Interaction of electrons and hybridons in a free quantum wire

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.

INTERACTION OF ELECTRONS AND HYBRIDONS

INTERACTION OF ELECTRONS AND HYBRIDONS

IN A

IN A FREE QUANTUM WIRE FREE QUANTUM WIRE FREE QUANTUM WIRE

Ta Anh Tan 1 , Đang Tran Chien 2 Pham Van Quang 3

1 Hanoi Metropolitan University 2

University of Natural Resources and Environment

3

Vietnam Commander officer training college

Abstract:

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:

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

2

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

i z il

m mn

m mn

e e

R L J

ϕ

ϕ

=

k

r k (1)

and the energy of the electron in the quantum wire

2

* 2

m

with , mn

m n

r

κ

=

k

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)

δ

M

ℏ (2) with: Mi f, = a f ( )t 2= a if ( )t 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 spin-conducting 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 ωj( )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 ,σ

±

k q We now determine the probability of electron transfer by the optical oscillator

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, ( ) ; f, ( )'

i = k Nq f = k Nq The state after absorbing an optical phonon (the end state of the process) is given by

( )

f

f = k Nq − and I have f i

= −

k k q, E f =E i+ℏω( )q The probability of state transition for phonon absorption is given by [3]

2

i i

+ +

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

( )

f

f = k Nq + and I have f i

= −

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

The probability of state transition for phonon emission is given by:

2

i i

+

−

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:

int

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

'

Nq are the phonon distributions in the absorption and phonon emission, according to the Bose-Ensten distribution, Hint is the Hamiltonian interaction of electrons and phonons ,

f i

T T

E E is the energy of the electron at state x and y with:

,

* 2

E m

= ℏ k + k ; 2 ( 2 ( )2)

,

* 2

E m

= ℏ k + k (6)

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:

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

e

e

i

m

− ℏ∇ With m e is the weight of the electron, the scalar Φ is related to mode LO and the vector Ais 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 m e 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:

Inside

*

e

V

  in the cylindrical coordinates of the

expression gradΦ is given by the expression:

1

grad

We obtain the following equations:

z

z

z

L

s p is i z L

s s p z

i z

s s p z

i z

s s p

i

r

s

z

ϕ

ϕ

ϕ

ρ

ρ ϕ

ρ

∂

=

∂

∂

= −

∂

∂

= −

∂

q

q

q

q

q

q

(10)

From these equations, identify the Φ:

z

z

z

L

s p is i z L

s s p z

i z L is

m s p z

i z

is L

s s p

i

s

ϕ

ϕ

ϕ

ρ

ρ

=

= −

= −

∫

∫

∫

q

q

q

q

q

q

(11)

or:

z

i z

m s p z

i

Substitution of the normalization coefficient, we get the expression of scalar potential

2 2

0 0

, 2

z

s z is i z L

s s p L

M

ϕ

η

ρ ω

= Θ

2

q

I q

2.3.3 Potential vector

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

∂

−

∂

p p

A

(14)

whit:

2 2 0

2 2

0

2 2

2 2

0

( )

( )

( )

z

z

z

Z

s z

R

ϕ

ϕ ϕ

ϕ

η ρ

η η

ρ

η η ρ

η



−

=





−





2

q 2

2

q

2

q

q

p

p

p

(15)

in it,

0 0

2 0 1

p

L

ε

ε

ε ω ε

∞

∞

−

=

−

Identify the integral:

2 2 0

2 2

0

2 2

2 2

0

( ) ( )

( )

z

z

z

s z Z

s z

z s z

R

ϕ

ϕ ϕ

ϕ

η ρ

ρ

η ρ



−

= −





 = −





2

q

2

q

2

q

I q q

I q

I q

I q

p

p

p

A

A

A

(16)

Substituting the standardized coefficients into ones:

2 2 0

2 2

0

2 2

2 2

0

( ) ( )

( )

z

z

z

s z Z

s z

z s z

M

M

M

r e e

ϕ

ϕ ϕ

ϕ

η

η

η

Θ



−

= −



Θ



 = −



2

q

2

q

2

q

I q

I q

p

p

p

A

A

A

(17)

2.3.4 Hamiltonian interaction

Momentum P is defined as follows:

Have:

ie e

Find the Hamiltonian interaction as follows:

,

,

0

z

L

s s p L

s s p

i z is

z e

R

r R

M e e

R s

ϕ

η η

ϕ

+

∂

J q

J q X

q

q

ℏ p

(20)

put:

s p

L

π ω

+

int

Inside:

0

, 2

0

i z is

L

s s p L

z

R

e e

r

R s

η η

η

ϕ

ℏ

(22)

2.3.5 Scattering speed

From (2) and (5) we have:

2 '

,

2

2

s p

L

π

π ω

+

ℏ

(23)

An intrasubbling scattering implies that one electron from the beginning state absorbs

or emits one phonon and moves to the final state f

z

mnk There are no electrons N and ( )q

'

( )q

N 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:

2 2

,

,

s p sp

N

ω

For the phonon emission process we have:

2 2

,

,

1

s p sp

N

ω

+

matrix element

,

z

i z

s p

Inside:

1 2 2

1

( )

f

i z

m m n

L R

ϕ

−

− +

k

J

1 2

2 2

1

( )

i z

i z

m mn

LR

ϕ

k

Instead (27) to (26) we have:

2

1

( )

R L

m mn

LR

π

−

−

+

put:

2

L

(29)

it will be obtained:

( )

i

z z

ρ

ρ ω κ

+

ℏ p

(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:

2 2

( )

i

z z

R

ρ

k q 2Ξ

ℏ p

(31) Inside:

( )

1

1

L

R

ε ω ρ

η

−

2

I q

(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:

0 k R z 0 R0 z 1 z R0

( )

2 2

01

f i

N

ω

For the phonon emission process we have:

2

01

N

ω

With delta function:

δ − ∓ ℏω =δ − ∓ ℏω (35)

We find the general expression for the recovery time:

( )

0

*

,

0

*

2 1

2

1 2

N

m e

d N

m

τ πω

∫

q

q

ℏ

ℏ

ℏ

ℏ

To integrate by qz we proceed as follows:

From

2 2*( )2 2 2*( )2 01

ℏ

(36)

Or 2 2 2

01

2

i

z z z c

∓ ℏ trong đó ϑ is the angle between q z and k z i

According to [4] we have:

( )

( )

z 1

z 2

z z

z z

ϑ ϑ







Put:

2

x

ℏ

Pay attention to the distribution function of the phonon:

01

01 01

01

1

1 1

B B

B

B

k T

B

k T

k T

khi k T

e khi k T e

ω

ω

ω ω

ω

−



>>







−



ℏ

ℏ ℏ

ℏ

(39)

Consider for different temperature ranges:

At high temperatures (k T B >> ℏω01) or is kz >>x, We have:

q q k (40) Because of:

k T B >> ℏω01

1 k T B 1

ω

= ≈ + = >>

ℏ

( )

2

3 ,

i z

i f z z i

z

k

k

2k T B 1 k T B

ω + ≈ ω

2

3 ,

3 2 3 3

1 8

i z

B

i f z z i

z

k T

m e

d

k

k

At low temperature (k T B << ℏω0p) or is i

z <<x

k , 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:

z

z

min max

x x







q k k (42)

We have ( ) ( ) ( ) 1

sp B

k T

ω

−

ℏ

( )

( 2 2 )

z

2 2 z

2 ,

1 4

z

z

x

z

i f z z i

x

m e

+ +

+ −

−

k k

q

k k

q

ℏ

2 2

2 2 z

,

3 2 3 3

1 8

z B

z

x

k T

i

m e

ω

+ +

−

+ −

k k

k k

k

ℏ

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

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

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Trường Đại học Thủ đô Hà Nội, số 24, KHTN&CN, tháng 6/2018

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Quốc gia Hà Nội

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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 t ắắắắtttt: 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

T ừ ừừ ừ khóa khóa khóa: Hàm bao, t ốc độ tán xạ, thời gian phục hồi, lý thuyết nhiễu loạn