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3/5/2020 Amany Elgarawany PHD Researcher 1

Dynamical effect of laser bumped electron- hole semiconductor

By

Amany Zakaria Elgarawany

Assistant Lecturer of Mathematics, Department of Basic Sciences,

• Modern Academy for computer sciences

5th SPSP 5th Spring Plasma School at Port Said 1- 5 March 2020

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I’ m Mathematician not Physicist

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Outline

Introduction

Electromagnetic wave effect Objective of the Paper The wave Equation Dispersion Relation Sagdeev Potential Modified NLSE Appendix

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An Electron is defined as a negative charge or negative atomic particle. A Hole as a vacancy left in the valence band because of the lifting of an electron from the valence band to a conduction band.

Introduction

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• Hole is a positive-charge, positive-mass quasiparticle.
• Holes can move from atom to atom in semiconducting

materials as electrons leave their positions.

• The mobility of electrons is higher than that of the holes,

because the effective mass of electron is less than a hole. An electron-hole pair For every electron raised to the conduction band by external energy, there is one hole left in the valence band, creating what is called an electron-hole pair.

Introduction

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Recombination

Occurs when a conduction-band electron loses energy and falls back into a hole in the valence band.

• Recombination rate is controlled by the minority carrier

lifetime.

• Recombination

mechanisms for materials is highly important for the optimization of semiconductor devices such as solar cells and light emitting diodes.

Introduction

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Semiconductors

are the materials which have a conductivity between conductors (generally metals) and non-conductors or insulators (such ceramics).

Introduction

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Types of Semiconductors  Intrinsic Semiconductor ( holes = electrons )  Extrinsic Semiconductor ( excess or shortage of electrons )  N-Type Semiconductor ( Mainly due to electrons)  P-Type Semiconductor ( Mainly due to holes )

Introduction

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Semi-conductor materials

Material Symbol Usage

Germanium Ge radar detection diodes - first transistors Silicon S integrated circuits -insulation layers Gallium arsenide GaAs high performance RF devices Silicon carbide SiC yellow and blue LEDs.

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Material Symbol Usage Gallium Nitride GeN microwave transistors -microwave ICs-blue LEDs Gallium phosphide GaP produce a green light-(+N ) yellow- green- (+ZnO) red. Cadmium sulphide CdS photoresistors - solar cells

Semi-conductor materials

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EMW effect on Semi-Conductor E-H particles

• Recombination Process
• Increase in current flow
• Diffusion of charges
• Decrease the energy gap
• Decrease the mobility of

carriers

• Increase the collision rate
• Increase the conductivity

(Intrinsic)

• Decrease the conductivity

( Extrinsic)

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• This

study introduce the mathematical model

• f

the interaction between electromagnetic field and electron hole particles.

• Using Maxwell’s equations along with e-h fluid equations

that contain laser field effect, we derive an evolution wave equation describing the system, which called a modified nonlinear Schrödinger equation (mNLSE).

• The mNLSE is reduced to an energy equation containing the

Sagdeev potential describing the localized propagating pulses in semiconductor.

Objectives

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The wave equation for homogeneous plasma

2 2 2 2

1 4 (1) 4 (2) , (3)

k k

e e e k k k e e k k

A J A c t c when J en v q n v and en q n                   

 

• From Maxwell's Eqs. , we can extract the wave equation ,

which describe the propagation of laser field in the plasma.

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Laser - Electron – Hole Interaction

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Laser - Electron – Hole Interaction

• We can extract the wave equation , which describe the

propagation of laser field in the electron-hole plasma. (4)

• And the Poisson eq.

(5)

• Where the velocity with relativistic effect

(6)

2 2 h h e e 2 2

1 A 4 π e

• A = -

(n v n v ) c t c       2

e h

= - 4 π e(n - n )

e h e e h h

e A e A v = , v = - m c γ m c γ

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• Also from the relativistic fluid equation

(7) at (8) ,we derive the densities (a) for electron (9) (b) for hole (10)

1 ( . )

t e e

V B p v p e E P c n

     

       

      

2 e e e e0 e e e

m c q n = n exp

• (γ - 1) +

T T       

2 h h h h0 h h h

m c q n = n exp

• (γ
• 1) +

T T

( )

/ ( ) ,

e e

kT pressure

p e A c momentum P n   

Laser - Electron – Hole Interaction

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• The vector potential of the light takes the form

(11)

• The differential equation after substituting in eq. (4) takes

the form (12)

 

1 A = (r, t) (x + i y)exp i k.r - i ω t 2 

2 2 2 2 e h 2 2 2

N N i + i + 1 - + M 1 - = 0 2 2 1 + a 1 + M a

pe

c k c t                               

Laser - Electron – Hole Interaction

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2 e h 2 2 2

N N a 1 1 i + a + 1 - + M 1 - a = 0 τ 2 2 1 + a 1 + M a                     

• The resulting form from the wave equation after the scaling, the

nondimensional form can be reduced to a modified NLSE (13)

• Where

(14)

   

 

2 2

a , / , / ,

pe g pe g pe

m c t e r c u u c k             

Laser - Electron – Hole Interaction

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Modified NLSE

• Instability
• f system
• Solution of

NLSE by HPM

• The Sagdeev

potential.

• The light

amplitude.

Modified NLSE

• NL dispersion

relation

• The modulat-

ional instability growth rate

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The nonlinear dispersion relation - Growth rate

• At the potential takes the form

(15) Substituting in the differential eq., (16)

• The nonlinear frequency shift take the form

(17)

 

1 1

a = ( )exp i , a a where a a    

2 e h 2 2 2

N N a 1 1 i + a + 1 - + M 1 - a = 0 τ 2 2 1 + a 1 + M a                     

e h 2 2 2

N ( ) N ( ) 1 1 - + M 1 - 2 1 + 1 + M a a a a                      

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• We linearize the differential eq. (16) At

with respect to where is the frequancy of the low frequancy modulations, and X , Y are real constants.

• After Tayler series , and linearization

(18)  

1 = (

)exp i . a X iY k i     

1

a

* * * * e 1 1 1 1 2 2 2

N ( ) N ( ) ( ) , ( ) 1 + 1 +

h

a a a a a a a M a          

The nonlinear dispersion relation - Growth rate

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• After substituting in Eq. (16) with respect to , the

differential eq. takes the form (19)

• Let
• Assume the plane wave

(20)

1

a

 

2 * * 1 1 1 1

1 i + ( ) = 0 τ 2 2 a a a M a a        

1 1 *

a U i V a U i V     

. .

,

i k i i k i

U u e and V v e

       

 

The nonlinear dispersion relation - Growth rate

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• The nonlinear dispersion relation is

(21)

• The modulational instability growth rate

(22)

2 2 2 * 0 (

M ) 2 2 k k a            

2 * 0 (

M ) 2 2 k k i a               

The nonlinear dispersion relation - Growth rate

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• The nonlinear system at

(23) Substituting in the differential eq.,

• The integration of a modified NLSE gives this form

(24)

 

a( , ) = ( )exp z w z i    

 

2

1 2 ( ) ( ) w z w    

The Sagdeev potential

2 e h 2 2 2

N N a 1 1 i + a + 1 - + M 1 - a = 0 τ 2 2 1 + a 1 + M a                     

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• Where is the Sagdeev potential

(25)





 

2 3 2 4 3 3 4 6

1 1 1 8 1 8 ( ) ( ) ( ) ( )

e h

w w M E H M w E M H M M w O w              

( ) w 

The Sagdeev potential

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• Fig. (1)

is a function of light amplitude . It have the figure of Sagdeev potential in the (+ve) and (-ve) values

• f amplitude. At the values ;

( ) w 

w

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 





 

2 2 3 2 4 3 3 4 6

1 1 1 1 2 8 1 8 ( ) ( ) ( ) ( )

e h

w z w M E H M w E M H M M w O w               

• Then eq. (24) takes the form

(25)

• The light amplitude w is

(26)

 

2 3 2 3 3

2 1 8 1 1 ( ) ( )

e h

dw dz A w w at A M E H M E M H M M               

 

The wave amplitude (EMW)

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• Fig. (2)

The light amplitude is a function of the distance , which gives a soliton wave at all values of the distance . At the values ;

( ) w z

z

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Modified NLSE

2 e h 2 2 2

N N a 1 1 i + a + 1- + M 1- a = 0 τ 2 2 1 + a 1 + M a                     

• The last differential eq. takes the form

(27)

• After Tayler series, the differential eq. takes the form

(28)

2 2

a 1 i + a + a a = 0 τ 2    

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• Fig. (3)

This case, with positive, gives unstable envelope soliton, which has Bright soliton solutions.



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HPM for solving mNLSE

2 e h 2 2 2

N N a 1 1 i + a + 1 - + M 1 - a = 0 τ 2 2 1 + a 1 + M a                     

• The last differential eq. takes the form

(27)

• After Tayler series takes the form

(28)

• Using HPM

(29)

2 2

a 1 i + a + a a = 0 τ 2    

2 2

a 1 i + a + a a = 0 τ 2 p          

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HPM for solving mNLSE

• The solution take the formula

(30) ,where p is embedding parameter

• Substituting from the formula (30) in the eq. (29) ,and

compare the coefficients p , we find

2 3 1 2 3

a = ... a p a p a p a     1 p  

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1 2 2 1 2 2 2 1 1 1 1 3 2 3 2 2 2 2 1 1 1 1 1 1

: ( , 0) 1 : ( ) 2 1 : ( ) 2 1 : 2 ( )

i z

p i D a a a z e p i D a a a a p i D a a a a a a a a a a a p i D a a a a a a a a a a a a a a a a a a a a

   

                         

HPM for solving mNLSE

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1 1 2 1 1

, 1 2

j j i j j i k j i k i k

then a i a a a a 

        

        

 

(31)

2 1 2 3 3

1 1 3 , , ( ) , 2 2 2 1 3 1 ( ) 2 4 2

i z i z i z i z

a e a i e a i i e a i i e                                                

Then the series takes the four terms

HPM for solving mNLSE

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The potential by HPM is (32)

HPM for solving mNLSE

1 1 2 3

a = ...

p

Lim a a a a



   

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Appendix

 

2 2 2 2 2 2 2 2 2 2 2 3 3

4 4 , , , exp[ (1 1 )] exp[ (1 1 )] 1 1 1 1 1 1 (1 ) (1 ) 4

e eo ho pe ph h e h e h e h e h e e h h e h e h e h e h

m e n e n M m m m E H M M N a N M a M a M a M M E H M M M M E H M                                                                   

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1 2 2 2 1 1 2 2 2 2 1 2 2 * 1 1 2 2 2 2 2 2 1 2 2 2 3 3 2 2 2

(1 (1 ) (1 ) 1 2(1 ) (1 ) (1 (1 ) (1 ) 1 2(1 ) (1 ) (1 (1 ) (1 ) 2(1 ) 2(1 )

e e h h e e

E H M a E H a a ME MH a MH ME M a M a a E H H a M a a a                                                                          

2 2 1 1 2 2 2 2 2 2 1 2 3 2 2 * 3 3 1 1 2 2 2 2 2 2 2 2 2 2 2

2(1 ) (1 ) (1 (1 ) (1 ) 2(1 ) (1 ) 2(1 ) 2(1 )

e h h h

H a M M a a a M ME MH E a M a E a M M a a M a M a                                                       

Appendix

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