ér F g \-t e”> \-t pߖ %ò%i\"H f¨Ãº

>

º Urbach ZOgË:`¦ Ô 9, Õª s©œ %ò%i\"H f¨Ãº

>

º +þ)a Tauc d” Lorentz ”1lx —¸+þA_ Y¼–Ð



Ér¦ &ñÙþ¡. ÕªQ F g \-t e”> \-t ü

< °ú `¦ âĺ ƒ5Åq$í`¦ ˜Ð©œ l #>H éߖ&hs e”



.

‘ :

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º&h¼–Ð yŒ™™è H Dunstan [21] —¸+þA\ „¸@/\"f_ Ä

»ôÇôÇ „ ú"î`¦ “¦Ç Forouhiü< Bloomer d”`¦ &h 6

 

x #Œ, F g \-t e”> \-t s©œ x9 s  ¿º %ò

% i

¼–Ð yŒ•yŒ• º#Q F g \-t\ @Ç f¨Ãº >º_ d”

`



¦ K$3&h¼–Ð Ä»•¸ %i. ¢¸ôÇ ïr &hrd” (Cauchy integral formula) õ Ļú s:r (residue theorem) `¦ &h 6

 

x #Œ Kramer-Kronig ìߖ ›'a>d”`¦ :ŸxK ÏãJ]X¦\ @/ ô



Ç K$3&h d”`¦ •¸Ø¦ %i¦ #ŒQ q&ñ|9 ìøÍ•¸^‰\¦  ê



r 7H[þt\"f ƒ/å)a z´+«>&h \¦ ‘:r 7H\"f Ä»•¸ ô



Ç f¨ÃºÖ¦õ ÏãJ]X¦\ @Ç d”¼–Ð r3x?/l %i.

II. T Â]Ø

Dunstan [21]s ]ߖôÇ f¨Ãº >º\ @Ç F g \-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 tº&h¼–Ð yŒ™™è 





H \-t SX‰Ò¦ ìí\¦, α (E) = f (E − EgH „¸@/ ü

< „¸@/ s %ò%i\"f f¨Ãº >º–Ð n = 2“ f”]X …; s

 ìøÍ•¸^‰“ âĺ Tauc ]ߖôÇ d” (1)õ °ú “Ér ŸíÓü

†

<

Êú +þAIÐ ÅÒ#Qt¦, E0H e”> \-ts. „¸@/\

"

f_ Ä»ôÇôÇ „ ú"î`¦ “¦Ç Forouhiü< Bloomer [12]

d

”

_ ‚;Ÿ¤ †<Êú\¦ d” (3)\ &h6 x ¦, f¨Ãº >º αü< f¨ F

g >º κ çߖ_ κ = αc/2ω ›'a>\¦ s6 x #Œ >ߖ  f¨ F

g >º κH

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”. #Œl"f B = 2Epg, C = E2pg + Γ2, D = Ac¯hΓ2±

2π, Γ = ¯hγ/2, Epg = Eσ∗ − EσH Penn- gap \-t, γ−1H „_ ú"î, Eσ∗H „¸@/_ ìøÍ

½ +

Ë \-t, EσH „@/_ +Ë \-ts. F g \



-t Eü< e”> \-t E0 °ú `¦ âĺ f¨F g >º κH κ (E0) = 2D±

β2E02¡

E02− BE0+ C¢

–

Ð ÅÒ#Qt 9 f¨F g

>

º κ(E) F g \-t\ @/K"f ƒ5Åq&h †<Êú e”

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¦ ·ú˜ ú e”. Kramer-Kronig ìߖ ›'a>d”`¦ d” (4)\

&

h

6 x  ÏãJ]X¦“Ér

n (E)−n (∞) = 2 πP.V

Z

0

E0

E02− E2[κ (E0) − κ (∞)]dE0 (5)

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ü

< °ú s ÅÒ#. #Œl"f P.V H ÅÒu (principle vol- ume)\¦  ·p. F g \-t E e”> \-t E0 s



©

œ x9 s Ð &hr ½¨çߖ`¦ º“¦ F g \-t E e”>

\

-t E0˜Ð Œ•“Ér %ò%i\"f f¨Ãº >º_ l#Œ &h

“

¦ &ñ  (5)d”Ér

n (E) = n (∞) +2 πP.V

Z

E0

E0κ (E0)

(E02− E2)dE0 (6)

–

Ð r jþt ú e”. d” (6)_ &hr ½¨çߖ`¦ 8ŠôÇ Êê



ïr &hrd” Ļú s:r`¦ &h6 x  ÏãJ]X¦ n“Ér

n (E) = n (∞) +ED2

£(E−E0)2+2β(E−E0)+2

β2

¤

E2−BE+C

+

D

½¡

3B24 −C¢£

(E01β)2+β21¤

(B2+2C)(B2−8C)

16 B24 (2C−E2)−2BC(E0β1)

¾

(C−E2)2+B2¡

C−B24 ¢

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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#.

III. ÷mÇ]MöÊÝ8ý Rw‹ õmÍ ÄZØV Ä

f



¨F g >º\ @Ç d” (4)ü< ÏãJ]X¦\ @Ç d” (7)`¦ s 6

 

x #Œ r3x?/l ©œÃº[þt B, C, D, E0, Eg, β, n (∞)\¦ 

&

ñ

l 0AK Mathworks_ #36 x ™èáÔàÔJ?# Matlab á

ԖÐÕªÏþ›_ Curve Fitting Tool`¦ 6 x %i. q&ñ|9 ìøÍ

•

¸^‰ B|9_ {ߖ \-t EgH Dunstans ƒ/åÇ E0 1/β`¦ 6 x #Œ ½¨ %i.

Fig. 1“Ér pè „ ©œu, IªœF g „t, F gl2Ÿ¤ B^‰

\

 6 x H a-Si_ F g \-t\ @Ç f¨F g >ºü< ÏãJ ]

X

¦`¦  ·p. z´+«> X<s'H Philipp [22]s z´oBH





&ñ`¦ „c” 7£ÃÌZO¼–Ð ]Œ•ôÇ q&ñ|9 z´oBH ~ÃÌ}Œ•

`



¦ ìøÍ Û¼&7˜àÔ!3 \¦ s6 x #Œ %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 ìߖ ›'a>d”`¦ 6 x #Œ %3Ér s:r





s.

Fig. 2H Aspnes 1px [23]s $:r &ñ z´oBH`¦ ëߖ[þtl 0

AK $úš<Æl©œ7£ÃÌ (LPCVD : low pressure chemical vapor deposition) ZO¼–Ð ]Œ•ôÇ a-Si ìøÍ•¸^‰ B|9Ð, 

"

é

¶ ¼#F gZO`¦ s6 x #Œ 1.5 eV\"f 6.0 eVt F g \



-t %ò%i\"f f¨F g >ºü< ÏãJ]X¦`¦ 8£¤&ñôÇ < d” (4)ü< d” (7) _ s:_ q§\¦ ˜Ð#ŒÅғ¦ e”.

Fig. 3“Ér Piller [24] ]Œ•ôÇ a-Si ìøÍ•¸^‰ Óüt|9_ f¨F g

>

ºü< ÏãJ]X¦`¦  ·p ÕªaË>s.

Fig. 4H Joseph 1px [25]s ]Œ•ôÇ ¿ºa "Ð Ér q

&

ñ

|9 TiO2 ìøÍ•¸^‰ B|9_ f¨F g x9 ÏãJ]X¦`¦ "é¶ ¼#F g Z

O

¼–Ð 8£¤&ñ #Œ  ·p ÕªaË>s.

Fig. H Philipp [26]s SiH4ü< NH3 +ËÓüt`¦ €• 1000•¸\"f 0lq#Œ"f &ñ|9 z´oBH 0A\ 7£ÃÌ #Œ ]Œ•ôÇ

수치

Fig. 2. The experimental data of refractive index and extinction coefficient against photon energy for a-Si ( [23]) with theoretic curves.
Fig. 3. The experimental data of refractive index and extinction coefficient against photon energy for a-Si ( [24]) with theoretic curves.
Table 1. Curve fitting parameters for various amorphous semiconductor materials.

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