U ØT Ì ¦ R  ¹ Å 4 8 ý ° Ë Ñ ¹ Åy ¢y ¢; c 6 ” X ¢ ° Ë ÑV R ˃ ºÀ X Ø8 ý W _ Ë] ‚ §

Ç

U ØT  Ì ¦ R  ¹ Å 4 8 ý ° Ë Ñ ¹ Åy ¢y ¢; c 6 ” X ¢ ° Ë ÑV R ˃ ºÀ X Ø8 ý W _ Ë] ‚ §

~ ç

¡ ‘ ž ó j u ^∗

x 9

€ ª œ@ /† < Ɠ § “ §€ ª œõ & ñ  Ò, x 9 € ª œ 627-702 (2005¸   9 Z 4 30{ 9  ~ à Î6 £ §, 2006¸   1 Z 4 2{ 9  þ j7 á x‘ : r ~ à Î6 £ §)

€

ª œ : Ÿ x > % i † < Æ& h  ƒ  í ß –  @ /à º~ ½ ÓZ O `  ¦  6   x # Œ F  g$ í Ÿ í 7 H õ   © œ  ñ Œ •6   x   H ï  r s  " é ¶ „   > _  F  g„  

•

¸• ¸\  ¦ Ä »• ¸Ù þ ¡ .   õ \   H „   ü < Ÿ í 7 H _  ì  r Ÿ í† < Êà º & h ] X y  Ÿ í† < Ê÷ &# Q e ” # Q" f „    Ÿ í 7 H`  ¦ f  ¨ à º

¢

¸  H ~ ½ ÓØ  ¦ €  " f „  s    H õ & ñ `  ¦ K $ 3  l  ¼ # o † < Ê`  ¦ · ú ˜ à º e ” % 3  . ¢ ¸, à ºu > í ß –_    õ \  _  

€

 , Ä ºÓ ü t ¿ ºa  7 £ x  €   ì ø Í; Ÿ ¤ u  y Œ ™™ è “ ¦ “ : r • ¸ `  ¦  €   ì ø Í; Ÿ ¤ u  7 £ x    H  כ Ü ¼– Ð
 z Œ ¤ .

PACS numbers: 72.20.-i, 72.10.Di Keywords: „  l „  • ¸• ¸, Ÿ í 7 H í ß –ê ø Í

I. " e  ] Ø

þ

j   H \  ì ø ͕ ¸^ ‰ ] j› ¸l Õ ü t _  µ 1 τ  Ü ¼– Ð “  K 
” ¸ß ¼l _ 

$

 " é ¶ ì ø ͕ ¸^ ‰_  ] j Œ •s  0 p x >  ÷ &% 3  . s    $  " é ¶

>

_  : £ ¤$ í `  ¦ › ¸    H ~ ½ ÓZ O ×  æ 
 s [ þ t >  ? /_  „  



[ þ t _  à º5 Å x‰ & ³ © œ\  % ò † ¾ Ó`  ¦ p u   H Ÿ í 7 H s 
 Ô  ¦í  HÓ ü t 1 p x

\

 _ ô  Ç í ß –ê ø Í´ òõ \  ¦ ƒ  ½ ¨   H  כ s  . s ü < › ' aº   ) a ƒ  ½ ¨

×

 æ 
 ‚  + þ A6 £ x ² ú šs  : r s   [1-11]. € ª œ : Ÿ x > % i † < Æ& h  ~ ½ Ó Z O

`  ¦ & h 6   x # Œ  % ò ƒ  í ß –  ~ ½ ÓZ O  [2-11]`  ¦  6   x €   ‚  + þ A 6

£

x ² ú šs  : r \  _ K  Å Ò# Qt   H „  l „  • ¸• ¸\  ¦  € ª œô  Ç + þ AI – Ð

½

¨½ + É Ã º e ”  . s  ~ ½ ÓZ O `  ¦ & h 6   x €    % ò ƒ  í ß – \  ¦ & ñ _  



 H ~ ½ ÓZ O \     „  • ¸• ¸\  ¦  € ª œ >  „  > h½ + É Ã º e ”  .

õ

 \   H  % ò í ß – \  ¦  6   x # Œ „  l „  • ¸• ¸\  ¦ > í ß –½ + É M

: ŠҖ Ð é ß –{ 9  „   ³ ð‰ & ³Ü ¼– Ð > í ß –Ù þ ¡Ü ¼
 [2-6],  ^ ‰  

% ò

ƒ  í ß –  ~ ½ ÓZ O Ü ¼– Ð > í ß –   H  כ s  Ó ü t o & h Ü ¼– Ð K $ 3  l 

 ¼ # o   “ ¦ · ú ˜ 94 R e ”   [7-11]. ¢ ¸ô  Ç,  © œI 1 l qw n   

% ò

ƒ  í ß –  (state-independent projection operator) ~ ½ ÓZ O 

“ É

r  s 9 þ t – Ðà ԏ : r / B N" î õ  ° ú  “ É r \  -t  ç ß –  s  { 9 & ñ ô  Ç  â Ä

º\  ô  Ç # Œ & h 6   x ½ + É Ã º e ”   H ì ø ̀  \ ,  © œI _ ” > r  % ò ƒ  í ß –



 (state-dependent projection operator) ~ ½ ÓZ O “ É r \  -t  ï

 r 0 A s _  ç ß –  s  { 9 & ñ t  · ú § • ¸ & h 6   x s  0 p x  “ ¦

· ú

˜ 94 R e ”   [10, 11].

‘

: r  7 Hë  H \ " f  H  © œI 1 l qw n   % ò ƒ  í ß –  ~ ½ ÓZ O `  ¦ & h 6   x # Œ ï

 r s  " é ¶ „   > _  F  g„  • ¸• ¸\  F  g$ í Ÿ í 7 H s  p u   H % ò † ¾ Ó

`

 ¦ € ª œ : Ÿ x > % i † < Æ& h  ~ ½ ÓZ O Ü ¼– Ð > í ß –ô  Ç .

∗

E-mail: [email protected]

II. X N Ë M X ê s ; c" e8 ý Ç U Ø T  Ì ¦ R  ¹ Å 4 

&

ñ  l  © œ B = Bˆz K | 9  M : ï  r s  " é ¶ „   > _  K  x 9

ž Ðm î ß –“ É r 7 ˜' ( J $ ™[ > `  ¦ A = (0, Bx, 0) – Ð ‚  × þ ˜ €   H _e = ^(p+eA) _2m

²

+ U (z)

= _2m ¹ P

i=x,y,z p ² _i + ω _c x ^~ _{i ∂y} ^∂ + ¹ ₂ mω ² _c x ² + U (z) (1) s

 . # Œl " f m“ É r „   _  Ä »´ ò| 9 | ¾ Ós “ ¦ p  H „   _  î  r 1

l

x | ¾ Ós  9 ω c = eB/m  H  s 9 þ t – Ðà ԏ : r ”  1 l x à º\  ¦
  · p



. ï  r s  " é ¶ „   > _  ½ ¨5 Å q ( J $ ™[ >  U (z)  H > _  ß ¼l  ´ ò õ

\  ¦ “ ¦ 9½ + É Ã º e ” • ¸2 Ÿ ¤ ¿ ºa  L z “   Á ºô  Ç  y Œ •Ä ºÓ ü t`  ¦ × þ ˜ ô

 Ç .

d ”

 (1)_  “ ¦Ä »† < Êà º– Ð

Ψ(x, y, z) = s 2

L _y L _z exp(ik y y)Φ(x) sin(k z z) (2)

\

 ¦ ‚  × þ ˜ô  Ç . L y   H > _  y~ ½ ӆ ¾ Ó_  ß ¼l s “ ¦  à º 7 ˜'  k\ 

@

/K " f k ^z = nπL z s  9 n“ É r € ª œ à ºs  . d ”  (1)`  ¦ d ”  (2) \  & h 6   x €   \  -t  “ ¦Ä »u  E\  @ /K " f

H e Ψ(x, y, z) = EΨ(x, y, z)

=  ~ ² k _y ²

2m + ~ ² k _z ² 2m − ~ ²

2m

∂ ²

∂x ² + 1

2 mω ² _c x ² + ω _c x~k y



Ψ(x, y, z) (3) s

 ÷ &“ ¦, x ¹ = x + ~k ^y /eB – Ð ¿ º€   1

2 mω _c ² x ² ₁ = 1

2 mω _c ² x ² + ~ ² k _y ²

2m + ω _c x~k y (4)

-125-

s

 .   " f d ”  (2)-(4)\  ¦  6   x €   (− _2m ^~

²

_∂x ^∂

²2

1

+ ¹ ₂ mω _c ² x ² ₁ )Φ(x)

= (E − ^~ _2m

²

^k

^z2

)Φ(x) ≡ E 1 Φ(x 1 ) (5)

\

 ¦ % 3 `  ¦ à º e ”  . 0 A d ” “ É r › ¸ o”  1 l x  _  “ ¦Ä »u  ~ ½ Ó& ñ d ”  Ü

¼– Ð Õ ª K   H ¸ ú ˜ · ú ˜ 94 R e ”  .   " f d ”  (1)_  “ ¦Ä »u ü <

“

¦Ä »† < Êà º  H  6 £ § õ  ° ú   .

E N,n = E N +  n = (N + 1/2)~ω ^c + n ²  0 (6) N = 0, 1, 2, · · · , n = 1, 2, 3, · · ·

Ψ N,n,k

_y

(x, y, z)

= q ₂

L

_y

L

_z

exp(ik y y)Φ N (x − x 0 ) sin(k z z) (7) Φ N (x − x 0 ) = ¹

( ^{√ π2}

^N

^{N !l}

c

)

^1/2

× exp 

− ^(x−x _2l

₂⁰

⁾

²

c

 H _N 

x−x

₀

l

_c



(8)

#

Œl " f N“ É r & ñ  l  © œ\  _ K " f ì  r o   ) a  © œI \  ¦
 ? /



 H Landau t à º, n“ É r ½ ¨5 Å q ( J $ ™[ > \  _ K " f ì  r o   ) a  © œI 

\

 ¦
 ? /  H t à ºs  . H ^N   H N −  Hermite  † ½ Ód ” s 

“

¦  ⁰ = ~ ² π ² /2mL ² _z , x 0 = −~k ^y /eB, l ² _c = ~/eBs  . d ”  (6) \ " f · ú ˜ à º e ” 1 p w s  „   _  \  -t   H x-y ~ ½ ӆ ¾ ÓÜ ¼– Ѝ  H

&

ñ  l  © œ\  _ K " f € ª œ  o÷ &“ ¦ z ~ ½ ӆ ¾ ÓÜ ¼– Ѝ  H ½ ¨5 Å q ( J $ ™ [ >

\  _ K  € ª œ  o  ) a  . ] j III] X \ " f  H 0 Aü < ° ú  “ É r > _ 

„

 l „  • ¸• ¸\  ¦ Ÿ í 7 H õ _   © œ  ñ Œ •6   x s  e ”   H  â Ä º\  @ /K " f

½

¨ô  Ç .

III.  ¹ ÅM  ¹ Åy ¢y ¢8 ý ] k ùÅ k ÄX ì Äß Ã Å A 0

ü

@ Ò\ " f y Œ •”  1 l x à º ω– Ð ”  1 l x   H „  l  © œ E(t) = Ee ^iωt  K | 9  M : „  l „  • ¸• ¸ J $ ™" f  H ‚  + þ A6 £ x ² ú šs  : r \  _

K  [7-9]

σ xx (¯ ω) = −e lim

s→0

⁺

X

α,β

X

γ,δ

(x) αβ (j x ) γδ A αβ (¯ ω) (9)

–

Ð Å Ò# Q”   . # Œl " f

A _αβ (¯ ω) = T _R {ρ eq [(~¯ ω − L _eq ) ⁻¹ a ^† _γ a _δ , a ^† _α a _β ]} (10) s

 . ¯ ω ≡ ω + is(s → 0 ⁺ ) s “ ¦ x  H „   _  0 Au  7 ˜' _  x$ í ì  r, j x = (ie~/m)∂/∂x  H é ß –{ 9  „   _  „  À Óx 9 • ¸ ƒ  í ß –



, (x) αβ ≡< α|x|β > s “ ¦ L eq   H e ” _ _  ƒ  í ß –  X\  @ / K

 L ^eq X = [H eq , X] – Ð & ñ _ ÷ &  H H eq \  @ /6 £ x   H o Ä ºq 

ƒ

 í ß – s  . a ^† α (a α )  H \  -t  E ^α “   |α >  © œI \ " f_ 

„

  _  Ò q t$ í (™ èY > )ƒ  í ß – s  .

d ”

 (10)`  ¦ > í ß – l  0 AK   % ò ƒ  í ß – \  ¦  6 £ § õ  ° ú  s 

&

ñ _ ô  Ç .

P X ≡ ^<X>

<a

^†_γ

a

_δ

> a ^† _γ a δ (11) Q ≡ 1 − P (12)

#

Œl " f

< X >≡ T _R {ρ _eq [X, a ^† _α a _β ]} (13) s

 . ρ eq   H ¨ î + þ Aì  r Ÿ í_  x 9 • ¸, T R   H  ^ ‰ traces  .  

% ò

ƒ  í ß – _  & ñ _ – РÒ'  1 = (P + Q)s Ù ¼– Ð d ”  (10)\ " f L eq · 1 = L eq (P + Q) – Ð  Ë ¨“ ¦ † ½ Ó1 p xd ” 

(A − B) ⁻¹ = A ⁻¹ + A ⁻¹ B(A − B) ⁻¹ (14)

`

 ¦  6   x # Œ „  > h €  

1

~ ¯ ω−L

eq

a ^† _γ a _δ = ¹

~ ¯ ω a ^† _γ a _δ + ¹

~ ¯ ω−L

eq

Q L _eq a ^† _γ a _δ ^A

^αβ

<a

^†_γ

a

_δ

> (15) s

  ) a  . # Œl " f Qa ^† γ a δ = 0`  ¦  6   xÙ þ ¡ . s ] j L ^eq = L d + L v – Ð ì  r o  “ ¦ (L ^d ü < L ^v   H y Œ •y Œ • @ /y Œ • K x 9 ž Ðm î ß – H _d ü <  © œ  ñ Œ •6   x † ½ Ó V \  @ /6 £ x   H o Ä ºq  ƒ  í ß – )

L _d a ^† _γ a _δ = (E _γ − E _δ )a ^† _γ a _δ ≡ E _γδ a ^† _γ a _δ (16)

\

 ¦  6   x €  

A αβ (¯ ω) = ^<a

† γ

a

_δ

>

~ ¯ ω−E

_γδ

−B

αβ

( ¯ ω) (17) B αβ (¯ ω) =< (~¯ ω − L eq Q) ⁻¹ L v a ^† _γ a δ > ^{~ ¯ ω}

<a

^†_γ

a

_δ

> (18) s

  ) a  . E ^㠍  H  © œI  |γ >≡ |N, n >_  \  -t s  . # Œl 

\



< a ^† _γ a δ >= T R {ρ eq [a ^† _γ a δ , a ^† _α a β ]}

= (f _β − f α )δ _βγ δ _αδ (19)

\

 ¦ “ ¦ 9 “ ¦ d ”  (17)`  ¦ d ”  (9)\  @ /{ 9  €    6 £ §`  ¦ % 3   H



.

σ xx (¯ ω) = −e lim s→0

⁺

P

αβ (x) αβ (j x ) βα

× ^f

^β

^−f

^α

~ ¯ ω−E

_βα

−Γ

αβ

( ¯ ω) . (20)

#

Œl " f f α   H |α >  © œI _  „   _  ` …Ø Ôp -n Ï þ ˜_  ì  r Ÿ í† < Ê Ã

ºs “ ¦

Γ αβ (s) = T R {ρ eq



(~¯ ω − L eq Q) ⁻¹

× L v a ^† _β a _α , a ^† _α a _β

 } _f ^{~ ¯ ω}

β

−f

α

(21)

s

 . # Œl \  † ½ Ó1 p xd ” 

T R {ρ eq [L eq QX, a ^† _α a β ]} = T R {ρ eq [L v a ^† _α a β , X]}

−T R {ρ eq [L v P X, a ^† _α a β ]} (22)

\

 ¦ “ ¦ 9 # Œ [s  כ “ É r T R (ABC) = T _R (BCA) ü < H eq   H ρ eq ü < “ § ¨ 8 Š 0 p x † < Ê`  ¦ s 6   x €   ~ 1 >  Ä »• ¸  ) a  ] X ≡ (~¯ ω−

L eq Q) ⁻¹ L v a ^† _β a α – Ð ¿ º€  

Γ αβ (¯ ω) = −T R {ρ eq [L v a ^† _α a β , (~¯ ω − L d ) ⁻¹ L v a ^† _β a α ]} 1

f _β − f _α (23) s

  ) a  . # Œl " f  © œ  ñ Œ •6   x _  [ jl  €  •  “ ¦ & ñ # Œ L v ( ¢ ¸  H V ) _  ] jY  L † ½ Ó t ë ß – “ ¦ 9Ù þ ¡ . d ”  (23)“ É r V ( ¢ ¸



 H L v )  ½ ¨^ ‰& h Ü ¼– Ð Å Ò# Qt €   > í ß –s  0 p x ô  Ç + þ AI s  .

]

j IV ] X \ " f  H s  כ `  ¦ „   -Ÿ í 7 H > \  @ /K " f ½ ¨ô  Ç .

IV.  ¹ Å -ƒ ºÀ X Ø4 8 ý  ¹ ÅM  ¹ Åy ¢y ¢

„

  _  à º5 Å x‰ & ³ © œ\  Ÿ í 7 H s  › ' a # Œ   H  â Ä º, ¨ î + þ A © œI 

\

" f_  K x 9 ž Ðm î ß – H ^eq   H  6 £ § õ  ° ú  s  „   _  K x 9 ž Ðm  î

ß –(H ^e ) ü < Ÿ í 7 H _  K x 9 ž Ðm î ß –(H ^p ), „   ü < Ÿ í 7 H _   © œ  ñ



Œ

•6   x(V ) Ü ¼– Ð s À Ò# Q”   .

H eq = H e + H p + V

= P

α E _α a ^† _α a _α + P

q ~ω q b ^† _q b _q + V (24) V = P

q

P

α,µ C α,µ (q)a ^† _α a µ (b q + b ^† _−q ) (25)

#

Œl " f E ^α   H d ”  (6)\  Å Ò# Q”   „   _  \  -t s “ ¦ b ^† _q (b q )  H \  -t  ~ω ^q “   |q > © œI _  Ÿ í 7 H _  Ò q t$ í (™ èY > )

ƒ

 í ß – s  9, ω q   H  à º 7 ˜'  q“   Ÿ í 7 H _  y Œ •”  1 l x à ºs  .

Ÿ

í 7 H _   à º 7 ˜'  qü < ì  rF G t à º s\  @ /K " f q = (q, s)– Ð

&

ñ _ ÷ &“ ¦ C α,µ ≡< α|C(q)|µ >  H „   ü < Ÿ í 7 H _   © œ  ñ Œ • 6

 

x ƒ  í ß –  C(q)_  ' Ÿ § > =כ ¹™ è\  ¦
  · p .

d ”

 (24)ü < (25)\  ¦  6   x # Œ d ”  (23)`  ¦ > í ß – €    6 £ § õ 

° ú   .

Γ αβ (¯ ω)(f β − f α ) = P

q

P

γ |C βγ (q)| ²



(1+N

_q

)f

_α

(1−f

_γ

)

~ ¯ ω−E

γα

−~ω

q

− ^N

^q

^f

^γ

^(1−f

^α

⁾

~ ¯ ω−E

_γα

−~ω

q

+ ^N

^q

^f

^α

^(1−f

^γ

⁾

~ ¯ ω−E

_γα

+~ω

q

− ^(1+N

^q

^)f

^γ

^(1−f

^α

⁾

~ ¯ ω−E

_γα

+~ω

q

 + P

q

P

γ |C αγ (q)| ²

×

 (1+N

_q

)f

_γ

(1−f

_β

)

~ ¯ ω−E

_βγ

−~ω

q

− ^N

^q

^f

^β

^(1−f

^γ

⁾

~ ¯ ω−E

_βγ

−~ω

q

+ ^N

^q

^f

^γ

^(1−f

^β

⁾

~ ¯ ω−E

_βγ

+~ω

q

− ^(1+N

^q

^)f

^β

^(1−f

^γ

⁾

~ ¯ ω−E

_βγ

+~ω

q

 (26)

#

Œl " f N ^q   H Ÿ í 7 H \  @ /ô  Ç e  ¦ | ½ Óß ¼ ì  r Ÿ í† < Êà ºs  . d ”  (26) _  Ó ü t o & h  _ p   H  6 £ § õ  ° ú   . ' Í † ½ ӓ É r „    Ÿ í



7 H`  ¦ f  ¨ à º €  " f % ƒ6 £ §  © œI  α\ " f ×  æ ç ß – © œI  γ– Ð „  s  



 H  כ `  ¦ _ p ô  Ç . 7 £ ¤, 1 + N q   H Ÿ í 7 H _  ~ ½ ÓØ  ¦`  ¦ _ p  “ ¦ f α (1 − f γ )  H α → γ „  s \  ¦ _ p ô  Ç . ' Í † ½ Ó_  ì  r — ¸  H

~ω + E α = E γ + ~ω ^q s Ù ¼– Ð \  -t  ˜ Д > rZ O g Ë :s  $ í w n † < Ê

`

 ¦ _ p ô  Ç .
 Qt  † ½ Ó[ þ t • ¸ ° ú  “ É r ~ ½ ÓZ O Ü ¼– Ð K $ 3 ½ + É Ã º e ” 

“

¦, — ¸Ž  H † ½ Ó\ 
 
  H f β − f α   H œ íl  © œI  β\ " f þ j7 á x



© œI  α– Ð „  s † < Ê`  ¦ _ p ô  Ç . s X O >  ¼ # o ô  Ç Ó ü t o & h  K 

$

3 s  0 p x ô  Ç   õ \  ¦ % 3 `  ¦ à º e ” % 3 ~   s Ä »  H # Œl \ " f   6

 

x ô  Ç ƒ  í ß –   H   É r s  : r[2-8] \ " f • ¸{ 9  ) a  כ [ þ t õ   H   Ø

ԓ ¦, > í ß –õ & ñ \ " f : £ ¤Z > ô  Ç l Z O  [\ V\  ¦ [ þ t # Q d ”  (22)]\  ¦



6   x % i l  M :ë  H s  .

V. • ¤V 4  ˜ m

„

  ü < F  g$ í Ÿ í 7 H _   © œ  ñ Œ •6   x s  „  l „  • ¸• ¸\  p u   H

% ò

† ¾ Ó`  ¦ ½ ¨^ ‰& h Ü ¼– Ð › ¸  l  0 AK   6 £ §`  ¦  6   x # Œ à º u

> í ß –`  ¦ ô  Ç .

(x) αβ = [x 0 δ N

_α

,N

_β

+ l c

q N

_β

+1

2 δ N

_α

,N

_β

+1

+l _c q N

_β

2 δ _N

_α

_,N

_β

₋₁ ]δ _n

_α

_,n

_β

δ _k

_yα

_,k

_yβ

(27) (j _x ) _βα = − _mil ^e~

c

[ q N

_α

2 δ _N

_β

_,N

_α

₋₁

− q

N

α

+1

2 δ N

_β

,N

_α

+1 ]δ n

_α

,n

_β

δ k

_yβ

,k

_yα

(28)

|C αγ (q)| ² = |V q | ² δ k

_yα

,k

_yγ

+q

_y

|A n

_α

,n

_γ

(q z , L z )| ²

×K 1 (N α , N γ : t) (29) A n

α

,n

γ

(q z , L z ) = _L ²

z

R L

_z

0 sin(n α πz/L z )

× exp(iq z z) sin(n γ πz/L z ) (30) K ₁ (N _α , N _β : t) = ^N _N

^<

^!

>

! t ^∆N e ^−t [L ^∆N _N

<

(t)] (31)

#

Œl " f (x) ^αβ =< α|x|β >, j x   H é ß –{ 9 „   _  „  À Óx 9 • ¸

ƒ

 í ß – , t = ~q ⊥ ² /2eB s “ ¦ L ^∆N N

<

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(Associated Laguerre polynomial)s  . q ^z   H Ÿ í 7 H _    Ã

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|V q | ² = e ² ~ω l

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 1

(∞) − 1

(0)

 q

(q ² + q ² _d ) ² (32) s

 . # Œl " f V   H > _   Òx , (∞)ü < (0)  H y Œ •y Œ • F  g Ä »„   Ö

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&

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(20) _  z  ´Ã ºÂ Òì  r`  ¦ ½ ¨ €    6 £ § õ  ° ú   .

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²

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⁺

P

α,β N

_α

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_β

−f

α

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+

P

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(f

β

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c

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#

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_β

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_βy

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+q

_z

× < [(1 + N q )f (N α , n α ){1 − f (N γ , n γ )} − N q f (N γ , n γ ){1 − f (N α , n α )}]

×(−π)δ{~ω − (N γ − N _α )~ω c − (n ² _γ − n ² _α ) ₀ − ~ω q }

|[N q f (N α , n α ){1 − f (N γ , n γ )} − (1 + N q )f (N γ , n γ ){1 − f (N α , n α )}]

×(−π)δ{~ω − (N γ − N α )~ω ^c − (n ² _γ − n ² _α ) 0 + ~ω ^q } >

+ X

q

X

γ

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_α

_,n

_γ

(q _z , L _z )| ² K ₁ (N _α , N _γ ; t)δ _k

_αy

_,k

_γy

_+q

_z

× < [(1 + N q )f (N γ , n γ ){1 − f (N β , n β )} − N q f (N β , n β ){1 − f (N γ , n γ )}]

×(−π)δ{~ω − (N β − N γ )~ω ^c − (n ² _β − n ² _γ ) 0 − ~ω q }

|[N q f (N _γ , n _γ ){1 − f (N _β , n _β )} − (1 + N _q )f (N _β , n _β ){1 − f (N _γ , n _γ )}]

×(−π)δ{~ω − (N β − N γ )~ω ^c − (n ² _β − n ² _γ ) 0 + ~ω ^q } > (34)

s

 . d ”  (34)\  ¦ d ”  (33)\  @ /{ 9  €   „  l „  • ¸• ¸\  ¦ ½ ¨½ + É Ã

º e ”  .

GaAs“    â Ä º F  g$ í Ÿ í 7 H s  „  l „  • ¸• ¸\  p u   H % ò † ¾ Ó

`

 ¦ à ºu & h Ü ¼– Ð › ¸  l   6 £ § _  Ó ü t o  © œÃ º[ þ t`  ¦ “ ¦ 9ô  Ç



[12, 13].

m = 0.067m 0 , m h = 0.51m 0 , E g = 1.424eV,

Fig. 1. Well width depedence of optical conductivity half-width due to electron-LO phonon scattering in pure GaAs for several photon energies.

(0) = 12.53, (∞) = 10.9, ω _l = 2π × 8.76 × 10 ¹² Hz

#

Œl " f m 0   H  Ä »„   _  | 9 | ¾ Ó, m h   H ½ ¨" í (hole)_  | 9 

|

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

    H t \  ¦ F  g  _  \  -t \  @ /K  ˜ Ð# ŒÅ ҍ  H Õ ªa Ë >s  .

㍠ H F  g„  • ¸• ¸ Re{σ ^xx (ω)} _  ì ø Í; Ÿ ¤ u  7 £ ¤, 2㍠ H þ j@ /u _ 

Fig. 2. Temperature dependence of optical conductivty

half-width due to electron-LO phonon scattering in pure

GaAs for several photon energies.

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Y

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[1] R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).

[2] H. Mori, Progr. Theor. Phys. 34, 399 (1965).

[3] A. Kawabata, J. Phys. Soc. Jpn. 23, 999 (1967).

[4] A. Lodder and S. Fujita, J. Phys. Soc. Jpn. 25, 774 (1967).

[5] S. D. Choi and O. H. Chung, Solid State Commun.

46, 717 (1983).

[6] J. Y. Ryu, Y. C. Chung and S. D. Choi, Phys. Rev.

B 32, 7769 (1985).

[7] S. Badjou and P. N. Argyres, Phys. Rev. B 35, 5964 (1987).

[8] Y. J. Cho and S. D. Choi, Phys. Rev. B 47, 9273 (1993-I).

[9] N. L. Kang, Y. J. Cho and S. D. Choi, Progr. Theor.

Phys. 96, 307 (1996).

[10] N. L. Kang, Y. J. Lee and S. D. Choi, J. Korean Phys. Soc. 44, 1535 (2004).

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Korean Phys. Soc. 46, 1040 (2005).

[12] C. M. Wolfe and G. E. Stillman, Physical Properties of Semiconductors (Prentice-Hall, Englewood Cliffs, New Jersey, 1989).

[13] D. K. Ferry, Semiconductors (Macmillan, New

York, 1991).

Effect of Optical Phonon Scattering on the Optical Conductivity in a Quasi-Two-Dimensional Electron System

Nam Lyong Kang ^∗

Faculty of Liberal Arts, Miryang National University, Miryang 627-702 (Received 30 September 2005, in final form 2 January 2006)

Utilizing the quantum statistical operator algebra technique, we derive the intraband linewidth function in the optical conductivity for a system of electrons interacting with longitudinal optical phonons in a quantum well. The electron and the phonon distribution functions are included in the conductivity, so we can explain the phonon emission and absorption in an organized way for all electron transition processes. The numerical result shows that the width decreases with the well width and increases with the temperture.

PACS numbers: 72.20.-i, 72.10.Di

Keywords: Conductivity, Scattering by phonons

∗

E-mail: [email protected]