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**A PATH FOR HORIZING YOUR INNOVATIVE WORK **

**A THEORETICAL EVALUATION OF ENHANCEMENT FACTOR Q OF ELECTRON **

**MOBILITY USING DIFFERENT SCATTERING MECHANISM FOR QUANTUM **

**WELL **

**RAMANI SHANKAR PANDEY1 _{, L. K. MISHRA}2**

1. Government Basic School, Babhangawn (Bhojpur) 802313 (Bihar) 2. Department of Physics, Magadh University, Bodh Gaya-824234 (Bihar)

**Accepted Date: 02/01/2019; Published Date: 01/03/2019 **

**\ **

**Abstract: - ****Using the theoretical formalism of T. T. Hai etal, D. N. Quang etal and T. Wang etal , we have theoretically evaluated mobility enhancement factor Q using three **
type of scattering mechanisms for quantum well. We have also evaluated 2DEG electron mobility using LDEG transport model for AlGaAs/GaAs quantum well. The evaluation of mobility
enhancement factor Q has been performed using three scattering mechanisms namely (i) *Q _{T}* (Top-interface scattering), (ii)

*Q*(Top-bottom interface scattering) and (iii)

_{TB}*Q*(Bottom-interface scattering). The enhancement factor Q is calculated in three different cases: (a) as a function of QW well width

_{B}*L A*( 0) keeping correlation length (

*A*0) and sheet hole density

*p cm*( 2) constant (b) as a function of sheet hole density

_{s}*p cm*( 2) keeping

_{s}*L A*( 0)and (

*A*0),constant (c) as a function of correlation length (

*A*0),keeping

*L A*( 0)and

*p cm*( 2) constant. In all these three cases enhancement factor Q increase for

_{s}*Q*and

_{T}*Q*but decrease for

_{TB}*B*

*Q* . The value of Q is large in *Q _{T}* and small for

*Q*. Using LDEG transport model, we have evaluated POP (polar optical phonon) scattering rate and ACP (acoustic phonon) scattering rate as a function of QW well width. Here, two scattering rate increase and decrease as a function of well width. The magnitude of POP scattering rate is larger than ACP scattering rate. Our obtained results of 2DEG electron mobility as a function of energy difference between first and second sub-bands along with as a function of well width indicate that 2DEG electron mobility increase attain maximum value and then decrease for both the cases. However, the magnitude of electron mobility is larger in first case than in the second case. Our evaluated results of electron energy distribution using Boltzmann distribution function show that energy distribution decrease for equilibrium distribution and also for different fields. Our evaluated results of 2DEG electron scattering rate as a function of electron energy for different well width indicate that scattering rate decrease with electron energy for all the well width taken. The magnitude is large for small well width and small for large well width. This type of scattering is due to inter sub-bands scattering. Our evaluated results of POP scattering rate as a function of electron energy for bulk electrons, 2DEG and 1DEG show that POP scattering rate increase and decrease with electron energy. Its value is large for bulk electrons and small for 1DEG.

_{B}The entire evaluation in this paper is based on two approaches namely Variational method and LDEG transport model. The theoretical findings in this paper is quite useful and helpful in orderto understand the structural and optical properties of metal organic phase epitaxy-grown multi-QW particularly light emitting diodes (LEDs)

**Keywords: **Surface roughness scattering, Top-interface scattering, Top-bottom interface scattering, Bottom-interface scattering, mobility enhancement factor, doping-induced
band bending, 2DEG electron mobility, 2DEG electron scattering rate, POP (polar optical) scattering rate, ACP (acoustic phonon) scattering rate, Misfit deformation potential, Variational
approach, LDEG transport model, Correlation length, Sheet hole density

**Corresponding Author: RAMANI SHANKAR PANDEY **
**Access Online On: **

www.ijpret.com

**How to Cite This Article: **

Ramani Shankar Pandey, IJPRET, 2019; Volume 7 (7): 1-25

**Available Online at www.ijpret.com ** 2

**INTRODUCTION **

Recent advances in crystal growth techniques like fine line lithography, metal organic chemical vapour deposition (MOCVD) and molecular beam epitaxy (MBE) method have made possible the fabrication of low dimensional semiconductor structure such as quantum well (QW),

quantum wire (QWR) and quantum Dots (QD)1-5_{ .A quantum well is formed when a thin layer of }

lower band gap semiconductor is sandwiched between two layers of higher band gap

semiconductors6,7._{ In the quantum well structure the electrical and optical properties of the }

semiconductors are totally different from those in the bulk material due to quantum effects8,9_{. }

Due to modulation doping in QW structures, carriers are separated from ionised impurity thereby increasing the mobility carrier due to reduced ionised impurity scattering. As is well

known10_{ enhanced mobility of 2D carriers in QW is achieved by means of modulation of the }

decisive factors, such as electronic structure, scattering mechanisms and confining sources. For instance, doping is indispensible source of carrier supply to the sample. This is a scattering mechanism for the carriers moving in-plane. This is also a confining source along the growth direction. Doping as a scattering mechanism was more studied than as a confining source.

The role of scattering in the in-plane depends strongly on the carrier distribution along the quantization direction. This is done through the envelope wave function. It is fixed by confining

sources. It has been reported that roughness-related scattering11-20_{ dominates transport in }

many hetro structures, especially thin square QWs. This is determined by the wave function near the interfaces. It is obvious that remote one-side (IS) doping of square QWs leads to modulation of the wave function makes some essential changes in 2D transport.

In this paper using the theoretical formalism of T. T. Hai etal21_{, D. N. Quang etal}22_{ and T. Wang }

etal23_{, we have theoretically evaluated electron mobility enhancement factor Q using three }

different scattering mechanisms. These mechanisms are (i) QT (Top-interface scattering) (ii) QB

(Bottom –interface scattering) (iii) QTB (Top-Bottom-interface scattering). The evaluation is

performed in three different cases: (i) as a function of QW well width L (*A*0_{) keeping correlation }

length (*A*0) and sheet hole density

2

( )

*s*

*p cm* _{ constant (ii) as a function of sheet hole density }

2

( )

*s*
*p cm*

keeping QW well width L(*A*0_{) and correlation length } (*A*0)_{ constant and (iii) as a }

function of correlation length (*A*0) keeping QW well width L(*A*0_{) and sheet hole density }

2

( )

*s*
*p cm*

constant. The evaluated results show that in all the three cases mobility enhancement

factor Q increase for QT and QTB and decrease for QB. The value of Q is large for QT and small for

**Available Online at www.ijpret.com ** 3

transport model, we have calculated POP (polar optical phonon) scattering rate (sec-1_{) and ACP }

(acoustic phonon) scattering rate (sec-1_{) as a function of well width L(}* _{A}*0

_{). We observed that }

both these scattering rate increase and decrease with well width. The magnitude of POP scattering rate is large compared to ACP scattering rate. Our evaluated results of 2DEG electron

mobility (*cm*2/*V*sec) as a function of energy difference between first and second sub-band at

T=300K show that 2DEG electron mobility increases attains maximum value and then decreases. We repeated the calculation of 2DEG electron mobility as a function of well width. The similar results were observed in this case also. However, the values of 2DEG electron mobility in first case are large than in the second case. We have evaluated electron energy distribution function using Boltzmann distribution function in three cases: (i) for equilibrium distribution function (ii) for field 1KV/sec (iii) for 2KV/sec. Our evaluated results of electron energy distribution function decrease in all these three cases as a function of electron energy. The values of distribution function in case of equilibrium distribution function are large

compared to 2KV/sec. Our evaluated results of 2DEG electron scattering rate (sec1) as a

function of electron energy (meV) for different values of well width indicate that 2DEG scattering rate decrease with electron energy for all the well width taken. The values are large for small well width and small for large well width. We have also evaluated 2DEG electron scattering rate as a function of energy separation between first and second sub-band for triangular like QW. In this case also the scattering rate increases attains maximum value and then decreases similar like rectangle QW. However, the magnitude of scattering rate in case of rectangle QW is large than triangular QW. Our evaluated results of POP scattering rate as a

function of

0

(*E* *E* )

in case of bulk electrons, 2DEG and 1DEG indicate that POP scattering rate decrease and increase in all these three cases. Here also, the value of POP scattering rate in case of bulk electrons is large than in the case of 1DEG. Our evaluated results are in good

agreement with those of the other theoretical workers24-26_{. }

**MATERIALS AND METHODS **

One starts with the effect from doping-induced band bending on the carrier distribution along the growth direction. For high enough barriers one may take asymmetric envelope wave

function *A*( )*z* _{ and symmetric }*S*( )*z* _{ for carriers (electrons or heavy holes) in the lowest }

sub-band of the QW.

**Available Online at www.ijpret.com ** 4

###

1/ 1 / cos( / )

*c z L*

*A* *z* *B* *L* *L e*

_{}

for *z* *L*/ 2

=0 for *z* *L*/ 2 (1)

Two-side doping (S)

*S*( )*z* 2*B*2 /*L*cosh(*c z L*2 / )_{ for } *z* *L*/ 2

=0 for *z* *L*/ 2 (2)

Here L is well width and B1, B2 and c1 ,c2 are variational parameters to be determined.

**Evaluation of enhancement of the mobility using different scattering mechanism **

First of all, one calculates the mobility of two dimensional Hole gas (2DHG) in p-channel QWs is one of the most important parameters which fixes its performance limited by various scatterings. Within the linear transport theory, the mobility at very low temperatures are determined by the transport lifetime

*
*e*
*m*
(3)

Here, m*_{ is the in-plane effective mass of the carrier. The transport lifetime is represented in }

terms of the autocorrelation function (ACF) for each disorder27

2 2
2
0 0
1 1
(2 )
*F*
*k*
*F*
*dq d*
*E*

##

2 2 1 22 _{2 2}

( )

( )

(4 * _{F}* )

*U q*
*q*

*q*

*k* *q*

(4)

**Here, q=(q,**) is the 2D momentum transfer due to a scattering event in the x-y plane (in polar

coordinate). *q* *q* 2*kF*sin( / 2)

with is the scattering angle. The Fermi energy is given by

2 2
*
2
*F*
*F*
*k*
*E*
*m*

with *kF* 2*ps* _{ and p}

s is the sheet density, is the dielectric constant of the

material. The ACF in equation (4)

2

( )

*U q*

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surface roughness (SR) and (ii) misfit deformation potential (DP). The overall lifetime *tol*_{ is then }

determined by the ones for individual disorders scattering to the Matthiessen rule

( ) ( ) ( ) ( )

1 1 1 1 1

*t* *b* *t* *b*

*tol* *SR* *SR* *DP* *DP*

_{ (5) }

Here the superindices (t) and (b) refer to the top and bottom interface respectively. According

to equation (4) one specifies the autocorrelation function in wave vector space

2

( )

*U q*

for these scattering sources.

(i)Surface roughness (SR)

This scattering deals with 2DHG from a rough potential barrier. The scattering potential is due

to roughness-induced fluctuations in the position of the barrier28_{. The autocorrelation function }

for surface roughness scattering in a square QW of an arbitrary depth was derived29_{. The result }

is given by

2

( / ) 2

0 ( ),

( ) ( )

*t b*

*SR* *A S*

*U* *q* *V*

(6)

With (*z* *L*/ 2).

(ii) Misfit deformation potential (DP)

This is an interface roughness to produce fluctuations in a strained field in a lattice-mismatched

hetrostructure. These in turn act as a scattering source on charge carriers. It has been shown

30-32_{ that the misfit deformation potentials for two kinds of carrier are quite different. The one for }

electrons are fixed by single normal diagonal component of the strained field, whereas for one for holes for its components. One applies the 2D Fourier transform of the misfit DP for cubic crystals. The scattering potential associated with the top interface (z=-L/2) is used for

electrons33_{. One may obtain the ACFs for misfit DP scattering for holes in the following form }

2
( / )
( )
*t b*
*DP*

*U* *q*

3

2 2

/ / 2 2

1 1

( ) [ ( / 2) ( / 2)

4

*t b* *t bB* *t*

*xt e* *c t* *c t*
*L*
+
2
1( / 2)]*t*

2 4 4

2 2 1

1

{3 / 2[ ( 1)] (1 sin cos )
(1 *t* / 4 )*n* *n* *x* *b Ks*

2

44

( ) (1
4
*s*
*d G*
*c*
2 2

sin cos )}_{ }

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In the well ( *z* *L*/ 2) and zero elsewhere. In equation (7) *bs*_{ and } *ds*_{ are the shear }

deformation potential constants of the well laye

r and is the lattice mismatch specified by the Ge content and the widths of the well and

barrier. Its anisotropy ratio is yielded by

44
11 12
2 *c*
*c* *c*

_{ (8) }

The elastic constants are given by

12
11
2*c*
*K*
*c*
,
12
11 12
11

2( 2 )(1 *c* )

*G* *c* *c*

*c*

(9)

Here, *cij*_{ act as elastic stiffness constants. It can be seen from equation (7) that the deformation }

potential related to rough interface decays rapidly (exponentially) with the increase of the distance measured.

**Calculation of mobility enhancement factor Q **

Now one considers the case that the roughness related scattering (SR and misfit DP) dominate the low temperature transport in remote-doped square QWs. As a measure of the advantage of the symmetric modulation of the square QW over its asymmetric modulation, one introduces an enhancement factor Q. This is defined as the ratio of the overall mobility in the 2S-doped

QW

,
*s BT*

*tot*

to that in the IS-doped counterpart (

,
*a BT*

*tot*

_{)with the same sheet carrier density }

and the same interface profile.

The mobility enhancement factor QBT (bottom-top interface scattering) is given by

,
,
( , ; , )
( , , )
( , ; , )
*s BT*
*tot* *s*

*BT* *s* *a BT*

*tot* *s*
*L p*

*Q* *L p*

*L p*

_{ (10) }

Since the roughness amplitude drops out of the ratio, this depends on the well width, sheet carrier density and correlation length. This is shaped by the features of the QW structure.

**Available Online at www.ijpret.com ** 7
,
,
( , ; , )
( , , )
( , ; , )
*s B*
*tot* *s*

*B* *s* *a B*

*tot* *s*
*L p*
*Q L p*

*L p*

_{ (11) }

The enhancement factor due to top-interface scattering (*QT*_{) is given by }

,
,
( , ; , )
( , , )
( , ; , )
*s T*
*tot* *s*

*T* *s* *a T*

*tot* *s*
*L p*
*Q L p*

*L p*

_{ (12) }

Here, L is the well width, *ps*_{ is sheet carrier density, }_{ is the energy gap parameter for the }

band and is the correlation length. The results of these three enhancement factor are shown

in table T1, T2 and T3 respectively.

**Evaluation of Electron mobility using low-dimensional electron gas (LDEG) transport model **

In order to calculate electron mobility, one uses LDEG transport model. In this case, two sorts of 2DEG quantum well structures and rectangular 1DEG quantum wire structures are studied. Electron concentrations in these devices are sufficiently low to maintain the non-degenerate condition. The sub-band structure and the electron wave function in the quantum well/wires are obtained by solving the Schrodinger equation under an effective mass approximation. The 2DEG and the 1DEG wave functions are expressed as

2

1

( )

*ik r*

*D* *e* *z*

*A*
13(a)
1
1
( , )
*x*
*ik x*
*D*
*x*

*e* *y z*

*L*

13(b)

Here, represents an electron envelope wave function in the quantization direction, A is the

area of a quantum well. *k* is the electron wave vector component parallel to quantum well. Lx

is the length of the quantum wire. The quantization is in the z-direction for 2DEG and in both y and z directions for 1DEG. Using the wave functions in equation (13a) and (13b), one obtains the square of the matrix elements between the m’th and nth sub-bands.

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

*mn* *D* *D* *x* *y* *z* *mn* *D* *z* *z*

*M* *M* *q q q* *x I* *q* *dq*

##

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2 2
2 _{2}
3 ,2
1
1
( ) ( , , ) ( , )
2

*mn* *D* *D* *x* *y* *z* *mn* *D* *y* *z* *y* *z*

*M* *M* *q q q* *x I* *q q* *dq dq*

##

14(b)

Here *M*3D_{ is the corresponding matrix element for a bulk electron and q is a phonon wave }

vector. The overlap integral *Imn*_{ in the above equations is defined by }

,2 ( ) ( ) ( )

*z*

*iq z*

*mn* *D* *z* *m* *n*

*I* *q*

##

*z*

*z e*

*dz*

14(c)

1 ( , ) ( ) ( )

*y* *z*

*iq y iq z*

*mn D* *y* *z* *y* *n* *z*

*I* *q q*

##

*y z*

*y z e*

*e*

*dydz*

14(d)

The coupling coefficient for 2DEG has been derived34

,2 ( ) 1 2 ( )1 ( )1 ( )2 ( ) exp(2 2 1)

*mn* *D* *m* *n* *m* *n*

*H* *q*

##

*dz dz*

*z*

*z*

*z*

*z*

*q z*

*z*

15(a)

The coupling coefficient for 1DEG has been obtained which is written as

2 2

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

*mn D* *x* *m* *n* *m* *n* *x*

*H* *q*

##

*dy dy dz dz*

*y z*

*y z*

*y z*

*y z x K q*

*y*

*y*

*z*

*z*

15(b)

Here, *q* is the phonon wave vector component parallel to the quantum well and *K*0_{ is the }

modified Bessel function of second kind of order 0. Using the matrix elements, the 2DEG and 1DEG POP (polar optical phonon) scattering rates are given by

2 ,2 ' ,2 0 ( ) 1 1

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

8
*mn* *D*
*POP*
*mn* *D*
*s*
*H* *q*
*e*

*S* *E* *N* *x* *k* *k* *q*

*q*

_{}

_{}

##

_{'}

( '*E* *E* )*dk*

15(c) 2 ' ' ,1 ,1 0 1 1

( ) ( )( 1 / 2 1 / 2) ( ) ( ) ( ' )

8

*POP*

*mn D* *mn D* *x* *x* *x* *x* *x*

*s*
*e*

*S* *E* *N*_{} *x H* *q* *k* *k* *q* *E* *E* *dk*

_{}

##

15(d) Here 1exp( ) 1

**Available Online at www.ijpret.com ** 9

Here, _{and } *s*_{ are optical and static dielectric constants, } *N*_{ is the phonon occupation }

number, is the polar optical phonon energy. sign stand for phonon emission and

absorption respectively. *E*'_{ and E denote initial-state and final-state electron energies. }

Acoustic –phonon scattering for LDEG is treated similarly to polar optical phonon scattering. The 2DEG and 1DEG ACP scattering rates are given by

* 2

2 2

,2 ( ) 2 2 ( ) ( )

*A*
*acp*

*mn* *D* *m* *n*

*l*
*m k TD*

*S* *E* *z* *z dz*

*S*
_{} _{}

_{}

16(a)
2
2 2
,1 ( ) 2 2 ( , ) ( , ) 1 ( )

*A*
*acp*

*mn D* *m* *n* *D*

*l*
*k TD*

*S* *E* *y z* *y z g* *E dydz*

*S*

_{} _{}

_{}

16 (b)

Here, *DA*_{ is the deformation potential, } _{ is the density of mass, } *Sl*_{ is longitudinal sound }

velocity and *g*1D_{ is the density of states for 1DEG with a broadening factor}35_{ of 2.5meV at }

T=300K .

The parameters for GaAs material used in this calculation is the following;

_{}_{=10.92, }*s*_{=12.90, } _{=36 meV, }*DA*_{=7.5eV, }_{=5.36g/cm}3

The results are shown in table T4 to T10.

**RESULTS AND DISCUSSION; **

Using the theoretical formalism of T. T. Hai etal21_{, D. N. Quang}22_{ and T. Wang etal}23_{, we have }

theoretically evaluated mobility enhancement factor Q using three different scattering mechanisms. The evaluation is based on variational approach with one-side and two-side

doped square QW. The three scattering mechanisms are (i) *QT*_{ (Top-interface scattering) (ii) }

*TB*
*Q*

(Top-bottom interface scattering) and (iii) *QB*_{ (Bottom-interface scattering). The evaluation }

is also performed in three different cases (i) as a function of QW well width L(*A*0) keeping

correlation length (*A*0) and sheet hole density

2

( )

*s*

*p cm* _{ constants. (ii) as a function of sheet }

hole density

2

( )

*s*

*p cm* _{ keeping correlation length } 0

(*A* )

_{ and well width L} 0

(*A* )_{ constant (iii) as }

**Available Online at www.ijpret.com ** 10

2

( )

*s*

*p cm* _{ constants. In table T1, we have shown the evaluated results of mobility enhancement }

factor Q as a function of well width L (*A*0) keeping correlation length (*A*0) and sheet hole

density

2

( )

*s*

*p cm* _{ constant for all the three scattering mechanism } *QT*_{, } *QTB*_{ and } *QB*_{. Our }

evaluated results show that enhancement factor Q increase in *QT*_{ and }*QTB*_{ but decrease in}*QB*_{. }

The magnitude is large in *QT*_{ but small in}*QB*_{. The values of }*QTB*_{ lie in between }*QT*_{ and }*QB*_{. In }

**table T2 and T3, we repeated the calculation of mobility enhancement factor Q as a function of **

sheet hole density and correlation length keeping two others parameter constant. The similar
**results were observed in these two tables also. In table T4, we have presented the evaluated **
results of POP (polar optical phonon) scattering rate and ACP (acoustic phonon) scattering rate
as a function of well width for low dimensional QW. Our obtained results show that both
scattering rate increase and decrease with well width. The values of scattering rate are large in
**case of POP scattering compared to ACP scattering. In table T5, we have shown the evaluated **
results of 2DEG electron mobility as a function of energy difference between first and second

sub-band (*E*1*E*0)_{ meV. Our obtained results show that 2DEG electron mobility increase, }

attain maximum value and then decrease as a function of (*E*1*E*0)_{. In table T6, we repeated }

the calculation of 2DEG electron mobility as a function of QW well width and here also we
noticed that 2DEG electron mobility increase attain maximum value as a function of well width
**and finally decrease and becomes flat. In table T7, we have shown the evaluated results of **
electron distribution function (arb Unit) as a function of electron energy (meV). The evaluation
is performed using Boltzmann distribution function in three different cases: (i) equilibrium
distribution function (ii) at field 1KV/sec (iii) at field 2KV/sec. Our evaluated results indicate that
electron energy distribution function in all these three cases decrease with electron energy. The
**value is large for equilibrium distribution function and small for field 2KV/sec. In table T8, we **

have presented the evaluated results of 2DEG electron scattering rate (sec )1 as a function of

energy separation (meV) for different QW well width L=80*A*0_{, 100}*A*0_{, 120}*A*0_{ and 150}*A*0_{. We }

observed that 2DEG scattering rate decrease as a function of

0

[*E* *E* ]

. The value is large for

L=80*A*0_{ and small for L=150}*A*0_{. This type of scattering is due to inter sub-band scattering. It is }

also noticed that 2DEG has peak mobility around *E*1*E*0 2 _{.In table T9, we repeated the }

calculation of 2DEG electron mobility (*cm*2/*V* sec) as a function of (*E*1*E*0)_{for triangular }

**Available Online at www.ijpret.com ** 11
then decrease as for rectangular QW. But the value of 2DEG electron mobility is less than the
**mobility found for rectangular QW. In table T10, we have shown the evaluated results of POP **

scattering rate

1

(sec ) _{ as a function of }

0

[*E* *E* ]

for (i) bulk electrons (ii) 2DEG (iii) 1DEG. The

2DEG structure has 130*A*0_{ and 1DEG structure has 110} 0

*A* _{x110} 0

*A* _{. Our evaluated results show }

that POP scattering rate is dominant in the case of bulk electrons its value is largest and

smallest for 1DEG. The results also indicate that POP scattering rate decrease with

0

[*E* *E* ]

.

There is some recent calculations36-42_{ which also reveals the similar facts. }

**CONCLUSION **

From the above theoretical investigations and findings, we have come across the following conclusions

1) We have evaluated mobility enhancement factor Q using three different scattering

mechanisms. (i) *QT*_{(Top-interface) scattering, (ii) }*QTB*_{ (Top-bottom interface) scattering (iii) }*QB*

(Bottom -interface) scattering. The evaluation is performed in three different cases: (i) as a

function of QW well width *L A*( 0)keeping correlation length (*A*0) and sheet hole density

2

( )

*s*

*p cm* _{constant. (ii) as a function of sheet hole length } *p cms*( 2)

keeping *L A*( 0) and (*A*0)

constant. (iii) as a function of correlation length (*A*0)keeping *L A*( 0)and

2

( )

*s*

*p cm* _{ constant. }

Our obtained results show that in all these cases the value of Q increases for *QT*_{ and }*QTB*_{ but }

decreases for*QB*_{. The magnitude of Q is large in }*QT*_{ and small in }*QB*_{. The entire evaluation is }

based on variational approach with one-side and two-side doping of QW.

2)We have evaluated electron mobility of low dimensional electron gas (LDEG) using transport

model. Our evaluated results of POP (polar optical phonon) scattering rate and ACP (acoustic phonon) scattering rate as a function of QW well width increase and decrease with well width. The magnitude of POP scattering rate is large compared to ACP scattering rate. The results also indicate that POP scattering is the dominant scattering in low dimensional electron gas (LDEG).

3)Our evaluated results of 2DEG electron mobility as a function of energy difference between

**Available Online at www.ijpret.com ** 12

4)Our evaluated results of electron distribution function as a function of electron energy for

three different cases (i) equilibrium distribution function (ii) at field 1KV/sec (iii) at field 2KV/sec show that electron distribution function decrease in all these cases. The electron energy distribution function has been calculated using Boltzmann distribution function. The magnitude of electron energy distribution is large in the case of equilibrium distribution function and small for field 2KV/sec.

5)Our evaluated results of 2DEG electron scattering rate as a function of electron energy for

different QW well widths show that 2DEG electron scattering rate decrease with electron energy for all the well width. The value is large for small well width and small for large well width. This type of electron scattering is due to inter sub-band structure in electron energy difference between sub-bands becomes equal to twice the polar optical phonon energy.

6)Our evaluated results of POP scattering rate as a function of electron energy for bulk

electrons, 2DEG and 1DEG show that POP scattering rate increase and decrease with electron energy. The value is large for bulk electrons and small for 1DEG.

7)The entire evaluation in this paper is based on two approaches: (i) variational approach with

one-side and two-side doping. (ii) LDEG transport model. The theoretical findings in this paper is quite useful and helpful in order to understand the structural and optical properties of metal organic vapour phase epitaxy-grown multiple-QW particularly for light-emitting diodes (LEDs).

**Available Online at www.ijpret.com ** 13

**Table T1: An evaluated results of mobility enhancement factor Q for the p-type Square QW as **
**a function of well width L (A0 _{) using three different type of scattering mechanism. (i) }**

**Top-bottom interface scattering (QTB) (ii) ***QB*** (Bottom-interface scattering (iii) ***QT***(Top-interface **

**scattering). Here correlation length ****=10***A*0_{ and sheet hole density }*ps* (1012*cm*2)

_{ are kept }

**fixed. **

** L (***A*0_{) }_{ }*QT* _{ }*QTB* _{ }*QB*

**20 ** 1.286 1.085 **1.045 **

**30 ** 1.475 1.153 **1.017 **

**40 ** 1.589 1.205 **0.995 **

**50 ** 1.764 1.239 **0.976 **

**60 ** 1.953 1.335 **0.885 **

**70 ** 2.106 1.472 **0.783 **

**80 ** 2.238 1.587 **0.699 **

**90 ** 2.475 1.692 **0.564 **

**100 ** 2.596 1.738 **0.475 **

**110 ** 2.774 1.846 **0.396 **

**120 ** 2.897 1.975 **0.324 **

**130 ** 3.105 2.054 **0.276 **

**140 ** 3.276 2.178 **0.185 **

**150 ** **3.358 ** **2.225 ** **0.156 **

**Available Online at www.ijpret.com ** 14

**Table T2: An evaluated results of mobility enhancement factor Q for the p-type Square QW as **

**a function of sheet hole density ** *ps*_{ using three different type of scattering mechanism. (i) }

**Top-bottom interface scattering (QTB) (ii) ***QB*** (Bottom-interface scattering (iii) ***QT*

**(Top-interface scattering). Here correlation length ****=10***A*0_{ and well width L=80 }*A*0_{are kept fixed. }

** Sheet density ***ps* _{ }*QT* _{ }*QTB* _{ }*QB*

**2.58x**1011 1.356 1.487 **1.257 **

**4.87x**1011 1.479 1.593 **1.204 **

**7.32x**1011 1.584 1.674 **1.186 **

**8.55x**1011 1.695 1.785 **1.084 **

**9.43x**1011 1.783 1.889 **0.965 **

**1.58x**1012 1.995 1.965 **0.843 **

**2.76x**1012 2.083 2.132 **0.762 **

**3.39x**1012 2.187 2.329 **0.675 **

**4.76x**1012 3.293 2.476 **0.597 **

**5.95x**1012 4.375 2.843 **0.486 **

**6.67x**1012 5.657 3.559 **0.392 **

**8.32x**1012 6.732 3.678 **0.305 **

**9.57x**1012 8.954 3.896 **0.267 **

**1.32x**1013 **9.107 ** **4.059 ** **0.219 **

**Available Online at www.ijpret.com ** 15

**Table T3: An evaluated results of mobility enhancement factor Q for the p-type Square QW as **

**a function of correlation length **(*A*0)** using three different type of scattering mechanism. (i) **

**Top-bottom interface scattering (QTB) (ii) ***QB*** (Bottom-interface scattering (iii) ***QT*

**(Top-interface scattering). Here sheet hole density **

13 2

2.10 10

*s*

*p* *x* *cm*

** and well width L=130 ***A*0_{are }**kept fixed. **

** Correlation ** **length **
0

(*A* )

** **

*T*
*Q*

** ***QTB* _{ }*QB*

**20 ** 3.462 1.986 **0.458 **

**30 ** 3.107 1.874 **0.479 **

**40 ** 2.984 1.775 **0.514 **

**50 ** 2.867 1.678 **0.532 **

**60 ** 2.755 1.604 **0.568 **

**70 ** 2.874 1.583 **0.723 **

**80 ** 2.387 1.509 **0.796 **

**90 ** 2.309 1.473 **0.814 **

**100 ** 2.267 1.402 **0.873 **

**120 ** 2.205 1.358 **0.922 **

**140 ** 2.167 1.306 **0.976 **

**160 ** 2.108 1.275 **1.054 **

**180 ** 2.075 1.187 **1.097 **

**200 ** **2.032 ** **1.145 ** **1.142 **

**Available Online at www.ijpret.com ** 16

**Table T4: An evaluated results of POP (polar phonon scattering rate) (**sec1**) acoustic phonon **
**scattering (ACP) rate (**sec1**) as a function of QW well width L (***A*0_{) for electron energy }

**100mev at T=300K **

**Well width L(***A*0_{) }**POP scattering rate **

**X **1012 sec1

**ACP scattering rate **

**X **1011 sec1

**50 ** 8.762 **4.239 **

**100 ** 12.486 **6.586 **

**150 ** 10.215 **3.768 **

**200 ** 9.345 **5.897 **

**250 ** 8.226 **3.972 **

**270 ** 9.567 **4.586 **

**300 ** 8.328 **3.149 **

**350 ** 8.047 **5.687 **

**370 ** 7.329 **4.142 **

**400 ** 8.356 **3.215 **

**420 ** 7.489 **3.986 **

**440 ** 8.667 **3.147 **

**460 ** 7.286 **2.875 **

**480 ** 7.184 **1.986 **

**500 ** 7.987 **2.764 **

**550 ** 6.586 **2.547 **

**Available Online at www.ijpret.com ** 17

**Table T5: An evaluated result of 2DEG electron mobility **(*cm*2 /*V*sec)** as a function of energy **
**difference between first sub band and second sub band at T= 300K **

**Energy difference **(*E*1*E mev*0) ** _{Electron mobility }**
2

(*cm* /*V*sec)

**40 ** **7980 **

**50 ** **8032 **

**60 ** **8156 **

**70 ** **8305 **

**75 ** **8397 **

**80 ** **8416 **

**85 ** **8487 **

**90 ** **8508 **

**95 ** **8569 **

**100 ** **8425 **

**105 ** **8348 **

**110 ** **8300 **

**115 ** **8296 **

**120 ** **8117 **

**125 ** **8077 **

**130 ** **7954 **

**Available Online at www.ijpret.com ** 18

**Table T6: An evaluated result of 2DEG electron mobility **(*cm*2 /*V*sec)** as a function of well **

**width (***A*0_{) for ALGaAs/GaAs QW at T=300K }

**Well width (***A*0_{) }** _{Electron mobility }**(

*cm*2 /

*V*sec)

**50 ** **7269 **

**100 ** **7348 **

**150 ** **7456 **

**170 ** **7582 **

**200 ** **7637 **

**250 ** **7748 **

**270 ** **7996 **

**300 ** **8059 **

**350 ** **8232 **

**370 ** **8147 **

**400 ** **8035 **

**450 ** **7986 **

**470 ** **7809 **

**500 ** **7786 **

**550 ** **7645 **

**570 ** **7518 **

**Available Online at www.ijpret.com ** 19

**Table T7: An evaluated result of electron distribution function (arb. Unit) as a function of **
**electron energy (meV) for different fields using Boltzmann distribution function **

**Electron ** **energy **
**(meV) **

** Electron energy distribution function **

Equilibrium Boltzm.

Distr.

Distribu. Funct. at 1KV/cm

**Distribution function **
**at 2KV/cm **

**20 ** 10.432 9.327 **8.057 **

**25 ** 9.678 8.864 **7.998 **

**30 ** 8.527 7.589 **7.385 **

**35 ** 7.486 6.955 **6.873 **

**40 ** 6.229 6.472 **6.602 **

**45 ** 5.862 6.148 **6.357 **

**50 ** 5.142 5.863 **5.954 **

**55 ** 4.697 5.157 **5.326 **

**60 ** 4.087 4.863 **5.108 **

**65 ** 3.986 4.697 **4.956 **

**70 ** 3.708 4.385 **4.822 **

**75 ** 3.432 3.957 **4.674 **

**80 ** 3.169 3.492 **4.238 **

**85 ** 2.867 3.325 **3.496 **

**90 ** 2.648 3.087 **3.148 **

**95 ** 2.509 2.886 **2.954 **

**Available Online at www.ijpret.com ** 20

**Table T8: An evaluated result of 2DEG scattering rate (**sec1**) as a function of electron energy [ **

0

*E* *E*

**] for different well width (***A*0_{) for low dimensional QW }

0

*E* *E*

** _{ --- 2DEG scattering rate (}** 1

sec _{)--- }

L=80 *A*0 _{L=100 }*A*0 _{L=120 }*A*0 _{L=150 }*A*0

**0.20 ** 3.147 2.886 2.059 **2.007 **

**0.40 ** 2.984 2.532 1.867 **1.762 **

**0.60 ** 2.768 2.149 1.329 **1.409 **

**0.80 ** 2.109 1.867 1.108 **1.172 **

**1.00 ** 12.693 10.765 8.358 **7.986 **

**1.20 ** 11.487 10.586 7.986 **7.704 **

**1.40 ** 10.886 10.205 7.654 **7.532 **

**1.60 ** 10.695 9.847 7.326 **7.247 **

**1.80 ** 10.432 9.632 6.967 **7.059 **

**2.00 ** 10.149 9.315 6.705 **6.798 **

**2.20 ** 9.868 9.059 6.148 **6.504 **

**2.40 ** 9.598 8.865 6.329 **6.227 **

**2.60 ** 9.372 8.327 7.846 **6.486 **

**2.80 ** 9.205 8.056 7.532 **7.949 **

**3.00 ** 8.867 9.484 7.408 **7.532 **

**3.20 ** 8.486 9.305 7.327 **7.407 **

**Available Online at www.ijpret.com ** 21

**Table T9: An evaluated result of 2DEG electron mobility **(*cm*2 /*V*sec)** as a function of energy **

**separation between two sub bands **(*E*1*E*0)_{ (meV) for triangular QW at T=300K }

**Energy separation **(*E*1*E*0)_{ meV }** _{2DEG mobility }**
2

(*cm* /*V*sec)

**40 ** **8122 **

**45 ** **8167 **

**50 ** **8198 **

**55 ** **8209 **

**60 ** **8297 **

**65 ** **8305 **

**70 ** **8329 **

**75 ** **8347 **

**80 ** **8278 **

**85 ** **8265 **

**90 ** **8236 **

**95 ** **8208 **

**100 ** **8179 **

**105 ** **8160 **

**110 ** **8152 **

**120 ** **8137 **

**Available Online at www.ijpret.com ** 22

**Table T10: An evaluated result of POP (polar optical phonon) scattering rate (**sec1**) of bulk **
**electrons, 2DEG and 1DEG, the 2DEG structure has a 130 ***A*0_{ well width and 1DEG has }

**structure 110***A*0_{x110}*A*0_{. The scattering rate has been evaluated as a function of electron }

**energy **

0

[*E* *E* ]

0

[*E* *E* ]

** _{ POP scattering rate (}** 1

sec _{) }

Bulk electrons 2DEG **1DEG **

**0.20 ** _{4.57x} 12

10 _{3.86x}1012 ** _{2.22x}**1012

**0.40 ** _{3.62x} 12

10 _{3.47x}1012 ** _{2.47x}**1012

**0.60 ** _{3.06x} 12

10 _{3.20x} 12

10 ** _{3.48x}** 12

10

**0.80 ** _{2.97x} 12

10 _{2.78x}1012 ** _{4.76x}**1012

**1.00 ** _{5.87x} 12

10 _{8.16x} 12

10 ** _{9.48x}** 13

10

**1.20 ** _{6.23x} 12

10 _{7.23x}1012 ** _{8.42x}**1013

**1.40 ** _{7.46x} 12

10 _{6.94x} 12

10 ** _{7.29x}** 13

10

**1.60 ** _{8.27x} 12

10 _{8.86x}1012 ** _{6.46x}**1013

**1.80 ** _{9.96x} 12

10 _{7.22x}1012 ** _{5.55x}**1013

**2.00 ** _{1.27x} 13

10 _{9.14x}1012 ** _{4.86x}**1013

**2.20 ** _{2.28x} 13

10 _{2.23x} 13

10 ** _{3.29x}** 13

10

**2.40 ** _{2.56x} 13

10 _{3.28x}1013 ** _{2.16x}**1013

**2.60 ** _{2.87x} 13

10 _{4.15x} 13

10 ** _{8.48x}** 12

**Available Online at www.ijpret.com ** 23

**2.80 ** _{2.95x} 13

10 _{5.54x}1013 ** _{7.79x}**1012

**3.00 ** _{3.05x} 13

10 _{5.97x}1013 ** _{6.64x}**1012

**3.50 ** _{3.17x} 13

10 _{6.23x}1013 ** _{5.32x}**1012

**4.00 ** ** _{3.65x}** 13

10 ** _{6.79x}**1013

**1012**

_{4.47x}**REFERENCES: **

1. **S K Sarkar and D chattopadhyay, Phys. Rev. B36, 264 (2000) **

2. **R Akimoto etal., Appl. Phys. Lett. 87, 181104 (2005) **

3. C Fujihashi etal., IEEE Transactions on Nanotechnology 6, 320 (2007)

4. G Dewey etal., IEEE Electron Device Lett. 29, 1094 (2008)

5. A. Gold, Appl. Phys. Lett. 92, 082111(2008)

6. **D N Quang and N H Tung, Phys Rev B77, 125335 (2008) **

7. **X Ni etal., Appl. Phys. Lett. 93, 121113 (2008) **

8. **D N Quang etal., Communication in Physics, 20, 193 (2010) **

9. **G C Bir etal., Appl Phys. Lett. 95, 031210 (2009) **

10. S Hooyong etal., Nanotechnology 21, 134026 (2010)

11. **Y H Xie etal., Appl. Phys. Lett. 63, 2263 (1993) **

12. K L Campman etal., Appl phys. Lett. 69, 2534 (1996)

13. H Celik etal., Semicon. Sci. Technol. 12, 389 (1997)

14. N Balkan etal., Superlattices Microstruct 22, 263 (1997)

15. M Cankurtaran etal., Phys. Status. Solidi B207, 139 (1998)

**Available Online at www.ijpret.com ** 24

17. C Gerl etal., Physica E32, 258 (2006)

18. **B. Rossner etal., Thin Solid Films 508, 351(2006) **

19. M Myronov etal., Appl. Phys. Lett. 88, 252115 (2006)

20. **F Szmulawicz etal., J. Appl. Phys. 101, 04706 (2007) **

21. **T T Hai etal., Proc. Natl. Conf. Theor. Phys. 35, 24 (2010) **

22. D N Quang etal., J. Appl. Phys. 104, 113711(2008)

23. T Wang etal., J. Appl. Phys. 74, 426 (1993)

24. J Y Zhang etal., Appl. Phys. Lett. 95, 161110 (2009)

25. J Wang etal., Appl Phys. Lett. 97, 201112 (2010)

26. ** C Lu etal., J. Appl. Phys. 113, 013102 (2013) **

27. **A Gold, Phys. Rev. 38, 10798 (1998) **

28. **T Ando etal., Rev. Mod. Phys. 54, 437 (1982) **

29. D N Quang etal., Phys. Rev. B68, 195316 (2003)

30. R. M. Feenstra etal., J. Appl. Phys. 78, 6091(1995)

31. D N Quang etal., Phys. Rev. B70, 195336 (2004)

32. G L Bir and G E Pikus, ‘Symmetry and Strain-induced Effects in Semiconductors (Wiley ,New

York 1974)

33. C G Van de Walle, Phys. Rev. B39, 1871(1989)

34. B K Ridley, J. Phys. C15, 5899 (1982)

35. S Briggs and J P Leburtow, Phys. Rev. B43, 4785(1991)

36. **C A Broderick etal., Phys. Rev. B90, 195301(2014) **

37. Wei Yi etal., Appl. Phys. Lett. 106, 142103 (2015)

**Available Online at www.ijpret.com ** 25

39. **Yan Liu etal., Nanoscale Res. Lett. (2017) 12, 120 **

40. **X Zhang etal., npj Quantum Materials, (2017) 2, 68 **

41. Son Tran etal., Science Advances, Vol3, 02 June (2017)