Một số hiệu ứng cao tần do tương tác electron - phonon trong dây lượng tử bán dẫn

E
n
1
,n
2
(k
z
) = E
n
1
,n
2
+

2
k
2
z
2m
e
, ψ
n
1

,n
2
,k
z
(x, y, z) =
e
ik
z
z
√
L
ψ
n
1
,n
2
(x, y),
L = L
z
z E
n
1
,n
2
ψ
n
1
,n
2
(x, y)

E
n
x
,n
y
=

2
π
2
2m
e

n
2
x
L
2
x
+
n
2
y
L
2
y

, ψ
n
x

,n
y
(x, y) =
2

L
x
L
y
sin

n
x
πx
L
x

sin

n
y
πy
L
y

.
E
n,
=


2
A
2
n,
2m
e
R
2
, ψ
n,
(r, ϕ) =
1
√
πR
1
J
n+1
(A
n,
)
J
n
(A
n,
r
R
)e
inϕ
,
A

n,
 n n =
0, ±1, ±2, )  = 1, 2, 3, J
n
(A
n,
) = 0
ψ(r) = 0 r = R
σ
µν
(ω) = lim
δ→0
∞

0
e
iωt−δt
(J
µ
, J
ν
(t))dt,
∞

0
e
−zt
(A, B(t))dt = ( A, B)[z −iη + (A, B)
−1
∞


0
e
−zt
(Q, R(t))dt]
−1
.
σ
µν
(ω) = lim
δ→+0
∞

0
e
iωt−δt
(J
µ
, J
ν
(t)) = lim
δ→+0
(J
µ
, J
ν
)

− i(ω + η)
+


i


2
(J
µ
, J
ν
)
−1
∞

0
e
iωt−δt
([U, J
µ
], [U, J
ν
]
I
) dt

−1
,
U
ω

E

Ω

F
A = T
R
(ρ(t)A) ρ(t)
∂ρ(t)
∂t
=
i

[ρ(t), H] ⇒
∂ρ(t)
∂t
=
i


ρ(t), H(t) + H
1
t

.
ρ(t)
A = A
0
−
i

t



t

H
1
t

(H
T
|t

, t), A

dt

, H
1
t

(H
T
|t

, t) = [H
T
, H
1
t


(t

, t)].
σ
k
(ω) = −
i
ω
lim
a→0
+
(ω −L)
−1
J
k
,
X ≡ T
R
{ρ
0
(H)[X, J

]}
P
αβ
X = X
αβ
J
k
/J

k

αβ
, Q
αβ
= 1−P
αβ
, X
αβ
≡ T
R
{ρ
0
(H)[X, a
+
α
a
β
]}.
P
αβ
X
J
k
α, β

E =

E
0

sin Ωt
H =

n
1
,n
2
,

k
ε
n
1
,n
2


k −
e
c

A(t)

a
+
n
1
,n
2
,


k
a
n
1
,n
2
,

k
+

q
ω
q
b
+
q
b
q
+

n
1
,n
2
,n

1
,n


2
,

k,q
M
n
1
,n
2
,n

1
,n

2
(q)a
+
n
1
,n
2
,

k
a
n

1
,n


2
,

k+q
(b
q
+ b
+
−q
).
t N
q
(t) =< b
+
q
b
q
>
t
∂N
q
(t)
∂t
= −
1

2

α,α


,

k
|M
α,α

(q)|
2
∞

s,s

=−∞
J
s
(
Λ
Ω
)J
s

(
Λ
Ω
) exp

i(s − s

)Ωt


×

t
−∞

f
α
(

k)f
α

(

k + q) + f
α

(

k + q)N
q
(t

) − f
α
(

k)(1 + N
q

(t

))

× exp

i


E
α

(

k + q) − E
α
(

k) − ω
q
− sΩ

(t − t

)

+

f
α


(

k −q)f
α
(

k) + f
α
(

k)N
q
(t

) − f
α

(

k −q)(1 + N
q
(t

))

× exp

−
i



E
α
(

k) − E
α

(

k −q) − ω
q
− sΩ

(t − t

)


dt

.
Λ =
e

E
0
q
m

e
Ω
.
P
k0
P

k0
≡ 1 − P
k0
∂J
k
(t)
∂t
= A
k0
J
k
(t)−
t

0
Q
k0
(τ)J
k
(t
1
)dt
1

−
i

Λ
k0
E

(t)−
i

J
1
k
(t).
J
k
(t)
m
P
m
k0
X = B
m
k0
(t)T
R

J
m
k

[E

(t)]
m
X

, P
m

k0
= 1 − P
m
k0
,
J
m
k
= J
k
(L


)
m
, B
m
k0
(t) = D
0
/Λ

m
k0
[E

(t)]
m
, Λ
m
k0
= T
R
(J
m
k
D
0
).
m = 1
J
k
(z) =
∞

m=0


m

j=0
T

j
k0

Λ
m
k0
E
m+1

(z)

.
m = 0
m ≥ 1
E(t) = E

(0) exp(−iωt)
E
m+1

(t)
E
m+1

(z) =

1 −
∞

n=0

(−1)
k

mω
z + ω

n

E
m

(0)E(z),
(2.13) (2.12)
J
k
(z) =
∞

m=0


m

j=0
T
j
k0

Λ
m

k0

1 −
∞

k=0
(−1)
n

mω
z + ω

n

E
m

(0)

E(z).
J
k
(z) = σ
m
k
(z)E(z)
σ
m
k
(z) =

∞

m=0


m

j=0
T
j
k0

Λ
m
k0

1 −
∞

n=0
(−1)
n

mω
z + ω

n

E
m


(0)

.
T
j
k0
= −
i


iz −A
j
k0
+ Q
j
k0
(z)

−1
. σ
m
k
(z)
H
int
(t) = lim
δ→0
+
e

3

j=1

α,α

(r
j
)
αα

a
+
α
a
α

E
j
e
i¯ωt
P
0
Q
0
P
0
X ≡
X
αα


a
+
β
a
β


αα

a
+
β
a
β

, Q
0
≡ 1 −P
0
.
σ
ij
(ω) = −e lim
∆→0
+

α,α

(r

j
)
αα

(j
i
)
α

α
f
α

− f
α
¯ω −E
α

α
− Γ
0
αα

(¯ω)
,
Γ
0
αα

(¯ω)

P
1
X ≡
X
ββ

αα

a
+
γ
a
γ


ββ

αα

a
+
γ
a
γ

, Q
1
≡ 1 − P
1
.

σ
ijk
(¯ω
1
, ¯ω
2
) = e
2

α,α


β,β


γ,γ

(r
j
)
αα

(r
k
)
ββ

(j
i
)

γγ


(f
α

− f
α
)
¯ω
2
− E
α

α
− Γ
0
αα

(¯ω
2
)

×

δ
γα

δ
β


α
δ
γ

β
¯ω
12
− E
α

β
− Γ
1
αα

β
(¯ω
12
)
−
δ
βα

δ
γ

α
δ
γβ


¯ω
12
− E
β

α
− Γ
2
αα

β

(¯ω
12
)

,
Γ
1
αα

β
(¯ω
12
) Γ
2
αα

β

(¯ω
12
)
(ω
12
− iδ) − E
αβ
± ω
q
δ → 0
+
δ(ω
12
− E
αβ
± ω
q
•
γ
0
αα

(ω)(f
α
− f
α

) =

qµ

|M
α

,µ
(q)|
2
[(1 + N
q
)f
µ
(1 − f
α
) − N
q
f
α
(1 − f
µ
)]δ(ω −E
µα
+ ω
q
)
− [(1 + N
q
)f
α
(1 − f
µ
) + N

q
f
µ
(1 − f
α
)]δ(ω −E
µα
− ω
q
)
+

q,µ
|M
µ,α
(q)|
2
[(1 + N
q
)f
α

(1 − f
µ
) − N
q
f
µ
(1 − f
α


)]δ(ω −E
α

µ
+ ω
q
)
− [(1 + N
q
)f
µ
(1 − f
α

) + N
q
f
α

(1 − f
µ
)]δ(ω −E
α

µ
− ω
q
)
•

γ
1
αα

β
(¯ω
12
)(f
α

− f
α
) =

q,µ
|M
β,µ
(q)|
2
[(1 + N
q
)f
α

(1 − f
µ
) − N
q
f
µ

(1 − f
α

)]δ(¯ω
12
− E
α

µ
+ ω
q
)
− [(1 + N
q
)f
α
(1 − f
µ
) − N
q
f
µ
(1 − f
α
)]δ(ω
12
− E
α

µ

+ ω
q
)
+ [(1 + N
q
)f
µ
(1 − f
α
) − N
q
f
α
(1 − f
µ
)]δ(¯ω
12
− E
α

µ
− ω
q
)
− [(1 + N
q
)f
µ
(1 − f
α


) − N
q
f
α

(1 − f
µ
)]δ(ω
12
− E
α

µ
− ω
q
)
−

q,µ
|M
µ,α

(q)|
2
[(1 + N
q
)f
α
(1 − f

µ
) − N
q
f
µ
(1 − f
α
)]δ(ω
12
− E
µβ
− ω
q
)
− [(1 + N
q
)f
µ
(1 − f
α
) − N
q
f
α
(1 − f
µ
)]δ(ω
12
− E
µβ

+ ω
q
).
|µ |α


E
α

= E
µ
+ ω
12
+ ω
q
γ
0
αα

(ω) =
π
(f
α
− f
α

)
V
(2π)
3

|M(q)|
2
×

I(q
1
)[(1 + N
q
)f
−
α
(q
1
)(1 − f
α
) − N
q
f
β
(1 − f
−
α
(q
1
))] + I(q
2
)
× [N
q
f

α
(1 − f
+
α
(q
2
)) − (1 + N
q
)f
+
α
(q
2
)(1 − f
α
)]

π
(f
α
− f
α

)
V
(2π)
3
|M(q)|
2
×


I(q3)[(1 + N
q
)f
+
α

(q
3
)(1 − f
α

) − N
q
f
α

(1 − f
+
α

(q
3
))]
+ I(q4)[N
q
f
−
α


(q
4
)(1 − f
α

) − (1 + N
q
)f
α

(1 − f
−
α

(q
4
))]

,
I(q
i
) =
4π
2
(n
β
x
)
2
/L

2
x
+ 4π
2
(n
β
y
)
2
/L
2
y
+ q
2
1

4π
2
(n
β
x
)
2
/L
2
x
+ 4π
2
(n
β

y
)
2
/L
2
y
+ q
2
1
+ q
2
d

2
+
2q
2
1
(q
2
1
+ q
2
d
)
2
, = 1 4.
500
γ
1

αα

β
(ω
12
)(f
α

− f
α
) = T
1

1 − f
+
β
(q
1
) + N
q

+ T
2

f
+
β
(q
1
) + N

q

− T
3

f
α
(1 + N
q
) − f
−
α

(q
3
)(f
α
+ N
q
)

+ T
4

f
+
α

(q
4

)(1 − f
α
+ N
q
) − f
α
N
q

,
T
1
=
e
2
ω

L
z
4(2π)
3
χ

1
χ
∞
−
1
χ
0



4π
2
(n
β
x
)
2
/L
2
x
+ 4π
2
(n
β
y
)
2
/L
2
y
+ q
2
1

4π
2
(n
β

x
)
2
/L
2
x
+ 4π
2
(n
β
y
)
2
/L
2
y
+ q
2
1
+ q
2
d

2
+
2q
2
1
(q
2

1
+ q
2
d
)
2

,
T
2
, T
3
, T
4
J
µ
 =
3

ν=1
{σ
µν
(ω, t)E
0ν
(t)},
σ
µν
(ω, t) = lim
δ→+0


∞
0
dt

e
iωt

−δt

(J
ν
, J
µ
(t, t − t

)),
J
µ
(t, t − t

) = exp

i


t
t−t

H(s)ds


J
µ
exp

−
i


t
t−t

H(s)ds

.
Jµ(t, t − t

) J
ν
σ
µν
(ω, t) =
lim
δ→+0
(J
ν
, J
µ
)

δ −i(ω + η) +


i


2
(J
ν
, J
µ
)
−1

∞
0
dt

e
iωt

−δt

([U, J
ν
], [U, J
µ
]
I
)

−1

,
[U, J
µ
]
I
= exp

i


t
t−t

H
0
(s)ds

[U, J
µ
] exp

−
i


t
t−t

H
0

(s)ds

.
σ
zz
(ω, Ω) =

α
e
β(E
F
−E
α
)
L
z
√
2π

2πm
e
β
2

1/2
Γ(ω, Ω) − iω
Γ
2
(ω, Ω) + ω
2

,
Γ(ω, Ω) = (j
z
, j
z
)
−1

2
π
L
z
e
2
m
e

2

α,α

,q,µ,ν
|M
α,α

(q)|
2
q
z
J

µ
(aq)J
ν
(aq)
×
1
ω −νΩ
[e
β(ω−νΩ)
− 1]e
i(ν−µ)Ωt
e
β(E
F
−E
α
)
N
q
exp

−
β
2
2m
e
(
λ
ν
q

z
−
q
z
2
)
2

,
α(ω, Ω) =
4πL
z
e
2
cN
∗
βm
e

2
ω
2

α,α

,q,µ,ν
| M
α,α

(q) |

2
q
z
J
µ
(aq)J
ν
(aq)e
β(E
F
−E
α
)
×
1
ω −νΩ
[e
β(ω−νΩ)
− 1] cos[(ν −µ)Ωt]N
q
exp

−
β
2
2m
e

λ
ν

q
z
−
q
z
2

2

,
N
∗
α
µν
(ω) = (4π/cN) [σ
µν
(ω)] =
4π
cN
(J
µ
, J
ν
)G
µν
(ω)
ω
2
+ G
µν

(ω)
2
,
G
µν
(ω)
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
x 10
−8
−0.04
−0.02
0
0.02
0.04
0.06
0.08
Radius of quantum wire (m)
Absorption coefficient (Arb.Unit.)
Dotted line: T = 82 K
Solid line: T = 77 K
Dashed line: T = 72 K
1.5 2 2.5 3 3.5
x 10
−8
−5
−4
−3
−2
−1
0

1
2
Radius of quantum wire (m)
Absorption coefficient (Arb.Unit.)
Dotted line: T = 82 K
Solid line: T = 77 K
Dashed line: T = 72 K
60 80 100 120 140 160 180 200 220
−8
−6
−4
−2
0
2
4
6
8
10
Temperature (K)
Absorptions coefficient (Arb.Unit.)
Dotted line: R=15,8nm
Solid line: R=16,3nm
Dash
line: R=16,8nm
0 0.02 0.04 0.06 0.08 0.1
0
0.05
0.1
0.15
0.2

0.25
0.3
0.35
Photon energy (eV)
Absorption coefficient (Arb.Unit.)
Dotted line: T = 273 K
Solid line: T = 77 K
Dashed line: T = 25 K
R = 25 nm
0.5 1 1.5 2 2.5 3 3.5 4
x 10
−8
0
2
4
6
8
10
12
Radius of quantum wire (m)
Absorption coefficient (Arb.Unit.)
Dotted line: T = 272 K
Solid line: T = 273 K
Dashed line: T = 274 K
0 0.02 0.04 0.06 0.08 0.1
0
50
100
150
200

250
Photon energy (eV)
Absorption coefficient (Arb.Unit.)
Dotted line: T = 272 K
Solid line: T = 273 K
Dashed line: T = 274 K
R ~ 200 A
o
∂N
q
(t)
∂t
= G
q
N
q
,
G
q
> 0
G
q
< 0
G
(±)
q
=
Lm
e
2

3
q

α,α

|M
α,α

(q)|
2

1
e
βA
(±)
+ 1
−
1
e
βB
(±)
+ 1

,
A
(±)
= E
α

− E

F
+

2
2m
e

m
e

2
q
(∆ε + ω
q
∓ Λ) +
q
2

2
,
B
(±)
= E
α
− E
F
+

2
2m

e

m
e

2
q
(∆ε + ω
q
∓ Λ) −
q
2

2
.
G
±
q
=
Lme
βE
F
2
3
q

α

α
|M

α

α
(q)|
2
× exp

− β

m
2
2
q
2

E
α
− E
α

+ ω
q
∓ Λ −

2
q
2
2m

2

+ E
α

e
−β(ω
q
∓Λ)−1

.
Λ > ω
q
eq

E
0
m
e
Ω
> ω
q
⇒
e

E
0
q
m
e
Ω
= v

d
q > ω
q
.
G(Ω)
G(Ω)
G(R)
0 50 100 150 200 250 300
, x 10
13
Hz
0
1
2
3
4
5
G, x 10
14
s
1
0
5
10
15
20 25 30
35
, x 10
13
Hz

-4
-2
0
2
4
Gs
1
Ω
20
40
60
80
100
L
x
nm
20
40
60
80
100
L
y
nm
0
2
4
6
Gx10
14

s
1
20
40
60
80
100
L
x
nm
20
40
60
80
100
L
y
nm
20
40
60
80
100
L
x
nm
20
40
60
80

100
L
y
nm
0
2
4
G x10
10
s
1
20
40
60
80
100
L
x
nm
20
40
60
80
100
L
y
nm
0 10 20 30 40 50
, x 10
13

Hz
0
5
10
15
20
25
30
G, x 10
14
s
1
0 20 40
60
80 100
, x 10
13
Hz
-30
-20
-10
0
10
20
30
G s
1
Ω
0
5

10
15
20 25 30
 x 10
13
Hz
0
2·10
9
4·10
9
6·10
9
8·10
9
1·10
10
G s
1
T 200K, R75nm q10
7
m
1
CQRAcoustic phononDeg El.
0 2.5
5 7.5
10 12.5
15 17.5
 x 10
13

Hz
0
0.002
0.004
0.006
0.008
0.01
0.012
G s
1
T 200 K, q10
7
m
1
,R75 nm
CQROptical phonon
0 50 100 150 200
R nm
0
0.2
0.4
0.6
0.8
1
1.2
GR x 10
5
s
1
T200K, q10

8
,10
14
CQRAcoustic Phonon
0 100 200 300 400 500
R nm
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
GR s
1
T 200 K, 10
13
Hz, q10
7
m
1
CQROptical phonon
∂
∂t
b
q

t
+ iω

q
b
q

t
=
1

2

α,α

,

k
|M
α,α

(q)|
2
∞

s=−∞
J
2
s
(
Λ
Ω
)


t
−∞
dt


f
α

(

k + q) − f
α
(

k)

× exp

i


E
α

(

k + q) − E
α
(


k) − ω
q
− sΩ

(t − t

)

+

f
α
(

k) − f
α

(

k −q)

× exp

−
i


E
α

(

k) − E
α

(

k −q) − ω
q
− sΩ

(t − t

)

.
b
q

t
c
q

t
ω
q
M
αα

(q) D

αα

(q) c
q

t
b
q

t
ν
q
D
αα

(q) M
αα

(q) f
α
(

k)
|α,

k >
b
q

t

c
q

t
[ω
2
− ω
2
q
−
2

2

α,α

|M
α,α

(q)|
2
ω
q
P
0
(q, ω)][(ω −sΩ)
2
− ν
2
q

−
2

2

α,α



D
α,α

(q)


2
×
ν
q
P
0
(q, ω −sΩ)] =
4

4

α,α

,s



M
α,α

(q)


2


D
α,α

(q)


2
ω
q
ν
q
P
s
(q, ω)P
s
(ω −sΩ, q),
P
s

(ω, q) =

∞

s=−∞
J
s
(Λ)J
s+s

(Λ)Γ
q
(ω + sΩ),
Γ
q
(ω + sΩ) = 


k
f
α

(

k) − f
α
(

k −q)
ε
α


(

k) − ε
α
(

k −q) − (ω + sΩ) − iδ
.
ω
q
±ν
q
= Ω
|M
α,α

(q)|
2
|D
α,α

(q)|
2
 1
ω
ac
(q) = ω
a
+ iτ
a

; | ω
a
| τ
a
ω
a
≈ ω
q
+
1

2

α,α

| M
α,α

(q) |
2
P
0
(q, ω); τ
a
≈ −
1

2

α,α


| M
α,α

(q) |
2
P
0
(q, ω
q
)
ω
op
(q) = ω
0
+ iτ
0
; |ω
0
|  |τ
0
|
ω
0
≈ ν
q
+
1

2


α,α

| D
α,α

(q) |
2
P
0
(q, ω), τ
0
≈ −
1

2

α,α

| D
α,α

(q) |
2
P
0
(q, ω
q
).
ω (ω

0
, q
0
)
ω
(±)
±
= ω
a
+
1
2

(v
a
± v
0
)∆q −i(τ
a
+ τ
0
) ±

[(v
a
∓ v
0
)∆q −i(τ
a
− τ

0
)]
2
±
2

2

α,α

| M
α,α

(q) || D
α,α

(q) |P
N
(q, ω
q
)
2

1/2

,
ω
(−)
+
ν

q
+ ω
q
= Ω
q = 0
ω
(−)
+
> 0
E
0
> E
th
=
2m
e
Ω
2
eq

γ(ω
q
) − γ(ν
q
)
[θ(ω
q
) − θ(ω
q
− Ω)]

2
+ [γ(ω
q
) − γ(ω
q
− Ω)]
2

1/2
,
γ(ω) =
Sm
3/2
e
(1 − e
βω
)
2
√
2πβ
2
q
e

β(E
F
−E
α

)−

βm
e
2
2
q
2
[E
α

,α
(ω)]
2

,
θ(ω) =
Sm
e
e
βE
F
2πβ

e
−βE
α

− e

−βE
α

+
β
2
q
2
2m
e

E
α

,α
(ω)

,
E
th
K
s
= C
q
(ν
q
)/B
q
(ω
q
)
s = 1 K
1

K
1
=

α,α

,

k
D
α,α

(−q)M
α,α

(q)P
−1
(q, ω
q
)/(δ + iγ
0
).
γ
0
=

α,α

,


k
|D
α,α

(−q)|
2
P
0
(q, ν
q
)
K
1
=




α,α

,

k
D
α,α

(−q)M
α,α

(q)λ Γ

q
(ω
q
)/2γ
0



= Γ/2γ
0
.