View of DIFFUSION INDUCED PARAMETRIC INTERACTIONS OF ABSORPTION AND AMPLIFICATION IN QUANTUM SEMICONDUCTOR PLASMAS WITH SDDC

1 DIFFUSION INDUCED PARAMETRIC INTERACTIONS OF ABSORPTION AND

AMPLIFICATION IN QUANTUM SEMICONDUCTOR PLASMAS WITH SDDC Nishchhal Yadav¹, P. S. Malviya², Manpreet Kaur³ and C.S. Sharma⁴ School of Studies in Physics, Vikram University, Ujjain 465010, India¹³⁴ Department of Physics, Govt. J.N.S. Post Graduate College, Shujalpur 465333, India² Abstract: We have presented an analytical investigation of diffusion induced parametric interactions in quantum semiconductors with strain dependent dielectric constant (SDDC) by using the quantum hydrodynamic (QHD) model. The diffusion induced third order (DITO) complex susceptibility and threshold pump due to a nonlinear current density have been determined by the coupled mode theory. It is found that diffusion induced parametric interactions (PIs) shows usual dispersion characteristics of absorption and amplification.

The real and imaginary part of DITO susceptibility illustrates variation with pump electric field

E

₀, wave number

k

and plasma frequency



_p, respectively. The qualitative behaviour of real part of DITO susceptibility  3



realfound to be in both negative and positive regime but imaginary part of DITO susceptibility



_imag^{ 3} _. Shows only negative regime at specific scale range of parameter.

1 INTRODUCTION

Nonlinear parametric interactions (PI) in semiconductors play a pivotal role in physical optics by providing key functions for a broad spectrum of devices. The nonlinear interactions of co propagating beams have been field of interest since origin of physical optics. These processes provide us useful information regarding the physical properties of the host medium. It is an important mechanism of nonlinear mode conversion from electromagnetic to electrostatic and from high to low frequency waves and vice versa [1-5].

Variety of nonlinear effects observed in the interaction of high-power electromagnetic waves with quantum semiconductor plasmas.

Acousto-optic (AO) materials are promising candidate in extensive research due to its potential applications in optoelectronics. AO interactions in high dielectric semiconductors are playing an increasing role in optical modulation and beam steering [6- 7]. The most direct approach to this problem is to tailor a new material with more desirable AO properties [8-9]. An alternative method for amplifying the acoustic wave within the existing device is also being pursued [10]. The acoustic wave diffracts the light beam within the active medium and provides an effective mechanism for a nonlinear optical response in AO devices. In most cases of nonlinear optical interaction, the diffusion is normally ignored specially in quantum plasma. It is found that the high mobility charge carrier become more relevant in semiconductor technology as they (charge carriers) travel significant distances before recombining. Therefore inclusion of carrier diffusion in theoretical studies of nonlinear PIs seems to be very important from the fundamental point of views as well as applications. So many researchers attracted this area of research in the last decades. It is expected that the inclusion of diffusion alter the third order optical susceptibility and hence significantly changes dispersion and transmission of the incident radiation in medium [11- 12].

Due to the electron density property of semiconductor materials, the great degree of miniaturization of devices and nano-scale objects, the quantum effects are important for the understanding electrostatic and electromagnetic mode in semiconductor plasmas. In quantum effects, the de Broglie wavelength of charge carriers can be comparable to the spatial variation of the doping profiles and the tunnelling effects play a central role in the behaviour of electronic components. From quantum information to ultrafast all-optical processing and pollution monitoring, a large number of applications might benefit from the full convergence of Optics and Electronics at chip-scale level [13]. Haas et al. and others have developed the quantum hydrodynamic model (QHD) for quantum plasmas [14-15]. The

2 QHD model for plasmas is used to derive a new dispersion relation for dust acoustic wave and dust acoustic solitary waves [16 -17]. The quantum picture of a system may include Fermi pressure, the quantum diffraction and the quantum statistics. The exchange and the correlation potentials due to spin effects in the plasmas gain importance, especially in the cold dense plasmas such as the ultra small electronic devices [18]. The inclusion of diffusion in QHD, it is found to alter the diffusion induced third order (DITO) optical susceptibility³ and significantly changes dispersion and transmission of the incident radiation in the medium [19].

It appear from the available literature that in most of the previous reported works in field of parametric dispersion in quantum plasma SDDC effect has not taken into account.

Thus, motivated by the above we have focused our attention on the modifications occurred by including the SDDC effect and we have analytically investigated the quantum effect on parametric process in material with high dielectric constant. The process is characterized by the third order susceptibility induced due to nonlinear current density in the high dielectric constant semiconductor plasma medium.

2 THEORETICAL FORMULATIONS

We have considered the well known model used in the analysis is the QHD model of the homogeneous one-component semiconductor crystal with SDDC satisfying the condition that k_al<<1 (where k_a is the acoustic wave number and l is the carrier mean free path).

This model proves to be suitable for the present study as it simplifies our analysis, without loss of significant information, by replacing the streaming electrons with an electron fluid described by a few macroscopic parameters like average carrier density, average velocity, etc. Authors assume a spatially uniform (k₀ 0) pumpExˆE₀exp



i



₀t



.The basic equations describing PI of the pump with the medium are as follows:

m eE t

v⁰

 v

₀

 

⁰







(1)

3 1 3

0 2 2 0 1

4 1 1 1 1 1 0

1

( ) [ ( )]

x n n x m P mn m

e x t

v f

B v E v

v

v

_^{ }_ ^_





        

^ (2)

2

0

1 2 1

1 1

0

0

  



^_ ^_ ^_





x n x

v x

n t

n

v n D

(3)

2 1 2

1 *

0 x

en u x

E

gE

^_





 

_



(4) and

x E x

u t

u

C

^_

gE

^_







2



^1*

2 2

2

( 

₀

)



(5) Equations (1) and (2) are the zeroth and first order oscillatory fluid velocities under influence of the respective fields,



and

m

represent the phenomenological momentum transfer collision frequency and effective mass of electrons. A one dimensional QHD model equation (2) includes two different quantum effects: (a) quantum diffraction and (b) quantum statistics. Quantum diffraction is taken into account by term proportional to ² .

Where



is Planck’s constant divided by 2 ? The contribution of this term may be interpreted alternatively as a quantum pressure term or quantum Bohm potential.

Quantum statistics is included in the model with the equation of which takes account the Fermionic character of the electrons. Fermi pressure-high density plasma and large wavelength-

x P mn

f





0

1 . Here ₂

0 3 1 2 3n

n mV

P ^f

f  , P_f stands for Fermi pressure with

m T V_f 2k^B

 as the Fermi speed in which k_B is Boltzmann constant and

T

is Fermi temperature of the electrons. n₀ and n₁ are the equilibrium and perturbed carrier concentration [20].

Equation (3) represents the continuity equation in which D^k_e^BT is diffusion coefficient with

 

_m^e_ is electron mobility. Equation (4) is the Poisson equation, in which



and g are scalar dielectric constant and the coupling constant. Equation (5) is the lattice

3 vibration equation in which

u

is the lattice displacement,



is mass density of the crystal,

C

is the elastic constant,

  

₀



_s ^,



_s is the dielectric constant in the absence of any strain,



₀is the dielectric constant of free space of the material.

2.1. Nonlinear Diffusion Current Density and Nonlinear Polarisation

Physically, the parametric interactions of acoustic waves (AWs) are possible because of the coupling that the driving pump wave introduces between the AW and the electron plasma wave (EPW). An acoustic perturbation in the lattice gives rise to an electron density fluctuation at the same frequency. The equation for density fluctuation of coupled electron plasma wave in n-type magnetized diffusive semiconductor with SDDC is obtained by equation (1) to (5) and using linearised perturbation theory as

x n m

gE ek n x p

n t

n t

n ^_

D

^_

n

^o

u E

^_





 

2

 

^{2 0}

  

¹

1 2 1 2

1 2

1



2





(6)

With

m

E 

eE⁰ ,



_p² 



²_p k²V_f^'2,V_f^'2 V_f



1e



, ^{mk T}

k

e 

₈^2 B²



and

 



a^{2 0}k²v¹a²



E g E

u

^ ik _



_{ }^  , ^v_a ^{ C}_

In the derivation of equation (6), we have neglected the Doppler shift under the assumption that



₀ 



kv₀,



_p 

 

ⁿ_m⁰^{e 12} is the plasma frequency. It is a known fact that the rotating wave approximation (RWA) is an integral part of the foundation of quantum optics, and is well discussed in both the old and the more modern text books and review articles. It concerns the interactions of a reservoir, consisting of an infinite numbers of oscillators, with either a two level atom or a cavity mode or a charged harmonic oscillator.

Hence the RWA should be confined only to the calculation of transition amplitudes for lower-order absorption and emission, where the known RWA terms do not contribute significantly. Above equation (6) leads to the following coupled equations:

t n m

gE ek n x n f

p t n t

n_{f f f s}

E u D

n

^_ ^_^









  

2



^{0 2 0}

  

¹

1 2 1

2 1 2

1

2







(7)

x

n x

n s

t p n t

ns ^_s

n D

_^ s

E

^_f^





  

2

 

¹

1 2

1 2

1

2







(8) Subscript

1 s

and 1f stands for the slow and fast components, respectively. An asterisk (*) represents the complex conjugate of the quantities. It can be inferred from above equations (7) and (8) that the slow and fast components of the density perturbation are coupled to each other. Thus it is observe that the presence of a pump field is the fundamental necessity for the PI to occur. Using above equation we obtain

 

      

¹

*

1 ^{2 2}

2 2 1 2 1 2

2 2 0

1 2 0 2

0

1

 











k E

i i

k E

E gE k n s

a a

a

n

_^_{ } ^{   }

(9) Where



₁² 

 

p²



₁²^k2



^D



, ₂² 



p² a² ^k2^D



The diffusion induced nonlinear current density may be expressed as

x n d

eD

s

J 

^_^1*

4



^{2 2 2}

  

^{12 1 2}



²²²

 

¹

0

1 2 0 2 0 3

1

 









^ik_E^{eDn g}_k^{E E i} _{k E} ⁱ

d

a a

a

J

_^ _ ^{   }

(10) The four wave parametric interaction involving the incident pump, electron density fluctuations and induced acoustic-optical idler wave characterised by cubic nonlinear third order susceptibility

 3

 . Hence the diffusion induced nonlinear polarisationrepresents as

1

i J d d

dt d

J

P 



 _ ^,



^{2 2 2}

  

^{12 12}



²²²

 

^{1 0 3 02 1}

0 1

1 2 0 2 0 3

1 E E

P

k E

i i k

E E E g n eDk d

a a

a



























 



_ ^{  }

(11) Which leads to the diffusion induced third order (DITO) susceptibility



_d³ in the coupled mode scheme as

 

³



^{2 2 2}

  

^{12 12}



²²²

 

¹

1 0 0

2 0 3

1

 









k E

i i

k E

g n eDk d

a a

a



























(12) Now rationalizing equation one obtains the real part and imaginary parts of the complex DITO susceptibility as

  ^ _{  } ^ _ ^













 ^{2 2}

2 1 2 1 2 1 2 2 2 2 1 2 2

1 2 2 2 2 1 2 2 2

2 2 2 1 0

0 2 5 3 0

) (

) . (

. 3 .











































a a

a a

a kE

E k k

m E g k e D n r

(13)

  ^ _

_

_{ } ^

_{ } ^ _

_

_ 2

_

²

_ ^

2 1 2 1 2 1 2 2 2 2 1 2 2

2 2 1 2 1 2

2 2 2 1 0

0 2 5 3 0

) . (

. 3 .









































a a a a

a k kE

m E g k e D n i

(14) The formula given in equation is reveals that the crystal susceptibility is influenced by quantum mechanical correction through^1, ^2 and strain dependent dielectric constant, Here the damping of acoustic wave arises due to its acousto-optic coupling with plasma waves. To compensate for the damping losses of the acoustic wave in the SDDC medium, one should apply a pump of certain minimum amplitude called the threshold pump amplitude.

The threshold pump amplitude ^E0th may be obtained by setting the  r

3

Equation is zero as



1 ²



²¹

2 2 2 1

0_{th ek}^m

   

_a



E  

(15) 3 RESULT AND DISCUSSION

On the basis of the theoretical formulations, we considered DITO nonlinearity both the real and imaginary part in quantum semiconductor plasma. The numerical study has been made by the specific irradiation of n-type doped semiconductor sample (BaTiO₃) medium by pulsed 10.6m CO₂ laser at77⁰K.

The following material parameters are taken as representative values of

BaTiO

₃ at

0K

77 to establish the theoretical values; m0.0145m₀,



_s 2000,v_a 310³ms^1, ,

10 5 .

3  ^{11 1}

 s

v



_a 1.610¹⁴s^1, ₀1.7810¹⁴s^1,_.

Being one of the principle objectives of the present analysis, the nature of diffusion induced parametric dispersion arises due to real and imaginary part of DITO susceptibility,

5 viz., ^{real 3} _andimag^{ 3} _. in quantum semiconductor plasmas with SDDC has been analysed. The threshold characteristics are interpreted in figure: 1. As usual, the threshold

E

₀^th decreases as the wave number

k

increases. Initially, at k 110⁶m^1, the threshold pump is

1 8

0 8.310 Vm^

E _th . The threshold pump decreases abruptly with

k

_{up to}k810⁶m^1_, and then decreases slowly. It is found that pump field threshold

E

₀^th is order of 10⁷Vm^1 near

k  4  10

⁶

to 2 . 4  10

⁷

s

^1_.

0 2 4 6 8 10 12 14 16 18 20 22 24

0 10 20 30 40 50 60 70 80

Wave number k(10⁶)m^-1 Threshold pump electric field E0th(107)Vm-1

Fig. 1 Variation of threshold pump electric field

E

_0th Vs wave number

k

The variation of real part of DITO susceptibility  3



real and imaginary part of DITO susceptibility



_imag^{ 3} _. on same scale of pump electric field

E

₀ is shown in figure: 2 and figure:

3, respectively. It may be inferred from figure: 2 that there exists easily seen anomalous parametric dispersion regime with positive and negative values. For

E

₀

 1 . 54  10

⁷

Vm

^1,

 3



real is negative (which show absorption) and decreases with increase

E

₀. However  3



real

falls and acquired minimum value at

 1 . 6  10

⁷

Vm

^1. A slight increase in

E

0 beyond this value of

 1 . 6  10

⁷

Vm

^1 very sharp rise in  3



real is positive (amplification), the resonance peak is at

1 . 65  10

⁷

Vm

^1. After this resonance condition



^(3)real increase sharply and again starts decreasing and become constant.

5 10 15 20 25

-130 -65 0 65 130 195 260 325 390 455

Pump field E₀ (10⁶) Vm^-1 Real Susceptibility (3)real (10-13)m2V-2

Fig. 2 Variation of real part of DITO susceptibility  3



real Vs pump electric field

E

₀ at

3 25 0 210 m^

n

6

5 10 15 20 25

-400 -350 -300 -250 -200 -150 -100 -50 0

Pump field E₀ (10⁶) Vm^-1 Imag. Susceptibility (3)imag. (10-13)m2V-2

Fig. 3 Variation of imaginary part of DITO susceptibility_imag^{ 3} _. Vs pump electric field E0atn₀210²⁵m^3

In figure: 3 the imaginary part of DITO susceptibility



_imag^{ 3} _. depicted the variation of pump electric field

E

₀ . It can be seen from figure that



_imag^{ 3} _. is in negative value on the given scale of

E

₀ . Initially when increase in

E

₀, the



_imag^{ 3} _.decreases and attains its minimum value-5.00210^-12V^2m² at E₀1.710⁷Vm^1 beyond this point



_imag^{ 3} _. increases with increase in

E

₀ . In both above figure: 2 and figure: 3 it is analysed that  3



real and



_imag^{ 3} _.

reaches its minimum value nearE₀1.610⁷to1.710⁷Vm^{1 .} The real part of DITO susceptibility  3



real and imaginary part of DITO susceptibility

 3 .

imag with same scale of wave number

k

is displayed in figure: 4 and figure: 5, respectively.

It can be observed that the real part of DITO susceptibility  3



real shows usual dispersive characteristics of a medium with an increase of wave number

k

. The  3



real profile is also similar to the dispersion characteristics of III-V semiconductors like InSb [21]. The DITO susceptibility  3



real decreases with increases

k

in negative (absorption) regime and reaches its minimum value-1.13510^-10V^2m²atk1.610⁷m^1.However when increase in

1

107

65 .

1  ^

 m

k , the  3



real increases in positive(amplification) regime and attains its maximum value4.18710^-10V^2m²atk1.710⁷m^1.After increase in

k

, _real^{ 3} falls abruptly and then increase in constant manner.

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 -100

-50 0 50 100 150 200 250 300 350 400

Wave number k(10⁷)m^-1 Real Susceptibility(3)real (10-12)m2V-2

Fig. 4 Variation of real part of DITO susceptibility _real^{ 3} Vs wave number

k

at

3 25 0210 m^ n

7

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 -1200

-1000 -800 -600 -400 -200 0

Wave number k(10⁷)m^-1 Imag. Susceptibility(3)imag. (10-13)m2V-2

Fig. 5 Variation of imaginary part of DITO susceptibility_imag^{ 3} _. Vs wave number

k

_at

3 25 0 210 m^ n

Similarly in figure: 5, the imaginary part of DITO susceptibility



_imag^{ 3} _.shows absorption and attains its minimum value-1.5910^-10V^2m^2at

k  1 . 7  10

⁷

m

^1. It is seen in this figure that the behaviour of



_imag^{ 3} _. is in negative regime and the minimum value of both susceptibilities



_real^{ 3 and}



_imag^{ 3} _. on same scale of

k

is near

 1 . 6  10

⁷

to 1 . 7  10

⁷

m

^1 .

Figure: 6 and figure: 7 are illustrating the variation of real part of DITO susceptibility

 3



real and imaginary part of DITO susceptibility_imag^{ 3} _. with same scale of plasma frequency



p (via carrier concentration

n

₀).

Figure: 6 shows the absorption and amplification dispersive characteristics of real part of DITO susceptibility_real^{ 3} . Initially when



_p

 

₁,  3



real is in positive (amplification) regime. As increase in



_p (via carrier concentration

n

₀),  3



real then abruptly increase and reaches its maximum value_real^{ 3} 5.18710^-13V^2m²at_p1.0610¹³s^1.

When a slight increase in

n

₀ beyond this point a sharp dip near

( 

_p

 

₁

)

, after this resonance condition_real^{ 3} abruptly decreases and changes the sign attains its minimum value_real^{ 3} 5.18710^-13V^2m^{2 at}_p2.1310¹³s^1attributing negative dispersion at

( 

_p

 

₁

)

(absorption), further increase in



_p,  3



real increases in negative values and saturates beyond the point.

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 -150

-125 -100 -75 -50 -25 0 25 50

Plasma frequency _p(10¹³) s^-1 Real Susceptibility(3)real (10-14)m2V-2

Fig. 6 Variation of real part of DITO susceptibility  3



real Vs plasma frequency



_p

8

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 -1000

-800 -600 -400 -200 0

Plasma frequency _p(10¹³) s^-1 Imag. Susceptibility(3)imag. (10-16)m2V-2

Fig. 7 Variation of imaginary part of DITO susceptibility



_imag^{ 3} _. Vs plasma frequency



p

Figure: 7 show same characteristics of imaginary part of DITO susceptibility ^{ 3}

.

imag

as usual when increase in_p (via carrier concentration

n

₀), ^{ 3}

.

imag decreases attains its minimum value 1.2310^13m²V^2 at_{p 2.1310}¹³_s^1further increase in



_p^,_imag^{ 3} _. increases abruptly and saturates beyond the point.

The present work deals with the analytical investigations of diffusion induced parametric absorption and amplification in quantum semiconductor plasmas with SDDC. The detailed analysis enables one to draw the following conclusions:

i. The above discussion makes it clear that QHD model successfully applied to study diffusion induced parametric interactions of absorption and amplification in quantum semiconductor plasmas above threshold pump field.

ii. It is found that diffusion induced PIs shows usual dispersion characteristics. The real and imaginary part of DITO susceptibility illustrates variation with pump electric field

E

0, wave number

k

and plasma frequency



_p, respectively.

iii. The qualitative behaviour of real part of DITO susceptibility_real^{ 3} found to be in both negative(absorption) and positive(amplification) regime but imaginary part of DITO susceptibility ^{ 3}

.

imag shows only negative(absorption) at specific scale range of parameter.

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