4/6 S. Hara@ [.ppt]
On Road 4/9 4/6 4/ 5/7 4 Scott 5/4 5 Off Road 5/ 5/8 6 7 S. Hara@ [.ppt]
... 4. [] S. Hara@ [.ppt]
: 4 ε s ε a ε b Anti-bonding orbital bonding orbital Anion Cation ε c ε a ε a ε b Anion: ανοδοζ Cation: S. Hara@ [.ppt]
5 Bonds and Bands εp εs 4 } 4 } Conduction band Anti-bonding orbital Empt states Valence band bonding orbital filled states or. sp IV. ε p -ε s E G ε p -ε s S. Hara@ [Bond-Band.ai]
6 Cation Anion ε c ε a { { Gap ε c p ε c s ε a p ε a s Anion: ανοδοζ : Cation: S. Hara@ [IonicBand-Bond.ai]
7 (doping S. Hara@ [.ls]
8 intrinsic i n p E G E F E F E C E V E F E F E F E G E G E F n i ~ 0 ~ 0 ~ 0 5 6 / cm / cm / cm for GaAs ( E for Si ( E G for Ge ( E G G.44eV.eV 0.66eV E C E V (Conduction band (Valence band S. Hara@ [.ppt]
4 8 9 ( ( M. Born(96: * t r * dτ dτ Scrödinger ( ( ( cos ( ( * π / λ ( cos ( cos π λ or (4 ( r r e i (5 ( r ( r i e i e ( r ω t ω ( r t v ω λω p pase π m m ω πν Euler ( cos isin e i ( e cos isin i S. Hara@ [.ppt]
4 (plane wave ( r ep( i r r (,, r (,, ep( i r cos ( r isin( r * S. Hara@ ( n, n, n π n,n,n,,,... [.ai]
9 ( ( V core ( core electron V ( V ( V ( ( S. Hara@ [.ppt]
5 9 0 de Broglie94 p λ ( (λ p m( E V m( E V / [ { } ] i m( E V / ( epi ep E E > V [ { } ] i m( E V / ( ep E < V ( ep V ( 0 (/e 0 [ { } ] i m( V E / [ { } ] ep m( V E / V ( S. Hara@ [.ai]
0 Bloc V (0 V ( 0 V ( 0 < < 0 Scrödinger S. Hara@ [.ppt]
0. ( 0 ( 0 0, ( 0 ( E Asin m ( n πn n n,,,... n A Q d 0. Scrödinger eq. ( m E ( n 0 n :. Pauli ( Wavefunctions Energ levels π/ π/ Energ E 5 4 6 0 π/ wave vector S. Hara@ [.ai]
. ( r 0 ( 0,, 0, (,, 0. Scrödinger eq. m ( r E ( r. Pauli (r ( r X ( Y ( Z( X '' Y '' Z'' E m X m Y m Z X '' Y '' Z'' E, E, E m X m Y m Z E E E E d X m d X ( E 8 ( r E Asin m X sin π n π n, ( sin π n sin πn n,,,... A Q X d 0, n,n,n,,,... ( E m S. Hara@ [.ppt]
(periodic boundar condition (,, (,, Y (, Z( Scrödinger d X ( d X ( X ( π m E n n n 0,,,,... E m m n X X ( ep( ± i ( Q λn 0 X ( d n 0 X ( n,,,... X ( ep( ± i ep( iω t X (, t epi( ± i ω t ω / ep( ± a cosa ± isin a cos( m sin( m ω ω t t S. Hara@ X ( sin, X ( cos π λ ω cos ( m t [.ai]
5 (Born-von Karman boundar condition (,, (,, (,, (,, (,, (,, Scrödinger m ( r E ( r E m ( r ep( i r V V π π π n, n, n n, n, n, 0,,,,... π / S. Hara@ ep( i ep( i ep( i [.ai]
6 0 6 Energ E 4 5 wave vector Energ E wave vector π n π n,, π n π π π n, n, n n,n,n,,,... n, n, n, 0,,,,... S. Hara@ [.ai]
7 (,, (,, (,, (,, (,, (,, S. Hara@ [Bloc.ai]
8 Bloc ( r ( r ep( i r u u ( r u ( r T V( (r u Bragg T ep( i r u (r T (r ( r ep( i r u Bloc S. Hara@ [Bloc.ai]
9 Kronig-Penne( R. de. Kronig and W. G. Penne, Proc. Ro. Soc. A0, 499 (9. Bloc u (r K Q 0 0 ( d d d d A B C D, ik(a - B Q (C - D ( ( 0<<a: -b Aep( ik ( -b<<0: C ep( Q 0 a ab Bep( ik Dep( Q U 0 0 0 a Ae ik ika ika Qb Qb i ( b Be ( Ce De e a ika ika Qb Qb i ( b ( Ae Be Q( Ce De e a [( Q K / QK] sinqbsin Ka cosqbcoska cos ( a b ( Bloc ab ( a < < a b ( b < < 0 ep( i( a b K, Q Bloc S. Hara@ i i sin cos ep( Euler's cos ( e e / sin ( e e / K nπ / a V ( Δ δ ( n( a b m n [KronigPenne.ai; KroningPenne.sg]
0 Kronig-Penne( Dirac δ-function potential b [( Q K / QK] sinqbsin Ka cosqbcoska cos( a b U 0 Q b~0 Q - K ~ Q, sin Qb ~ Qb, cosqb ~ 0 U 0 0 a ( P / Kasin Ka coska cosa P Q ba / Ka ( P / Kasin Ka cos Ka 0 Ka E K /m 4π π 0 π 4π 0 π π π 4π wave vector a S. Hara@ i i sin cos ep( Euler's cos ( e e / sin ( e e / K nπ / a V ( Δ δ ( n( a b m [KronigPenne.ai; KroningPenne.sg]
band dispersion E- etended Brillouin ones 4 reduced Brillouin one E K /m E K /m 0 π/a π/a π/a 4π/a wave vector π/a 0 π/a wave vector S. Hara@ [.ppt]
or n: E top n ( E F( E de E C (E E dn de ( E F( E F(E Fermi-Dirac E E C 0 (EF(E de E F E C (E F(E E E F 0 0.5 F (E S. Hara@ [.ppt]
4 (E densit of states (E: ( E E C ( E M C m π E C (Conduction band S. Hara@ Ω total : total 4 π F π V π F V π me total d V m ( E E de π Ec Ω V EF F m Ω π F mef F π Ω Fermi spere 4 π [Fermi.ai]
5 Fermi-Dirac n l m m s a b Fermi-Dirac F( E E E ep T F : Boltman T: [K] ( r, r ( r, r ( r, r ϕa( r ϕb( r ϕb( r ϕa ( r He... S. Hara@ [.ppt]
6 n Here, E top n ( E F( E de E C ( E E C ( E M C m π F( E E E ep T C C F E top π E C ( E EC E E ep T F π C E π mdet F Fermi-Dirac integral η dη F ( η f 0 ( η η f e E T M η ( E EC / T, η f ( EF EC / T C F C de η f < 0 (E F - E c <0 F ( η f π e η E n C ep E p ep V m de ( m m m m d V f / C * * * E T E T V * * ( ml m F F π mdt * * m de ( ml mt for Si * * m l : longitudinal, m t : transverse E C E V M C * * m l : ligt ole, m (Conduction band (Valence band : eav ole d E m* d S. Hara@ EV EF p V F π T [.ppt]
7 n p n i n i n i n i np C V e E G / T n i : [cm - ] E G : [ev] 4.9 0 5 mdem m 0 d 4 T e E G / T m de : [g] m d : [g] m 0 : [g] n i ~ 0 ~ 0 ~ 0 5 6 / cm / cm / cm for for for GaAs ( E Si ( E G Ge ( E G G.44eV.eV 0.66eV S. Hara@ [.ppt]
8 n p E F E D E F ( E F E A Si 4 s p s p 5 P s p s p Al s p s p S. Hara@ [.ai]
9 "" "dose densit" D incorp. A incorp. "doping densit"? "impurit concentration"? "donor concentration" ( ( D A "ionied donor" - D A D D incorp. "carrier concentration" n p fied but variable [] SiE G incorp. D D > D n n A D p E G incorp. D >> D >> D n mobile & variable S. Hara@ [.ppt]
0 Fermi D D E g ep F E T D g 0 0 0 9 0 8 V Si @00K n-tpe D 0 6 cm - C n A A D E g ep A E T F g 4 E D E A Carrier Concentration [cm - ] 0 7 0 6 0 5 0 4 0 0 D p D p n C ep E C E T F D E ep E F F E T D 0 0 0 E V n i E F 0 9 0 0. 0.4 0.6 0.8.0 E F [ev] E D E C S. Hara@ E E n ep C F C T [.ppt]
S. Hara@ Appendi
[.ppt] Scrödinger t i m ( ep, ( t i t ω r r ω t i i i i r i j i nabla p p i i p (aplace ( ( p p p p p m ω ω p m t i m p m ω E E m (, ( e t t i ω (, ( t E E m 9 9 S. Hara@
40 V ( r m E V (r : : E, E, E,... :,,... eigenvalue eigenfunction, V (r Wavefunctions Energ levels π/ π/ 0 π/,,,... S. Hara@ [.ppt]