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Interaction of electrons and hybridons in a free quantum wire

Scientific Journal − No27/2018

107

INTERACTION OF ELECTRONS AND HYBRIDONS
IN A FREE QUANTUM WIRE
Ta Anh Tan1, Đang Tran Chien2 Pham Van Quang3
1
Hanoi Metropolitan University
2
University of Natural Resources and Environment
3
Vietnam Commander officer training college
Abstract: In this work we used the wrapper method to solve the problem of interaction of
electrons and hybridons in a free quantum wire. Using the Dirac turbulence theory, we
established the expression for determining the rate of scattering and the recovery time for
the electrons in the wire
Keywords: Hybridons, rate of scattering, recovery time, turbulence theory.

Email: tatan@hnmu.edu.vn
Received 22 September 2018

Accepted for publication 15 December 2018

1. INTRODUCTION
In the publication [1], we have linear combinations of the oscillations in the quantum
wires that are pair 3 of the LO oscillators, IP1 and IP2. All of these oscillation modes
vibrate at the same frequency and vector. Quantization leads to the concept of a new
quantum that is hybridon hybrid. The interaction of electrons with these hybrid particles is
described as internal and external scattering in an infinite quantum well. Using the cone
method we solve the problem for electrons in quantum wires. Then using the Dirac
turbulence theory we established the expression for determining the rate of scattering and
the recovery time for the electrons in the wire.

2. CALCULATIONS
2.1. The state of the Electron in the quantum wire
Electrons moving in quantum wires are influenced by crystal circuitry and captive
power. The wave energy and energy of the electron in the quantum wire are the solution of
the Shrödinger equation


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 −ℏ 2 2

 2 m ∇ + U (r ) + V (r )  Φ (r ) = E Φ (r )



The equation of the above equation is found by the effective mass-mass method [2].
Expressions for electron wrappers in quantum wires:

F ( r, ϕ , z ) =

eik z z eilϕ
J (k r )
(π R02 L ) J m+1 (κ mn ) m mn

(1)


and the energy of the electron in the quantum wire
Erϕ = Emn =

with k m ,n =

ℏ2
( k 2z + k mn2 )
2 m*

κ mn
r

2.2. Probability of state transition
When studying the interaction of electrons with phonons, as well as the interaction of
electrons with other particle norms in solids, we need to study the probability of electron
state transfer under the effect of the small V(t)

M ml =
2


m V (t ) l


with: M i , f = a f ( t ) = aif ( t )

2

2

δ ( Em − El )

(2)

(i - is the initial state symbol) [5]. Such turbulence is

responsible for the transfer of the system from one quantum state to another. Electrons in
solids are granular and occupy single-electron states in the energy-domain structure. They
are described by the Block function, which is the area index, k is the wave vector, the spin
of the electron. In this section we only care about the electrons in the conduction band, so
the region index only appears in some cases. Furthermore, when the transfer in the spinconducting region of the electron is generally preserved, then we write the state function of
the electron normally through its wave vector. Phonon is a particle standard that describes
network oscillations. The number of phonons of the individual states is characterized by
the observable wave vectors and the j-branches of the diffusion spectrum ω j ( q ) .
The electron-phonon interaction is expressed by the phonon generation or phonon
removal (q, j) with the simultaneous transformation of the electron state k , σ

to the state

k ± q, σ . We now determine the probability of electron transfer by the optical oscillator.


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109

The probability of state displacement is determined by the formula (2), where the
disturbance is replaced by the Hamiltonian interaction between the electron and the phonon
optical. The initial states i and f are characterized by the number N(q) of the phonon and
the k-wave vector of the electron i = k i , N ( q ) ;

f = k f , N ('q ) .

The state after absorbing an optical phonon (the end state of the process) is given by
f = k f , N ( q ) − 1 and I have k f = k i − q , E f = Ei + ℏω ( q ) . The probability of state
transition for phonon absorption is given by [3]
M i , f = M k i + q ,k i = M + ( k , q ) =

2

k i + q, N ( q ) H int k i , N ( q ) − 1 δ ( E f − Ei − ℏω ( q ) ) (3)


Status after the emission of an optical phonon (end state of the process) given by

f = k f , N(q) + 1

and I have k f = k i − q , E f = Ei − ℏω ( q ) . The probability of state

transition for phonon emission is given by:
The probability of state transition for phonon emission is given by:
M i , f = M k i − q ,k i = M + ( k , q ) =


k i + q, N ( q ) + 1 H int k i , N (q )


2

δ ( E f − Ei + ℏω ( q ) ) (4)

2.3. Rate of scattering in quantum wires
From the theory for mass semiconductor we apply to calculate the scattering speed for
quantum wires. Here the wire system is a one-dimensional system so that the state of the
electron and the phonon optical are only represented by the wave vector in the z axis of the
wire.
From (3), (4), the probability that the electron's energy level in wire from i-state to
end-state in a time unit is determined as follows:
M i→ f =


k zf , N ('q ) H int N (q ) , k iz


2

δ ( ETf − ETi ± ℏωsp )

(5)

where M i → f is the scattering rate of the electron from the i-state to the f-state, N (q ) and

N('q) are the phonon distributions in the absorption and phonon emission, according to the
Bose-Ensten distribution, H int is the Hamiltonian interaction of electrons and phonons.
ETf , ETi is the energy of the electron at state x and y with:
ETi =

ℏ2
ℏ2
2
i 2
f
;
k
+
k
E
=
k m2 , n + ( k zf
(
)
m
,
n
z
T
*
*
2m
2m

(

)

(

)

2

)

(6)


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2.3.1. Hamiltonian interaction in the wires

The electron-phonon interaction in the wire is Fröhlich's interaction, so the
Hamiltonian interaction in the wire is defined as:

H = -eΦ + A P

(7)

The scalar Φ is related to mode LO and the vector A is related to mode IP and P =

−i

e
ℏ∇ . With me is the weight of the electron, the scalar Φ is related to mode LO and
me

the vector A is related to mode IP.
In this case, the scalar Φ is connected to the LO mode and the vector A is connected
to the IP mode and P is the operator. Where me is the mass of the electron, the scalar Φ is
related to the LO oscillation mode and the vector A is associated with the oscillation
mode IP.

2.3.2. The scalar Φ
The scalar Φ is defined in the relation only through the LO mode in the following
way:
EL = ρ 0u L = -gradΦ

Inside ρ 0 =

(8)

1 1
e*
; e* = MV0ω L2ε 02  −  in the cylindrical coordinates of the
ε 0V0
ε∞ ε0 

expression gradΦ is given by the expression:
gradΦ = e r

∂Φ
1 ∂Φ
∂Φ
+ eϕ
+ ez
∂r
r ∂ϕ
∂z

(9)

We obtain the following equations:

∂Φ
∂r

= Aρ0

iq Ls, p
qz

eisϕ eiq z z J 's (q sL, p r )

1 ∂Φ
s isϕ iq z z
= − Aρ0
e e J s (q sL, p r )
qz r
r ∂ϕ
∂Φ
∂z

= − Aρ0 eisϕ eiq z z J s (q sL, p r )

From these equations, identify the Φ :

(10)


Scientific Journal − No27/2018

Φ = Aρ 0

iq sL, p
qz

Φ = − Aρ 0

111

eisϕ eiq z z ∫ J 's (q sL, p r )dr

s iq z z
e J m (q sL, p r ) ∫ eisϕ dϕ
qz

(11)

Φ = − Aρ0 eisϕ J s (q Ls , p r ) ∫ eiq z z dz
or:

Φ = Aρ0

i isϕ iq z z
e e J m (q sL, p r )
qz

(12)

Substitution of the normalization coefficient, we get the expression of scalar potential
2
2 2
M ρ0ω η − s I s ( q z R0 )  isϕ iq z z
Φ=
Xe e J s (q Ls , p r )
π LΘ q 2z R0 η J s ( q sL, p R0 )

(13)

2.3.3. Potential vector
In the wire, the vector is determined by the IP mode according to the following
formula:


∂A
= E = ρp u p
∂t

(14)

whit:


 q 2Z R 02 + s 2  η 2 − s 2 I 2s ( q z R 0 ) 
 u rp = ρ p B 
I s ( q z r ) e isϕ e i q z z
2
2 2
q Z R0 η


2
2 2
 p
η − s I s ( q z R 0 )  I ( q r ) e isϕ e iq z z
 u ϕ = is ρ p B
s
z
q 2Z R 02η I s ( q z R 0 )


2 2
 2

 u p = i ρ B η − s I m ( q z R 0 )  I ( q r ) e isϕ e iq z z
z
p
s
z

q z R 0η I s ( q z R 0 )

in it,

ωsp2 −
ρ p = ρ0

Identify the integral:

ε∞ 2
ω
ε0 L

 ε∞  2
 1 −  ωL
 ε0 

(15)


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q 2Z R02 + s 2  η 2 − s 2 I s2 ( q z R0 ) 
 Ar = i ρp B
I s (q z r )eisϕ eiq z z
2 2 2
ωq Z R0η


 η2 − s 2I 2s ( q z R0 ) 

ρ
A
=

s
B
I s (q z r )eisϕ eiq z z
 ϕ
p
2 2
ω
η
q
R
I
q
R
Z 0
s ( z 0)


2
2 2
 A = − ρ B η − s I s ( q z R0 )  I (q r )eisϕ eiq z z
p
 z
ωq z R0η I s ( q z R0 ) s z


(16)

Substituting the standardized coefficients into ones:


q 2Z R02 + s 2  η 2 − s 2I 2s ( q z R0 ) 
M
 Ar = i ρ p
ωX
I s (k z r )eisϕ eiq z z
2 2 2
π
ω
L
Θ
R
q
η

Z 0

2
2 2
η − s I s ( q z R0 ) 
M

ωX  2 2
I s (q z r )eisϕ eiq z z
 Aϕ = − s ρp
π LΘ
ωq Z R0η I s ( q z R0 )


η 2 − s 2I 2s ( q z R0 ) 
M
 A = −ρ
ω
X
I s (q z r )eisϕ eiq z z
p
 z
π
ω
η
Θ
L
q
R
I
q
R
z 0
s ( z 0)


(17)

2.3.4. Hamiltonian interaction
Momentum P is defined as follows:
P=-

iℏe
iℏe  ∂

∂ 
∇=+ eϕ
+ ez 
 er
me
me  ∂r
∂ϕ
∂z 

(18)

ie 


∂
+ Az  Φ
 Ar + Aϕ
∂r
∂ϕ
∂z 
me 

(19)

Have:

H = -eΦ -

Find the Hamiltonian interaction as follows:

H = −e∑ Ξ
s, p

M X e isϕ e iq z z
π L q z R 2η

 ρ 0 ω R 0η
L
 J q L R J s (q s , p r ) +
 s ( s, p 0 )



2
2
2

 − i  q Z R 0 + s  I s ( q z r ) ∂ r +
ρ
i

p
+

 m 
q z R 0η



e
+
I (q r )
+
I (q r )

 I (q R ) s z ∂ϕ I (q R ) m z ∂z
s
z 0
s
z 0






 




 

(20)

put:

H int = −e∑
s, p


⌢ ⌢
ℚ {a + + a }
2π Lω

(21)


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113

Inside:


iℏ ρ p
e isϕ e iq z z  ρ 0ω R0η
ℚ=Ξ
J s (q sL, p r ) +

2
L
q z R0 η  J s ( q s , p R )
me





2
2
2
 − i  q Z R0 + s  I s (q z r ) ∂ r +



q z R0η


∂ 
+

 I ( q R ) I s (q z r ) ∂ϕ + I ( q R ) I m (q z r ) ∂z  
s
z 0
s
z 0



(22)

2.3.5. Scattering speed
From (2) and (5) we have:

Mi→ f



⌢ ⌢
mnk zf , N( q)' − e
ℚ{a + + a} N(q) , mnkiz
=


2π Lω s, p

2

δ ( ETf − ETi ± ℏωsp )

(23)

An intrasubbling scattering implies that one electron from the beginning state absorbs
There are no electrons. N ( q ) and

or emits one phonon and moves to the final state. mnk zf

N '( q ) is the function of the phonon distribution in phonon delivery and absorption. Here we

consider multiple systems so they follow the Bose-Einstein distribution. The quantum
transfer probability in (5) will be determined:
For phonon absorption we have:

Mi, f = e

2

2

N(q)

mnk

Lωsp

f
z

∑ℚ mnk

δ ( ETf − ETi − ℏωsp )

i
z

s, p

(24)

For the phonon emission process we have:

Mi, f = e

2

N(q) + 1
Lωsp

2

mnk

f
z

∑ℚ mnk
s, p

i
z

δ ( ETf − ETi + ℏωsp )

(25)

matrix element

G i , f = mnk zf

∑ ℚe ϕ e
is

iq z z

mnk zi

s, p

Inside:
1

m nk

f
z


2
1
− im ϕ − i k zf z
= 
e
 J m (kr )e
2
 π L R 0 2 J m + 1 (κ m n ) 

(26)


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1

mnk

i
z


2
1
imϕ ik iz z
=
 J n (k mn r )e e
2 2
 π LR0 J m +1 (κ mn ) 

(27)

Instead (27) to (26) we have:


1
Gi , f = ∑ 

2 2
s,p  π LR0 J m+1 (κmn ) 

R L 2π

∫∫ ∫ J

f

m

i

(k mn r )e−imϕ e−ik z z ℚeisϕ eiqz z J m (k mn r)eimϕ eik z z rdrdϕdz

(28)

0 0 0

put:
R

£1 = ∫ rJ s (q r )J (k mn r )dr;
L
s, p

2
m

0

R

£ 2 = ∫ I s (q z r )J 2m (k mn r )dr
0

R

R

0

0

(29)

£3 = ∫ rJ m (k mn r )I s (q z r )J m+1 (k mn r )dr; £ 4 = ∫ rI s (q z r )J 2m (k mn r )dr
it will be obtained:

G i, f = ∑
p

 ρ ω R η
ℏρ

0
0
£1 + p

42
L
q z ηR0 m+1 (κ mn )  J 0 ( q 0 p R0 )
me


 2 2
k iz q z R0 η  
2 4
m
q
R
£

k
q
R
£

£4  
 Z 0 2
mn Z 0 3
I 0 ( q z R0 )  


(30)

We consider, in approximate terms, the contribution of the first solution of the Bessel
function to the largest for the Hamiltonian interaction, and to examine the intrasubband
scattering for the electrons in the lowest energy region ie the regional index m = 0 and
n = 1:

G i, f =

 ρ ω R η
ℏρ p 
k iz q z R0η  

2 2
0
0
£

k
q
R
£

£4  


1
01 Z 0 3
q zη R02 J12 (κ 01 )  J 0 ( q0L p R0 )
me 
I 0 ( q z R0 )  



(31)

Inside:
L 2


 2 L
2 
 2 2

 M ω01  ( q 01 )
 J1 (q 01 R ) + ε ( ω ) ρ 2  q Z R + q z Rη  I 2 (q R ) 
Ξ=
+
1
p
1
z
2
L
 J 20 ( q 01
I 02 ( q z R ) 
η2
R)

 V0  q z



η = I 0 ( k z R0 ) − R0q z I1 ( q z R0 ) 



1
2

(32)

(30) into (24) and (25) we will find the electron scattering rate determining method for
the phonon absorption and emission states as follows:
For phonon absorption we have:


Scientific Journal − No27/2018

Mi+, f = e2

N( q)
Lω01

115

2

Gi , f δ ( ETf − ETi − ℏω01 )

(33)

For the phonon emission process we have:

N(q) +1

2

Gi, f δ ( ETf − ETi + ℏω01 )

(34)

δ ( ETf − ETi ∓ ℏω01 ) = δ ( ETf − ETi ∓ℏω01 )

(35)

Mi−, f = e2

Lω01

With delta function:

We find the general expression for the recovery time:

1

τ if

=

e2

2πω012 ∫

Gi, f



2
 ℏ2  i

i 2

N
δ
 (q)  * ( k z + q z ) − ( k z )  + ℏω0 p  +

2
 2m



 dq z
2
+ N + 1 δ  ℏ  k i + q 2 − k i 2  − ℏω  
( z )  0 p 
 * ( z
z)
(q )

 2m



(

)

To integrate by qz we proceed as follows:
From
2
2
ℏ2
ℏ2
i
k
+
q
=
k i ± ℏω01
(
)
z
z
*
* ( z)
2m
2m

Or

(36)

ℏ 2 2 2ℏ 2 i
qz ±
k z q z cosϑ ∓ ℏω01 = 0 trong đó ϑ is the angle between qzand kzi.
2 m*
2 m*

According to [4] we have:

q = −k cos ± k 2cos 2ϑ + x 2
z
z
 z (1)

q z ( 2) = k z cos ± k 2z cos 2ϑ − x 2

(37)

ℏ 2 x2
= ℏω01
2 m*

(38)

Put:

Pay attention to the distribution function of the phonon:

N (q ) = N( q )

 k BT
 ℏω
 01
ℏω
≈ N(q) + 1 = 
− 01
1
 kT
≈ e k BT
 ℏωB 01
−1
e

khi k BT >> ℏω01
(39)

khi k BT << ℏω01


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Consider for different temperature ranges:

At high temperatures ( kBT >> ℏω01 ) or is k z >> x , We have:
q z ( min ) = 0; q z ( max ) = 2k iz

(40)

Because of:

kBT >> ℏω01
N( q ) = N ( q ) ≈ N ( q ) + 1 =
m*
=
τ if 2πω012 8π 2 ℏ 2ω01k iz 3
1

e2

1

2 k iz

∫ ( 2 N( ) + 1) G
q

k BT
>> 1
ℏω01
2

q3z dq z attention to

i, f

0

m*e2
kT
= 3 2 3 i3 B
τ if 8π ℏ ω01k z ℏω01
1

2 k iz



2k BT
kT
+1 ≈ B
ℏω01
ℏω01

2

G i , f q3z dq z

(41)

0

At low temperature ( k BT << ℏω0 p ) or is k iz << x , The electron's energy is much
smaller. The Schrödinger equation with the energy of the phonon should only have a
significant phonon absorption. The phonon emission process is very small that can be
ignored. To match the above process, the word (37) is obtained:
q ( min ) = k 2 + x 2 − k
z
z
 z

2
2
q z ( max ) = k z + x + k z
We have

(42)

N ( q ) = N (q ) ≈ N (q ) + 1 ≈ e

m*e2
= 3 2
τ if 4π ℏ ω01k iz 3
1

1

τ if

* 2

=

me
3 2 3 i3
8π ℏ ω01k z

(

k 2z + x 2 + k z


k 2z + x 2 −k z

)

(

k 2z + x 2 + k z

)

(

k 2z + x 2

k BT

N( q )



e
−k z

ℏωsp

)

(





)

2
q 2z − x 2
G i , f q2z dq z
qz

ℏω01
k BT

Gi, f

2

(q

2
z

− x 2 ) q z dq z

(43)

3. CONCLUSION
Using the envelope method, we solved the problem for the electrons in the quantum
wires. Then using the Dirac turbulence theory we established the expression for


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117

determining the rate of scattering and the recovery time for the electrons in the wire;
however, the number of scattering rates and the recovery time for a particular quantum
wiring are not yet calculated. Therefore, we can further develop by applying numerical
calculations for scattering rates and recovery time, as well as applying dispersion lines to
other polarizable semiconductors. It is possible to use the scattering rate and the recovery
time of electrons in the wire to calculate the electron mobility, the dielectric constant.

REFERENCES
1.

Đặng Trần Chiến, Tạ Anh Tấn (2018), “Hạt lai trong dây lượng tử tự do”, - Tạp chí Khoa học,
Trường Đại học Thủ đô Hà Nội, số 24, KHTN&CN, tháng 6/2018.

2.

Nguyễn Văn Hùng (2000), Lý thuyết chất rắn, - Trường Đại học Quốc gia Hà Nội.

3.

Nguyễn Quang Báu (2001), Lý thuyết bán dẫn, - Trường Đại học Khoa học Tự nhiên - Đại học
Quốc gia Hà Nội.

4.

A. Anselm, “Introduction to semiconductor theory”, - Revised from the 1978 Russian edition.

5.

N. C. Constantinou and B.K. Redley (1989), Interaction of Electron the confined LO phonons
of a free-standing GaAs quantum wire,- Phys. Rev. B. 1989. 41: p.10627.

TƯƠNG TÁC ELECTRON-HYBRIDON TRONG DÂY LƯỢNG TỬ
Tóm tắ
tắt: Trong bài báo này chúng tôi sử dụng phương pháp hàm bao để giải bài toán
tương tác của electron trong sợi dây lượng tử. Sau đó dùng lý thuyết nhiễu loạn Dirac,
chúng tôi thành lập biểu thức xác định tốc độ tán xạ và thời gian hồi phục cho electron
trong dây.
Từ khóa:
khóa Hàm bao, tốc độ tán xạ, thời gian phục hồi, lý thuyết nhiễu loạn.



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