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INTERNATIONAL JOURNAL OF PURE AND
APPLIED RESEARCH IN ENGINEERING AND
TECHNOLOGY
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. MISHRA2
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) QT (Top-interface scattering), (ii) QTB (Top-bottom interface scattering) and (iii) QB (Bottom-interface scattering). The enhancement factor Q is calculated in three different cases: (a) as a function of QW well width L A( 0) keeping correlation length (A0) and sheet hole density p cms( 2) constant (b) as a function of sheet hole density p cms( 2) keeping L A( 0)and (A0),constant (c) as a function of correlation length (A0),keeping L A( 0)and p cms( 2) constant. In all these three cases enhancement factor Q increase for QT and QTBbut decrease for
B
Q . The value of Q is large in QT and small forQB. 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.
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:
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How to Cite This Article:
Ramani Shankar Pandey, IJPRET, 2019; Volume 7 (7): 1-25
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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 etal22 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 (A0) keeping correlation
length (A0) 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(A0) and correlation length (A0) constant and (iii) as a
function of correlation length (A0) keeping QW well width L(A0) 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
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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(A0). 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 (cm2/Vsec) 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.
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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 2B2 /Lcosh(c z L2 / ) 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 2kFsin( / 2)
with is the scattering angle. The Fermi energy is given by
2 2 * 2 F F k E m
with kF 2ps 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 2c 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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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 M3D 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 dz14(c)
1 ( , ) ( ) ( )
y z
iq y iq z
mn D y z y n z
I q q
y z y z e e dydz14(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 z15(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 z15(b)
Here, q is the phonon wave vector component parallel to the quantum well and K0 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
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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 g1D is the density of states for 1DEG with a broadening factor35 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/cm3
The results are shown in table T4 to T10.
RESULTS AND DISCUSSION;
Using the theoretical formalism of T. T. Hai etal21, D. N. Quang22 and T. Wang etal23, 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(A0) keeping
correlation length (A0) 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
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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 (A0) keeping correlation length (A0) 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 inQB.
The magnitude is large in QT but small inQB. 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 (E1E0) meV. Our obtained results show that 2DEG electron mobility increase,
attain maximum value and then decrease as a function of (E1E0). 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=80A0, 100A0, 120A0 and 150A0. We
observed that 2DEG scattering rate decrease as a function of
0
[E E ]
. The value is large for
L=80A0 and small for L=150A0. This type of scattering is due to inter sub-band scattering. It is
also noticed that 2DEG has peak mobility around E1E0 2 .In table T9, we repeated the
calculation of 2DEG electron mobility (cm2/V sec) as a function of (E1E0)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 130A0 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 (A0) 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 (A0)
constant. (iii) as a function of correlation length (A0)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 forQB. 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
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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 =10A0 and sheet hole density ps (1012cm2)
are kept
fixed.
L (A0) 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
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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 =10A0 and well width L=80 A0are kept fixed.
Sheet density ps QT QTB QB
2.58x1011 1.356 1.487 1.257
4.87x1011 1.479 1.593 1.204
7.32x1011 1.584 1.674 1.186
8.55x1011 1.695 1.785 1.084
9.43x1011 1.783 1.889 0.965
1.58x1012 1.995 1.965 0.843
2.76x1012 2.083 2.132 0.762
3.39x1012 2.187 2.329 0.675
4.76x1012 3.293 2.476 0.597
5.95x1012 4.375 2.843 0.486
6.67x1012 5.657 3.559 0.392
8.32x1012 6.732 3.678 0.305
9.57x1012 8.954 3.896 0.267
1.32x1013 9.107 4.059 0.219
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Table T3: An evaluated results of mobility enhancement factor Q for the p-type Square QW as
a function of correlation length (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 sheet hole density
13 2
2.10 10
s
p x cm
and well width L=130 A0are 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
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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 (A0) for electron energy
100mev at T=300K
Well width L(A0) 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
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Table T5: An evaluated result of 2DEG electron mobility (cm2 /Vsec) as a function of energy difference between first sub band and second sub band at T= 300K
Energy difference (E1E mev0) Electron mobility 2
(cm /Vsec)
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 (cm2 /Vsec) as a function of well
width (A0) for ALGaAs/GaAs QW at T=300K
Well width (A0) Electron mobility (cm2 /Vsec)
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 (A0) for low dimensional QW
0
E E
--- 2DEG scattering rate ( 1
sec )---
L=80 A0 L=100 A0 L=120 A0 L=150 A0
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 (cm2 /Vsec) as a function of energy
separation between two sub bands (E1E0) (meV) for triangular QW at T=300K
Energy separation (E1E0) meV 2DEG mobility 2
(cm /Vsec)
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 A0 well width and 1DEG has
structure 110A0x110A0. 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 3.86x1012 2.22x1012
0.40 3.62x 12
10 3.47x1012 2.47x1012
0.60 3.06x 12
10 3.20x 12
10 3.48x 12
10
0.80 2.97x 12
10 2.78x1012 4.76x1012
1.00 5.87x 12
10 8.16x 12
10 9.48x 13
10
1.20 6.23x 12
10 7.23x1012 8.42x1013
1.40 7.46x 12
10 6.94x 12
10 7.29x 13
10
1.60 8.27x 12
10 8.86x1012 6.46x1013
1.80 9.96x 12
10 7.22x1012 5.55x1013
2.00 1.27x 13
10 9.14x1012 4.86x1013
2.20 2.28x 13
10 2.23x 13
10 3.29x 13
10
2.40 2.56x 13
10 3.28x1013 2.16x1013
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 5.54x1013 7.79x1012
3.00 3.05x 13
10 5.97x1013 6.64x1012
3.50 3.17x 13
10 6.23x1013 5.32x1012
4.00 3.65x 13
10 6.79x1013 4.47x1012
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)
