Ç
U ØT Ì ¦ R W ë s ¤ ö n Ú º8 ý M «ã _ Ë ôV ê s; c 6 X ¢ Ö ¨] § º À X Ø8 ý W _ Ë] §
~ ç
¡ ó j u ∗
x 9
ª @ / < Æ § § ª õ & ñ Â Ò, x 9 ª 627-702
L
| ç ¡% ã < †
â
· ¡ ¤ @ / < Æ § Ó ü t o < Æõ , @ /½ ¨ 702-701 (2004¸ 10 Z 4 18{ 9 ~ Ã Î6 £ §)
ï
r s " é ¶ > _ 1 l x` ¦ Ðs H ª Ä ºÓ ü t ½ ¨ ¸_ # 4 \ Ã ºf Ü ¼ Ð l © s K | 9 M : 6 £ § ¾ Ó í 7 H õ © ñ 6 x H s > _ dc l ¸ ¸\ ¦ ª : x > % i < Æ& h ~ ½ ÓZ O Ü ¼ Ð Ä » ¸ % i . # l " f H l
> r_ s : r õ H É r í ß @ /Ã º~ ½ ÓZ O ` ¦ 6 x # Ó ü t o & h Ü ¼ Ð K $ 3 l ¼ # o ô Ç õ \ ¦ % 3 % 3 . F g
$ í
í 7 H_ â Ä ºü < H Ø Ô> 6 £ § ¾ Ó í 7 H © ñ 6 x \ @ /K " f H l © _ [ jl ½ + É M : dc l ¸ ¸
1 l x ¦ 1 l x; ¤ É r : r ¸_ 7 £ x \ & t H כ ` ¦ · ú Ã º e % 3 .
PACS numbers: 72.20.-i, 72.10.Di Keywords: l ¸ ¸, í 7 Hí ß ê ø Í
I. " e  ] Ø
þ
j H_ ì ø Í ¸^ ] j ¸l Õ ü t_ µ 1 Ï Ü ¼ Ð K \ ¦ ï r s
" é ¶& h Ü ¼ Ð ¹ ¡ §f s ¸2 ¤ ¿ º H ª Ä ºÓ ü t ½ ¨ ¸> ,
\
¦ ï r { 9 " é ¶& h Ü ¼ Ð ¿ º H ª ½ ¨ ¸> , ¸ H ~ ½ Ó ¾ Ó\
"
f ½ ¨5 Å q` ¦ Å Ò H ª & h ½ ¨ ¸> , Õ ªo ¦ ô Ç ~ ½ Ó ¾ Ó ¢ ¸ H ¿ º
~
½ Ó ¾ ÓÜ ¼ Ð ½ ¨5 Å q ) a í > 1 p x õ ° ú É r ¸ ß ¼l _ $
"
é
¶ ì ø Í ¸^ _ ] j s 0 p x > ÷ &% 3 . s Qô Ç $ " é ¶ ì ø Í
¸^ \ " f { 9 # Q H ª / B N" î & ³ © (1) l í 7 H/ B N" î
&
³ © (2) í 7 H/ B N" î & ³ © (3) Û ¼ 2 ; l í 7 H/ B N" î & ³ © (4) l $ ½ Ó 1 l x & ³ © 1 p x õ ° ú É r $ " é ¶ ì ø Í ¸^ _ ª Ã
º5 Å x & ³ © \ @ /ô Ç ½ ¨ H ì ø Í ¸^ _ Ã º5 Å x & ³ © ` ¦ ½ ¨
H X < Ä »6 xô Ç Ã ºé ß Ü ¼ Ð s 6 x ÷ &l M :ë H \ þ j H \ ´ ú § É r ' a d
_ @ / © s ÷ & ¦ e [1-10].
l í 7 H/ B N" î ´ òõ [1-3] H Ã ºz meV # 3 0 A\ e H F g
$ í
í 7 H_ \ -t & ñ l © \ _ K + þ A$ í ) a ê ø Í Ä º ï r 0
A_ & ñ Ã ºC | ¨ c M :\ µ 1 ÏÒ q t ¦, s Qô Ç l í 7 H ´ òõ ü
< x 9 ] X ô Ç ' aº s e H l $ ½ Ó 1 l x É r ´ ú § É r F G$ í Ó ü t| 9
\
" f 1 l x õ î rì ø Í _ Ã º5 Å x$ í | 9 ` ¦ ¸ l 0 A K
" f 6 x ÷ &% 3 . s 1 l x É r F g$ í í 7 H s ' a # Ù ¼ Ð ©
@
/& h Ü ¼ Ð Z } É r : r ¸% ò % i (T ∼ 100 − 180K)\ " f { 9 # Qè ß
.
∗
E-mail: [email protected]
†
E-mail: [email protected]
: r 7 Hë H \ " f H s Qô Ç l 1 l x & ³ © s 6 £ § ¾ Ó í 7 H s ' a
#
½ + É M : ¸ { 9 # Q± ú Ã º e 6 £ §` ¦ Ð . s & ³ © É r 6 £ § ¾ Ó í
7 H s ' a # Ù ¼ Ð © @ /& h Ü ¼ Ð ± ú É r : r ¸% ò % i (T < 70 K)
\
" f { 9 # Qè ß . s \ ¦ 0 AK : r ½ ¨ [ þ t s " é ¶ > _ Ã º5 Å x & ³ © ` ¦ ½ ¨ H X < ¸ ç ß 6 x K : r % ò í ß
~
½ ÓZ O [12-14]` ¦ & h 6 xô Ç . ] j II ] X \ " f H ï r s " é ¶
>
\ @ /K l Õ ü t ¦, ] j III ] X \ " f H l ¸ ¸\ @ /K 4
¤_ þ v ¦, ] j IV ] X \ " f H Ã ºu > í ß ` ¦ # õ \ ¦ % 3
¦, ] j V ] X \ " f : r` ¦ ë B H .
II. Ç U ØT Ì ¦ R ¹ Å 4
&
ñ l © B = B ˆ z K | 9 M : ü < í 7 H s © ñ 6
x H > _ K x 9 Ðm î ß H(t) É r
H(t) = H eq + H int (t) (1) s
. # l " f H int (t) H r ç ß _ > r ½ Ós ¦ H eq H r ç ß \ _
> r t · ú § H ¨ î + þ A © I _ K x 9 Ðm î ß Ü ¼ Ð
H eq = H d + V = H e + H p + V (2) s
. H e , H p , V H y y , í 7 H, ü < í 7 H_ © ñ
6 x \ @ /ô Ç K x 9 Ðm î ß Ü ¼ Ð H e = X
α
E α a † α a α (3)
-39-
H p = X
q
~ω q b † q b q (4)
V = X
q
X
α,µ
C α,µ (q)a † α a µ (b q + b † −q ) (5)
Ð Å Ò# Q . # l " f a † α (a α ) H \ -t E α |α > © I
\ " f_ _ Ò q t$ í ( èY > ) í ß s ¦ b q (b † −q ) H \ - t
~ω q |q > © I _ í 7 H_ Ò q t$ í ( èY > ) í ß s 9, ω q H à º 7 ' q í 7 H_ 1 l x à ºs . 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 < í 7 H_ © ñ 6 x í ß C(q)_ ' § > =כ ¹ è\ ¦ · p . ¢ ¸ô Ç, & ñ l © B = B ˆ z
K t H ¿ ºa L z Á ºô Ç Ä ºÓ ü t ½ ¨ ¸\ ¦ ï r s " é ¶ > Ð
¦ 9 ¦Ä » < Êà º H Ψ α (r) =
2 L y L z
exp(ik yα y)Φ N
α(x −x 0 ) sin(k zα z) (6)
Φ N
α(x −x 0 ) =
1
√ π2 N
αN α !l c
exp( −y 2 /2)H N
α(y) (7)
N α = 0, 1, 2, · · · , n α = 1, 2, 3, · · · s
¦, \ -t ¦Ä »° ú כ É r
E α = E N
α+ n = (N α + 1/2)~ω c + n 2 α 0 (8) s
. # l " f k zα = n α π/L z , 0 = ~ 2 π 2 /2mL z , x 0 =
−~k yα /mω c , y = (x − x 0 )/l c , l c = (~/eB) 1/2 s ¦ ω c = eB/m H s 9 þ t Ðà Ô : r 1 l x à º, H N H N \ Ø Ôp à Ô
½ Ód s .
III. DC ¹ ÅM ¹ Åy ¢y ¢
ü
@Â Ò\ " f l © E(t) = Ee st (s → 0 + )ˆ x K | 9 M : dc l ¸ ¸ J $ " f H + þ A6 £ x² ú s : r \ _ K [11-14]
σ xx = e lim
s →0
+X
α,β
X
γ,δ
(x) αβ (j x ) γδ A αβ (s) (9)
A αβ (s) = T R {ρ eq [(i~s + L eq ) −1 a † γ a δ , a † α a β ] } (10)
Ð Å Ò# Q . # l " f x H _ 0 Au 7 ' , j x = (ie~/m)∂/∂x H é ß { 9 _ À Óx 9 ¸ í ß , (x) αβ ≡
< α |x|β >s ¦ L eq H e _ _ í ß X\ @ /K L eq X = [H eq , X] Ð & ñ _ ÷ & H H eq \ @ /6 £ x H o Ä ºq í ß s
.
d
(10)` ¦ > í ß l 0 AK % ò 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)
#
l " f
< X > ≡ T R {ρ eq [X, a † α a δ ] } (13) s
. % ò í ß _ & ñ _ ÐÂ Ò' 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 # > h 1
i~s + L eq
a † γ a δ = 1
i~s a † γ a δ − 1
i~s + L eq Q L eq a † γ a δ A αβ
< a † γ a δ > . (15) s
) a . s ] j L eq = L d + L v Ð ì r o ¦(L d ü < L v H y y
H d ü < V \ @ /6 £ x H o Ä ºq 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 (s) = < a † γ a δ >
i~s + E γδ + B αβ (s) (17)
B αβ (s) =< (i~s + L eq Q) −1 L v a † γ a δ > i~s
< a † γ a δ > (18) s
) a . # l \
< 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 6 £ §` ¦ % 3 H .
σ xx = e lim
s →0
+X
αβ
(x) αβ (j x ) βα
f β − f α
i~s + E βα + Γ αβ (s) (20)
#
l " f f α H _ ` Ø Ôp -n Ï þ _ ì r í < ÊÃ ºs ¦ Γ αβ (s) = T R {ρ eq [(i~s + L eq Q) −1 L v a † β a α , a † α a β ] } i~s
f β − f α
(21) s
. # l \ 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 # [s כ É r T R (ABC) = T R (BCA) ü < H eq H ρ eq ü < § ¨ 8 0 p x < Ê` ¦ s 6 x ~ 1 > Ä » ¸ ) a ] X ≡ (i~s + L eq Q) −1 L v a † β a α Ð ¿ º
Γ αβ (s) = −T R {ρ eq [L v a † α a β (i~s+L d ) −1 L v a † β a α ] } 1 f β − f α
(23) s
) a . # l " f í 7 H õ _ © ñ 6 x_ [ jl ¦
& ñ # L v ( ¢ ¸ H V )_ ] jY L ½ Ó t ë ß ¦ 9Ù þ ¡ . d (3)-
(5)` ¦ 6 x # d (23)` ¦ > í ß 6 £ § õ ° ú .
Γ αβ (s)(f β − f α )
= X
q
X
γ
|C βγ (q) | 2 (1 + N q )f α (1 − f γ )
i~s + E γα + ~ω q − N q f γ (1 − f α )
i~s + E γα + ~ω q + N q f α (1 − f γ ) i~s + E γα − ~ω q
− (1 + N q )f γ (1 − f α ) i~s + E γα − ~ω q
+ X
q
X
γ
|C αγ (q) | 2 (1 + N q )f γ (1 − f β )
i~s + E βγ + ~ω q − N q f β (1 − f γ ) i~s + E βγ + ~ω q
+ N q f γ (1 − f β )
i~s + E βγ − ~ω q − (1 + N q )f β (1 − f γ ) i~s + E βγ − ~ω q
(24)
#
l " f N q H í 7 H \ @ /ô Ç e ¦| ½ Óß ¼ ì r í < ÊÃ ºs . d (24)_ Ó ü t o & h _ p H 6 £ § õ ° ú . ' Í ½ Ó É r í
7 H` ¦ ~ ½ ÓØ ¦ " f % 6 £ § © I α\ " f × æç ß © I γ Ð s
H כ ` ¦ _ p ô Ç . 7 £ ¤, 1 + N q H í 7 H_ ~ ½ ÓØ ¦` ¦ _ p ¦ f α (1 − f γ ) H α → γ s \ ¦ _ p ô Ç . ì r ¸ H s → 0 + { 9 M
: E α = E γ +~ω q s Ù ¼ Ð \ -t Ð > 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 ¦,
¸ H ½ Ó\ H f β − f α H íl © I β\ " f þ j7 á x © I
α Ð s < Ê` ¦ _ p ô Ç . s X O > Ó ü t o & h Ü ¼ Ð K $ 3
H כ s 8 ¼ # o ô Ç s Ä » H # l \ " f 6 xô Ç í ß H
É
r s : r[8,11][ þ t õ H Ø Ô ¦ [d (11), (12)], > 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
.
IV. ¤V 4 m
ü < 6 £ § ¾ Ó í 7 H © ñ 6 x s dc l ¸ ¸\ p u H
% ò
¾ Ó` ¦ ½ ¨^ & h Ü ¼ Ð > í ß l 0 AK 6 £ §` ¦ 6 x # Ã º u
> í ß ` ¦ ô Ç .
(x) αβ =
"
x 0 δ N
α,N
β+ l c
r N β + 1
2 δ N
α,N
β+1 + l c
r N β
2 δ N
α,N
β−1
#
δ n
α,n
β= δ k
yα,k
yβ(25)
(j x ) βα = − e~
mil c
"r N α
2 δ N
β,N
α−1 −
r N α + 1
2 δ N
β,N
α+1
#
δ n
α,n
βδ k
yβ,k
yα(26)
|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) (27)
A n
α,n
γ(q z , L z ) = 2 L z
Z L
z0
sin(n α πz/L z ) exp(iq z z) sin(n γ πz/L z ) (28)
K 1 (N α , N β : t) = N < !
N > ! t ∆N e −t L ∆N N
<(t)
(29)
#
l " f t = ~q ⊥ 2 /2eB s ¦ L ∆N N
<(t) H > Ø Ô ' a ½ Ód (Associated Laguerre polynomial)s . N < (N > ) H
Fig. 1. DC Conductivity as a function of magnetic field.
N α , N β × æ É r à º( H à º)s . 6 £ § ¾ Ó í 7 H â Ä º
|V q | 2 = E 2 1 ~ q
2ρ m v s V (30) s
. E 1 É r + þ A( J $ [ > © Ã º, ρ m É r | 9 | ¾ Óx 9 ¸, v s H ω q = v s q Ð & ñ ÷ & H 6 £ § ¾ Ó í 7 H_ 5 Å q§ 4 , V H > _ Â Òx s .
GaAs â Ä º, Ó ü t o © Ã º[ þ t É r[15-16]
m c = 0.067m 0 , m h = 0.51m 0 , E g = 1.424eV,
ρ m = 5360kg/m 3 , v s = 4030m/s, E 1 = 6.3eV s
. 2 " é ¶& h x 9 ¸ H n 2D = 2 × 10 12 cm −2 Ü ¼ Ð ¿ º l
Ð ô Ç .
Fig. 1 É r l ¸ ¸_ l © _ > r$ í ` ¦ Ð# ï r .
#
l " f H 6 £ § ¾ Ó í 7 H \ _ ô Ç l ¸ ¸_ o\ ¦ · ú ¦
Ù ¼ Ð í 7 H_ f ¨ Ã ºõ & ñ ë ß 2 [/ å L % i . Õ ªa Ë >\ " f Ð
H ü < ° ú s l © _ [ jl < Ê\ ¸ ¸
1 l x < Ê` ¦ · ú Ã º e ¦ Õ ª M : þ j@ /° ú כ É rB ∼ (2n + 1) × 0.1 [Tesla] (n = 0, 1, 3, · · ·) { 9 M : ¦ s כ É r ω q = v s q = ω c s Ù ¼ Ð
q = ω c
v s
= eB
v s m ∼ π(2n + 1) L z
(31)
\
_ K " f þ j@ /° ú כs ì ø Í4 ¤ K " f z ` ¦ · ú Ã º e . ¢ ¸ ô
Ç, 6 £ § ¾ Ó í 7 H_ \ -t © @ /& h Ü ¼ Ð l M :ë H \ F g < Æ
í 7 H Ð H © @ /& h Ü ¼ Ð É r l © % ò % i \ " f þ j@ /u
ß
¼> H כ ` ¦ · ú Ã º e . F g < Æ í 7 H_ â Ä º, { 9 & ñ ô
Ç F g < Æ í 7 H_ 1 l x à º ω l s s 9 þ t Ðà Ô : r 1 l x à º ω c _
&
ñ à ºC ü < ° ú ` ¦ M : [7 £ ¤, ω l = lω c (l = 1, 2, 3, · · ·)` ¦ ë ß 7 á ¤½ + É M
:] / B N" î & ³ © s { 9 # Qè ß ¦ · ú 94 R e . ì ø Í \ : r 7 Hë H
\
" f H 1 l x à º { 9 & ñ t · ú § É r 6 £ § ¾ Ó í 7 H( 1 l x à º ω q )_
â
Ä º, ω q = ω c { 9 M : ¸ l © _ [ jl \ / B N" î & ³ © s { 9
# Qz ` ¦ · ú Ã º e . ¢ ¸ô Ç, s Qô Ç / B N" î & ³ © É r d (31)\ _ K 2 " é ¶ > _ ß ¼l 7 £ ¤, L z _ % ò ¾ Ó` ¦ ~ Ã Î6 £ §` ¦ · ú Ã º e
.
V. + s Ç Â ] Ø
t
F K t ï r s " é ¶ > _ dc l ¸ ¸\ ¦ % ò í
ß ~ ½ ÓZ O Ü ¼ Ð Ä » ¸ % i . õ H Ó ü t o & h Ü ¼ Ð K $ 3 l
¼ # o < Ê` ¦ · ú Ã º e % 3 ¦ l © _ [ jl < Ê\
¸ ¸ 1 l x < Ê` ¦ · ú Ã º e % 3 . s Qô Ç 1 l x & ³ © É r t
F K t H Å Ò Ð F g$ í í 7 H í ß ê ø Í\ _ K " fë ß { 9 # Qè ß ¦
·
ú 94 R e % 3 Ü ¼ : r ½ ¨_ õ \ _ 6 £ § ¾ Ó í 7 H í ß ê
ø Í\ _ K " f ¸ q §& h ô Ç l © [ jl ü < ± ú É r : r ¸ % ò
%
i \ " f { 9 # Qz ` ¦ · ú Ã º e % 3 . s כ É r 6 £ § ¾ Ó í 7 H_ \ - t
q §& h ¦ ± ú É r : r ¸% ò % i \ " f 6 £ § ¾ Ó í 7 H s Ö ¸$ í
o ÷ &l M :ë H Ü ¼ Ð ó ø Íé ß ) a . s Qô Ç l © [ jl ü < : r ¸
% ò
% i \ " f ' K z ´+ « >u \ O # Q q § t 3 l wô Ç כ s
~ 1
.
Y c
p w à U Ø ô
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Effect of Acoustic Phonon Scattering for a Magnetophonon Resonance in a Quasi-Two-Dimensional Quantum Well Structure
Nam Lyong Kang ∗
Faculty of Liberal Arts, Miryang National University, Miryang 627-702 Sang Don Choi
Department of Physics, Kyungpook National University, Daegu 702-701 (Received 18 October 2004)
Based on a quantum-statistical operator-algebra technique, we derive the dc conductivity for a system of electrons interacting with acoustic phonons in a quantum well with a static magnetic field applied normal to its barriers. By utilizing the projection operator P and Q, which are quite different from the others, we obtain a result that is more convenient for interpreting the transitions physically. The dependence of the magnetic field for acoustic phonon scattering is found to be different from that for optical phonon scattering. We also can see that the dc conductivity oscillates with the intensity of the magnetic field.
PACS numbers: 72.20.-i, 72.10.Di
Keywords: Conductivity, Scattering by phonons
∗
