Excerpt
Index
i
Nanostructure Physics of nonisolated Quantum Well of symmetric double barrier:
analytical calculation of equation obeyed by quasibound energy levels of the
Quantum Well
Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash
Department of Physics, Shahjalal University of Science and Technology,
Sylhet 3114, Bangladesh, www.sust.edu, schowdhuryphy@yahoo.com
Chapter
I
Background on Quantum Mechanics
124
1.1 Wave equation of a free particle: Schrödinger equation
2
1.2 Schrödinger equation of a particle subject to a conservative mechanical
force
3
1.3 Conservation of probability and probability current density
5
1.4 Timeindependent Schrödinger equation and stationary state
6
1.5 Continuous and discontinuous function
9
1.6 Finite and infinite discontinuity
11
1.7 Admissibility conditions on wavefunction
12
1.8 Free particle: eigenfunctions and probability current density
14
1.9 Single rectangular tunnel barrier
15
1.9.1 Calculation of transfer matrix and investigation of its properties
(E < V
0
)
16
1.9.2 Calculation of transmission coefficient
21
1.10 Further topics on Quantum Mechanics
24
Index
ii
Chapter
II
Background on Microelectronics
25
35
2.1 Intrinsic
semiconductor 26
2.2 Semiconductors: elemental and binary
27
2.3 Alloy semiconductors (ternary and quaternary)
30
2.4 Bandgap
engineering
31
2.5 Semiconductor heterojunction and heterostructure
33
2.6 Effective
mass
34
2.7 Further topics on Microelectronics
35
Chapter
III
Background on Nanostructure Physics
36
58
3.1 Single rectangular tunnel barrier
37
3.2 Transmission coefficient of a single rectangular tunnel barrier
38
3.3 Quantum
well
(QW)
44
3.4 Double
potential
barrier
48
3.5 Transmission coefficient of double potential barrier
49
3.6 Transmission coefficient of double barrier if two barriers are identical
49
3.7 Profile of transmission peak
51
3.8 T versus E curve of symmetric rectangular double barrier
52
3.9 Further study of double barrier structures
58
Index
iii
Chapter
IV
Analytical calculation of transcendental equation
obeyed by quasibound energy levels of the nonisolated Quantum Well
of symmetric rectangular double barrier
59
74
4.1 Description of the problem
60
4.2 Methodology and Physics of calculation of the transcendental equation
61
4.3 Calculation of transfer matrix of the left hand rectangular tunnel barrier 64
4.4 Calculation of inverse transfer matrix of the right hand rectangular
tunnel barrier
68
4.5 Calculation of transcendental equation obeyed by quasibound energy
levels
72
4.6 Resonant transmission peaks obey the same condition
73
Chapter
V
Taking effective mass inequality into account:
analytical calculation of transcendental equation
obeyed by quasibound energy levels
of the nonisolated Quantum Well
of symmetric rectangular double barrier:
75
90
5.1 Introduction
76
5.2 Calculation of transfer matrix of the left hand rectangular tunnel
barrier: taking effective mass inequality into account
77
5.3 Calculation of inverse transfer matrix of the right hand rectangular
tunnel barrier: taking effective mass inequality into account
82
5.4 Calculation of transcendental equation obeyed by quasibound energy
levels: taking effective mass inequality into account
88
5.5 Resonant transmission peaks obey the same condition:
taking effective mass inequality into account
90
Index
iv
Chapter
VI
Derivation of WKB solution of Schroedinger equation
and introduction to WKB connection formulae
91
100
6.1 WKB
approximation
92
6.2 WKB solution of (1 dimensional) Schroedinger equation
92
6.3 Classical turning point
96
6.4 WKB connection formula: case I
98
6.5 WKB connection formula: case II
99
Chapter
VII
Analytical calculation of transfer matrix and transmission coefficient
of single tunnel barrier of general shape using WKB method
101
112
7.1 Calculation of transfer matrix of single tunnel barrier of general shape
using WKB method
102
7.2 Calculation of transmission coefficient of single tunnel barrier of
general shape using WKB method
110
Chapter
VIII
Symmetric double barrier of general shape:
analytical calculation of condition or equation
obeyed by quasibound energy levels of nonisolated Quantum Well
using WKB method
113
130
8.1 Description of the problem
114
8.2 Methodology and Physics of calculation of the condition or equation
115
8.3 Calculation of inverse transfer matrix of right hand tunnel barrier
118
8.4 Calculation of transcendental equation obeyed by quasibound energy
levels using WKB method
127
8.5 Resonant transmission peaks obey the same condition
130
References
131
133
Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:
Nanostructure Physics of nonisolated Quantum Well of symmetric double barrier:
analytical calculation of equation obeyed by quasibound energy levels of the Quantum Well
1
Chapter I
Background on Quantum Mechanics
Chapter I Background on Quantum Mechanics
2
1.1 Wave equation of a free particle: Schrödinger equation
If we associate the wave packet
)
t
,
x
(
=
2
1
dk
e
)
k
(
a
)
t
kx
(
i
³
+


where a(k) =
2
1
dx
e
)
t
,
x
(
)
t
kx
(
i
³
+



with a free material particle, we can write
)
t
,
x
(
=
!
2
1
dp
e
)
p
(
a
)
Et
px
(
i
³
+


!
where a(p) =
!
2
1
dx
e
)
t
,
x
(
)
Et
px
(
i
³
+



!
using de Broglie's equations p =
! k and E =
! . In three dimensions, we have
)
t
,
r
(
&
=
3
)
2
(
1
!
³

p
d
e
)
p
(
a
)
Et
r
.
p
(
i
&
&
&
&
!
__________ (1.1)
where r
.
p
&
&
=
z
p
y
p
x
p
z
y
x
+
+
. Equation (1.1) gives
x
=
3
x
)
2
(
p
i
!
!
p
d
e
)
p
(
a
)
Et
r
.
p
(
i
&
&
&
&
!
³

and
2
2
x
=
3
2
2
x
)
2
(
p
!
!

p
d
e
)
p
(
a
)
Et
r
.
p
(
i
&
&
&
&
!
³

.
Similarly,
2
2
y
=
3
2
2
y
)
2
(
p
!
!

p
d
e
)
p
(
a
)
Et
r
.
p
(
i
&
&
&
&
!
³

and
2
2
z
=
3
2
2
z
)
2
(
p
!
!

p
d
e
)
p
(
a
)
Et
r
.
p
(
i
&
&
&
&
!
³

.
all momentum space
Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:
Nanostructure Physics of nonisolated Quantum Well of symmetric double barrier:
analytical calculation of equation obeyed by quasibound energy levels of the Quantum Well
3
2
=
3
2
2
z
2
y
2
x
)
2
(
/
)
p
p
p
(
!
!
+
+

p
d
e
)
p
(
a
)
Et
r
.
p
(
i
&
&
&
&
!
³

=
3
2
2
)
2
(
p
!
!

p
d
e
)
p
(
a
)
Et
r
.
p
(
i
&
&
&
&
!
³

or,
2
=
2
2
p
!

or,
m
2
2
!

)
t
,
r
(
2
&
=
m
2
p
2
)
t
,
r
(
&
_________(1.2)
Again, equation (1.1) gives
t
=
!
iE

3
)
2
(
1
!
p
d
e
)
p
(
a
)
Et
r
.
p
(
i
&
&
&
&
!
³

=
!
iE

or, i
!
t
= E
_________(1.3)
For a free particle, E =
m
2
p
2
. Hence equation (1.2) and (1.3) give
m
2
2
!

2
= i
!
t
_________(1.4)
Equation (1.4) is the differential equation for the matter wave of a free particle and
equation (1.4) is called wave equation or Schrödinger equation for a free particle.
Equation (1.4) is a linear equation and hence a monochromatic wave such as
)
Et
r
.
p
(
i
e
)
p
(
a

&
&
!
&
as well as a wave packet given by equation (1.1) satisfy it.
1.2 Schrödinger equation of a particle subject to a conservative mechanical
force
1) A comparison of
< x > =
³
x
*
dx
and
< p > =
³
x
i
*
!
dx
shows that expectation value of momentum < p > of a particle having wavefunction
associated with it can be computed in the same way as that of position < x > if the
Chapter I Background on Quantum Mechanics
4
operator
x
i
!
is substituted in place of x. This statement introduces operator
formalism in quantum mechanics. Thus the operator of x is x, operator of p is
x
i
!
.
By "operator of p is
x
i
!
", we in fact mean, the operator we need to calculate the
expectation value of p is
x
i
!
.
2) The Schrödinger equation of a free particle is

2
2
2
x
m
2
!
= i
t
!
.
This equation can be obtained from the classical equation
m
2
p
2
= E by the operator
correspondence of p and E as
x
i
!
and i
t
! respectively, and letting the operators
operate on the wavefunction
. Thus the operator of E is i
t
! .
3) If a conservative force acts on a particle, the total energy E =
m
2
p
2
+ V where V is
potential energy and hence V is a function of position only. Hence
>
<
)
x
(
V
=
³
+

dx
)
t
,
x
(
)
x
(
V
2
=
³
+

dx
)
x
(
V
*
Thus the operator of V(x) is V(x).
4) Using the operator correspondences of E, p and V,
m
2
p
2
+ V = E gives
m
2
x
i
2
¸
¹
·
¨
©
§
!
+ V = i
t
!
or,

2
2
2
x
m
2
!
+ V = i
t
!
or,

2
2
2
x
m
2
!
+ V
= i
t
!
or,
»
»
¼
º
«
«
¬
ª
+

V
x
m
2
2
2
2
!
= i
t
!
.
Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:
Nanostructure Physics of nonisolated Quantum Well of symmetric double barrier:
analytical calculation of equation obeyed by quasibound energy levels of the Quantum Well
5
This is the Schrödinger equation of a particle moving in a potential V. In three
dimensions, [
m
2
2
!

2
+ V]
= i
t
!
is the Schrödinger equation of a particle
moving in a potential V. The operator [
m
2
2
!

2
+ V ] is the operator of total energy
of a conservative system and is called Hamiltonian operator.
1.3 Conservation of probability and probability current density
Let us consider a particle under the action of a conservative mechanical force.
The wave equation of the associated matter wave is given by the Schrödinger
equation

m
2
2
!
2
+ V = i!
t
(1.5)
where V is the potential associated with the force. Complex conjugate of equation
(1.5) is

m
2
2
!
2
*
+ V
*
=  i!
t
*
V is real.
(1.6)
Multiplying equation (1.5) by
*
, we get

m
2
2
!
*
2
+ V
*
= i!
*
t
_______(1.7)
Multiplying equation (1.6) by
, we get

m
2
2
!
2
*
+ V
*
=  i!
t
*
_______(1.8)
Equation (1.7) (1.8)

m
2
2
!
[
*
2

2
*
] = i! ¨
©
§
t
*
+
¸
¸
¹
·
t
*
or,

m
2
2
!
&
· (
*
&

&
*
) = i!
t
(
*
)
or,
&
· [
mi
2
!
(
*
&

&
*
)] = 
t
2
_______(1.9)
Chapter I Background on Quantum Mechanics
6
or,
&
· S
&
=

t
where
=
2
and S
&
=
mi
2
!
(
*
&

&
*
)
_______(1.10)
Equation (1.9)
³
·
v
d
S
&
&
=

t
³
v
d
over a volume v.
or,
³
· A
d
S
&
&
=

t
³
v
d
_______(1.11)
Using
Gauss's
divergence
theorem
The surface integral is over the closed surface that encloses the volume v. In equation
(1.11),
³
v
d is the probability of the presence of a particle in the volume v.

t
³
v
d is the time rate of decrease of the probability of the presence of the particle
in the volume v. Since this rate is equal to
³
· A
d
S
&
&
, we can say that
S
&
is the time rate
of flowout of the probability per unit area through the surface enclosing the volume
v. The probability changes or decreases because of the change of
with time.
Hence
S
&
is called probability current density. For a system of a large number of
particles, < S > is average particle current density, i.e. number of particles crossing
per unit time through a unit area perpendicular to the flow.
1.4 Timeindependent Schrödinger equation and stationary state
The Schrödinger equation for a particle under conservative force is

m
2
2
!
2
)
t
,
r
(
&
+ V
)
t
,
r
(
&
= i
!
t
)
t
,
r
(
&
______(1.12)
which is called timedependent Schrödinger equation. If V( r
&
) is independent of time,
the Hamiltonian is also timeindependent and equation (1.12) simplifies
considerably.
Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:
Nanostructure Physics of nonisolated Quantum Well of symmetric double barrier:
analytical calculation of equation obeyed by quasibound energy levels of the Quantum Well
7
Let us try
)
t
,
r
(
&
= u( r
&
) f(t) as a solution of equation (1.12). u is a function
of space coordinates only and f is a function of time only. Equation (1.12)

m
2
2
!
2
[u( r& ) f(t)] + V( r& ) u( r& ) f(t) = i!
t
[u( r
&
) f(t)]
or,

m
2
2
!
f(t)
2
u( r& ) + V( r& ) u( r& ) f(t) = i! u( r& )
t
f(t)
or,

m
2
2
!
( )
( )
r
u
r
u
2
&
&
+ V( r
&
) =
t
)
t
(
f
)
t
(
f
i
!
_______(1.13)
Dividing by u( r
&
) f(t)
The LHS of equation (1.13) is a function of position only and the RHS of equation
(1.13) is a function of time only, because V( r
&
) is a function of position only. Since
space (coordinates) and time are independent (ignoring theory of relativity), equation
(1.13) makes sense only if both sides of equation (1.13) are equal to a constant, say
C.
Thus

m
2
2
!
( )
( )
r
u
r
u
2
&
&
+ V( r
&
) = C
______(1.14)
and
t
)
t
(
f
)
t
(
f
i
!
= C
______(1.15)
Equation (1.14)

m
2
2
!
2
u( r& ) + V( r& ) u( r& ) = C u( r& )
or, [

m
2
2
!
2
+ V( r& )] u( r& ) = C u( r& )
______(1.16a)
or,
op
H
u( r
&
) = C u( r
&
)
______(1.16b)
op
H
is Hamiltonian operator, i.e. operator of total energy. Equation (1.16) is
eigenvalue equation of total energy.
C is an eigenvalue of total energy (an
observable).
C is real. Let us denote C by E.
Equation (1.16)
Chapter I Background on Quantum Mechanics
8
[

m
2
2
!
2
+ V( r& )] u( r& ) = E u( r& )
______(1.17a)
or,
op
H
u( r
&
) = E u( r
&
)
______(1.17b)
Equation (1.17) is called timeindependent Schrödinger equation. Equation (1.15)
i
!
t
f(t) = E f(t)
or,
dt
d
f(t) =
!
i
)
t
(
Ef
Let f(t) =
nt
e
n
nt
e
= E
!
i
1
nt
e
or, (n
 E
!
i
1
)
nt
e
= 0 ,
n  E
!
i
1
= 0
nt
e
0
or, n = E
!
i
1
=

!
iE
=
 i
f(t) =
t
i
e

=
Et
i
e
!

( r& , t) = u( r& ) f(t) = u( r& )
Et
i
e
!

= u( r
&
)
t
i
e

______(1.18)
2
)
t
,
r
(
&
=
*
( r& , t) ( r& , t)
= u*( r
&
)
t
E
i
*
e
!
u( r
&
)
Et
i
e
!

= u*( r
&
) u( r
&
)
E is real,
E* = E.
=
2
)
r
(
u
&
_______(1.19)
From equation (1.19), we find that
2
)
t
,
r
(
&
is independent of time. Thus the states
given by equation (1.18) are stationary states.
The expectation value of any observable A is
< A > =
³
d
)
t
,
r
(
A
)
t
,
r
(
op
*
&
&
=
³

d
]
e
)
r
(
u
[
A
e
)
r
(
u
Et
i
op
t
E
i
*
*
!
!
&
&
using equation (1.18)
=
³

d
)
r
(
u
A
e
e
)
r
(
u
op
Et
i
Et
i
*
&
&
!
!
E is real.
If
op
A
does not explicitly contain the variable t (time).
=
³
d
)
r
(
u
A
)
r
(
u
op
*
&
&
For stationary states, the expectation value of any observable A is independent of
time, provided the operator
op
A
itself does not depend explicitly on t.
Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:
Nanostructure Physics of nonisolated Quantum Well of symmetric double barrier:
analytical calculation of equation obeyed by quasibound energy levels of the Quantum Well
9
In hydrogen atom, e.g., the potential V(r) =
0
4
1

r
e
is a function of position
only. Thus the solutions of equation (1.12) will give stationary states. Thus
2
for
the electron and the expectation value of all observables (e.g. energy) of the electron
remains independent of time. This explains why hydrogen atom is stable; thus we get
an explanation of one of the ad hoc assumptions of Bohr that the electron in
hydrogen atom stays in stationary state.
1.5 Continuous and discontinuous function
f(x) above is a continuous function. It is singlevalued at every value of x.
dx
df
is also
continuous and singlevalued.
x
f(x)
O
Chapter I Background on Quantum Mechanics
10
f(x) above is a discontinuous function of x. The discontinuity is at x = x
0
where the
function is many valued. f(x
0
) is unspecified.
1
f < f(x
0
) <
2
f .
0
Lt
f(x
0

) =
2
f
0
Lt
f(x
0
+
) =
1
f .
dx
df
at x = x
0
 is tan
1
.
dx
df
at x = x
0
+
is tan
2
. Here 0.
1
and
2
may or may not be equal, depending on the nature of f(x). Thus the
derivative
dx
df
may or may not be continuous. At x = x
0
,
dx
df
= tan 90
°=
.
x
f(x)
O
f
1
f
2
x
0
x
f(x)
O
x
0
1
2
Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:
Nanostructure Physics of nonisolated Quantum Well of symmetric double barrier:
analytical calculation of equation obeyed by quasibound energy levels of the Quantum Well
11
1.6 Finite and infinite discontinuity
The discontinuity considered above is finite discontinuity, because
1
f
and
2
f
are finite. If a < f(x
0
) < b where either a or b (or both) is +
or

, the discontinuity
is infinite discontinuity.
0
Lt
f(x
0
+
) = a
0
Lt
f(x
0
 ) = +
a < f(x
0
) <
.
x
f(x)
O
x
0
a
x
f(x)
O
90
°
x
0
Chapter I Background on Quantum Mechanics
12
1.7 Admissibility conditions on wavefunction
1.
(x , t)
0 as x
± , because
³
d
2
= 1 is finite.
2.
(x , t) must be finite, singlevalued and continuous function of x for all time t;
this is because of probability interpretation of
.
3.
t
is a continuous function of x, because otherwise
t
= c where
1
c
< c <
2
c
at
say x = x
0
.
This means
= c t which is manyvalued at x = x
0
. But
must be singlevalued,
according to condition (2).
4.
x
must be continuous function of x if V(x , t) is continuous. Because, if V(x, t)
is continuous, V(x, t)
(x, t) is continuous.
t
is also continuous (condition (3))
function of x. Hence Schrödinger equation
i
!
t
t)
(x,
=

m
2
2
!
2
2
x
(x , t) +V(x , t) (x , t)
x
t
O
c
1
c
2
x
0
all space
Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:
Nanostructure Physics of nonisolated Quantum Well of symmetric double barrier:
analytical calculation of equation obeyed by quasibound energy levels of the Quantum Well
13
gives
2
2
x
is continuous. Thus
x
is continuous function of x; otherwise
2
2
x
becomes infinite at points (x) where
x
is discontinuous.
5.
x
must be continuous function of x if V does not have infinite discontinuity. In
Schrödinger equation i
!
t
=

m
2
2
!
2
2
x
+ V(x, t) (x, t),
t
is always
continuous function of x (condition 3). If
x
has any discontinuity say at x = x
0
, the
x
versus x curve becomes vertical at x = x
0
and hence its slope
2
2
x
becomes
infinite at x = x
0
. Thus

m
2
2
!
2
2
x
=

. This forces V to become + to keep the
Schrödinger equation valid. Since
is finite, continuous and single valued
everywhere, V is forced to be +
at x = x
0
. Thus unless V has an infinite
discontinuity,
x
must be a continuous function of x.
Any finite discontinuity of V makes V
finite discontinuous. This is adjusted
by a finite discontinuity of
2
2
x
because
t
is always continuous [see Schrödinger
equation]. Finite discontinuity of
2
2
x
means
x
is continuous, otherwise
2
2
x
gets infinite discontinuity there.
Chapter I Background on Quantum Mechanics
14
1.8 Free particle: eigenfunctions and probability current density
If the potential is constant, V(x) =
0
V , the force acting on the particle F(x) =

x
V
= 0; so the particle is free.
0
V can be taken to be zero, without any loss of
generality.
The timeindependent Schrödinger equation becomes

m
2
2
!
2
2
dx
)
x
(
u
d
+ 0 = E u(x)
or,
2
2
dx
u
d
+
2
mE
2
!
u = 0
or,
2
2
dx
u
d
+
2
k
u = 0.
If
0
V
0, k
=
)
V
E
(
m
2
0
2

!
.
u(x) = A
ikx
e
(x , t) = A
)
t
kx
(
i
e

= A
)
Et
px
(
i
e

!
k =
2
mE
2
!
, E =
m
2
k
2
2
!
The probability current density corresponding to the free particle eigenfunction
=A
)
Et
px
(
i
e

!
is
S =
mi
2
!
[
x
*

x
*
]
=
mi
2
!
[
*
A
)
Et
px
(
i
e


!
x
(A
)
Et
px
(
i
e

!
)
 A
)
Et
px
(
i
e

!
x
(
*
A
)
Et
px
(
i
e


!
)]
=
mi
2
!
2
A [
!
i
p (

!
i
p)] =
mi
2
!
2
A 2
!
i
p
=
m
p
2
A
Thus S = v
2
A and
2
=
2
A .
Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:
Nanostructure Physics of nonisolated Quantum Well of symmetric double barrier:
analytical calculation of equation obeyed by quasibound energy levels of the Quantum Well
15
1.9 Single rectangular tunnel barrier
a
+a x
If we have variation of potential V(x) as shown in the above figure, we have a
onedimensional, single, rectangular potential barrier. If the width and height of the
barrier are finite and small, we have a tunnel barrier of width 2a and height V
0
. The
barrier is defined as
V(x) = 0 for
a
x
>
= V
0
for
a
x
a
+
<
<

There are two finite discontinuities of V(x), one at x =
a and another at x = +a.
There are three regions as shown. According to the choice of origin, V(x) is zero in
two of the three regions and is constant (V
0
) in region II. We now proceed to obtain
transfer matrix of the barrier and find its properties. We shall use waves and match
them at the potential discontinuities using the boundary conditions described in
section 1.7.
V(x)
(0, 0)
Region I
Region II
Region III
V
0
Chapter I Background on Quantum Mechanics
16
1.9.1 Calculation of transfer matrix and investigation of its properties (E < V
0
)
a
+a x
Solutions of timeindependent Schrödinger equation
[

m
2
2
!
2
2
dx
d
+ V(x)] u(x) = E u(x)
or,
2
2
dx
u
d
+
0
u
))
x
(
V
E
(
m
2
2
=

!
in the three regions are u
1
, u
2
and u
3
given by
ikx
ikx
1
Be
Ae
)
x
(
u

+
=
where
2
2
mE
2
k
!
=
x
x
2
De
Ce
)
x
(
u

+
=
where
2
0
2
)
E
V
(
m
2
!

=
ikx
ikx
3
He
Ge
)
x
(
u

+
=
The expressions for u
2
and
2
imply that we are considering free electrons of kinetic
energy less than V
0
impinging on the barrier from the left.
Using the boundary condition
u
1
= u
2
at x =
a, we get
ika
ika
Be
Ae
+

=
a
a
De
Ce

+
(1.20)
Again, the boundary condition
dx
du
dx
du
2
1
=
at x =
a gives
ik
ikx
ikx
ikBe
Ae


=
x
x
De
Ce


at x =
a
or, ik
ika
ika
ikBe
Ae


=
a
a
De
Ce


V(x)
(0, 0)
Region I
Region II
Region III
V
0
Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:
Nanostructure Physics of nonisolated Quantum Well of symmetric double barrier:
analytical calculation of equation obeyed by quasibound energy levels of the Quantum Well
17
or,
ika
ika
Be
ik
Ae
ik


=
a
a
De
Ce


(1.21)
The boundary condition u
2
= u
3
at x = +a gives
a
a
De
Ce

+
=
ika
ika
He
Ge

+
(1.22)
And
dx
du
dx
du
3
2
=
at x = +a gives
x
x
De
Ce


=
ikx
ikx
ikHe
ikGe


at x = +a
=>
a
a
De
Ce


=
ika
ika
ikHe
ikGe


=>
a
a
De
ik
Ce
ik


=
ika
ika
He
Ge


(1.23)
From equation (1.20) + (1.21), we have
ika
ika
a
Be
)
ik
1
(
Ae
)
ik
1
(
Ce
2

+
+
=


=>
a
ika
a
ika
Be
)
ik
1
(
2
1
Ae
)
ik
1
(
2
1
C
+
+


+
+
=
From equation (1.20)
 (1.21), we have
ika
ika
a
Be
)
ik
1
(
Ae
)
ik
1
(
De
2
+
+

=

=>
a
ika
a
ika
Be
)
ik
1
(
2
1
Ae
)
ik
1
(
2
1
D



+
+

=
Now in the form of matrix we can write
¸¸
¸
¸
¸
¹
·
¨¨
¨
¨
¨
©
§
¸¸
¸
¸
¸
¸
¹
·
¨¨
¨
¨
¨
¨
©
§
+


+
=
¸¸
¸
¸
¸
¹
·
¨¨
¨
¨
¨
©
§

+


+

B
A
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
D
C
a
ika
a
ika
a
ika
a
ika
From equation (1.22) + (1.23), we have
ika
a
ika
a
a
a
ika
De
)
ik
1
(
2
1
Ce
)
ik
1
(
2
1
G
De
)
ik
1
(
Ce
)
ik
1
(
Ge
2





+
+
=
=>

+
+
=
Chapter I Background on Quantum Mechanics
18
From equation (1.22)
 (1.23), we have
ika
a
ika
a
a
a
ika
De
)
ik
1
(
2
1
Ce
)
ik
1
(
2
1
H
De
)
ik
1
(
Ce
)
ik
1
(
He
2
+

+


+
+

=
=>
+
+

=
Now in the form of matrix we can write
¸¸
¸
¸
¸
¹
·
¨¨
¨
¨
¨
©
§
¸
¸
¸
¸
¹
·
¨
¨
¨
¨
©
§
+


+
=
¸¸
¸
¸
¸
¹
·
¨¨
¨
¨
¨
©
§
+



+

D
C
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
H
G
ika
a
ika
a
ika
a
ika
a
¸¸
¸
¸
¸
¹
·
¨¨
¨
¨
¨
©
§
¸
¸
¸
¸
¹
·
¨
¨
¨
¨
©
§
+


+
¸
¸
¸
¸
¹
·
¨
¨
¨
¨
©
§
+


+
=
¸¸
¸
¸
¸
¹
·
¨¨
¨
¨
¨
©
§

+


+

+



+

B
A
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
H
G
a
ika
a
ika
a
ika
a
ika
ika
a
ika
a
ika
a
ika
a
or,
¸¸¹
·
¨¨©
§
=
¸¸¹
·
¨¨©
§
22
12
21
11
M
M
M
M
H
G
¸¸¹
·
¨¨©
§
B
A
(1.24)
The M matrix is called transfer matrix. The elements of the matrix are calculated in
the following.
a
ika
ika
a
11
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
M
+


+
+
=
a
ika
ika
a
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1






+
ika
2
a
2
a
2
ika
2
a
2
a
2
e
]
e
)
1
ik
ik
1
(
e
)
1
ik
ik
1
[(
4
1
e
]
e
)
ik
1
)(
ik
1
(
e
)
ik
1
)(
ik
1
[(
4
1




+


+
+
+
+
=


+
+
+
=
Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:
Nanostructure Physics of nonisolated Quantum Well of symmetric double barrier:
analytical calculation of equation obeyed by quasibound energy levels of the Quantum Well
19
ka
2
i
a
2
a
2
a
2
a
2
e
]
2
/
)
e
e
)(
ik
ik
(
2
1
)
e
e
(
2
1
[




+
+
+
=
ka
2
i
e
]
a
2
sinh
)
k
k
(
2
i
a
2
[cosh


+
=
(1.25)
a
ika
ika
a
22
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
M
+
+


=
a
ika
ika
a
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1

+

+
+
+
ika
2
a
2
a
2
e
]
e
)
ik
1
)(
ik
1
(
e
)
ik
1
)(
ik
1
[(
4
1

+
+
+


=
ka
2
i
a
2
a
2
a
2
a
2
ka
2
i
a
2
a
2
e
]
2
/
)
e
e
)(
ik
ik
(
2
1
)
e
e
(
2
1
[
e
]
e
)
1
ik
ik
1
(
e
)
1
ik
ik
1
[(
4
1




+

+
=
+
+
+
+
+


=
ka
2
i
e
]
a
2
sinh
)
ik
ik
(
2
1
a
2
[cosh
+

=
ka
2
i
e
]
a
2
sinh
)
k
k
(
2
i
a
2
[cosh


=
(1.26)
a
ika
ika
a
12
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
M
+


+
=
]
e
)
ik
1
)(
ik
1
(
e
)
ik
1
)(
ik
1
[(
4
1
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
a
2
a
2
a
ika
ika
a




+

+

+
=
+

+
]
e
)
1
ik
ik
1
(
e
)
1
ik
ik
1
[(
4
1
a
2
a
2



+
+

+

=
2
/
)
e
e
)(
ik
ik
(
2
1
a
2
a
2



=
a
2
sinh
)
ik
ik
(
2
1

=
a
2
sinh
)
k
k
(
2
i
+

=
(1.27)
Chapter I Background on Quantum Mechanics
20
a
ika
ika
a
21
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
M
+

+
+

=
]
e
)
ik
1
)(
ik
1
(
e
)
ik
1
)(
ik
1
[(
4
1
e
)
ik
1
(
2
1
e
)
ik
1
(
2
1
a
2
a
2
a
ika
ika
a



+


+
+
+

=

+
+
a
2
sinh
)
ik
ik
(
2
1
2
/
)
e
e
)(
ik
ik
(
2
1
]
e
)
1
ik
ik
1
(
e
)
1
ik
ik
1
[(
4
1
a
2
a
2
a
2
a
2

=


=

+

+


+
=


a
2
sinh
)
k
k
(
2
i
+
=
(1.28)
We find that the elements of the transfer matrix M obey the following properties.
11
*
22
M
M
=
22
*
11
M
M
,
or
=
(1.29)
21
*
12
M
M
=
or,
12
*
21
M
M
=
(1.30)
We now evaluate the determinant of the M matrix.
12
*
12
11
*
11
21
12
22
11
22
21
12
11
M
M
M
M
M
M
M
M
M
M
M
M

=

=
ka
2
i
ka
2
i
e
]
a
2
sinh
)
k
k
(
2
i
a
2
[cosh
e
]
a
2
sinh
)
k
k
(
2
i
a
2
[cosh


+


=
]
a
2
sinh
)
k
k
(
2
i
[
+

]
a
2
sinh
)
k
k
(
2
i
[
+

1
a
2
sinh
a
2
cosh
a
2
sinh
]
k
k
4
[
4
1
a
2
cosh
a
2
sinh
]
)
k
k
(
)
k
k
[(
4
1
a
2
cosh
a
2
sinh
)
k
k
(
4
1
a
2
sinh
)
k
k
(
4
1
a
2
cosh
2
2
2
2
2
2
2
2
2
2
2
2
2
=

=

+
=
+


+
=
+


+
=
Thus
1
M
M
M
M
22
21
12
11
=
(1.31)
i.e. the determinant of the transfer matrix is unity.
Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:
Nanostructure Physics of nonisolated Quantum Well of symmetric double barrier:
analytical calculation of equation obeyed by quasibound energy levels of the Quantum Well
21
1.9.2 Calculation of transmission coefficient
a
+a x
We have solutions of timeindependent Schrödinger equation
[

m
2
2
!
2
2
dx
d
+ V(x)] u(x) = E u(x)
or,
2
2
dx
u
d
+
0
u
))
x
(
V
E
(
m
2
2
=

!
in the three regions given by u
1
, u
2
and u
3
ikx
ikx
1
Be
Ae
)
x
(
u

+
=
where
2
2
mE
2
k
!
=
x
x
2
De
Ce
)
x
(
u

+
=
where
2
0
2
)
E
V
(
m
2
!

=
ikx
ikx
3
He
Ge
)
x
(
u

+
=
.
To calculate transmission coefficient of the tunnel barrier, we need to
recognize that A is amplitude of the plane wave incident on the barrier from the left.
B is that of the wave reflected from the barrier. G is that of transmitted plane wave. H
is that of the reflected wave (if any) in region III. We have equation (1.24) relating
amplitudes of the four plane waves.
¸
¸
¸
¹
·
¨
¨
¨
©
§
¸
¸
¸
¹
·
¨
¨
¨
©
§
=
¸
¸
¸
¹
·
¨
¨
¨
©
§
B
A
M
M
M
M
H
G
22
21
12
11
The matrix equation is equivalent to the following two equations.
B
M
A
M
G
12
11
+
=
(1.32)
and B
M
A
M
H
22
21
+
=
.
(1.33)
V(x)
(0, 0)
Region I
Region II
Region III
A
B
V
0
G
H
C
D
Chapter I Background on Quantum Mechanics
22
To obtain an expression for transmission coefficient, we set H = 0 recognizing that
we expect no reflection in region III. As such, equation (1.33) gives
A
M
M
B
22
21

=
(1.34)
which we can put in equation (1.32) to get
A
M
M
M
A
M
G
22
21
12
11

=
A
M
M
M
M
M
22
21
12
22
11

=
A
M
1
22
=
(1.35)
Using
equation
(1.31)
Now transmission coefficient of the single barrier is given by
2
2
1
A
G
T
=
.
With the aid of equation (1.35), T
1
reduces to
2
22
1
M
1
T
=
(1.36)
Reflection coefficient of the single barrier is given by
2
2
1
A
B
R
=
.
With the aid of equation (1.34), R
1
reduces to
2
22
2
21
1
M
M
R
=
(1.37)
1
2
21
T
M
=
(1.38)
using equation (1.36).
We now use equation (1.36) to obtain an analytic expression for T
1
for E < V
0
,
The transmission coefficient is given by
2
22
1
M
1
T
=
22
*
22
M
M
1
=
ka
2
i
ka
2
i
e
]
a
2
sinh
)
k
k
(
2
i
a
2
[cosh
e
]
a
2
sinh
)
k
k
(
2
i
a
2
[cosh
1



+
=

Using
equation
(1.26)
a
2
sinh
)
k
k
(
4
1
a
2
sinh
1
1
a
2
sinh
)
k
k
(
4
1
a
2
cosh
1
2
2
2
2
2
2

+
+
=

+
=
Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:
Nanostructure Physics of nonisolated Quantum Well of symmetric double barrier:
analytical calculation of equation obeyed by quasibound energy levels of the Quantum Well
23
a
2
sinh
]
2
k
k
4
[
4
1
1
1
a
2
sinh
]
)
k
k
(
4
1
1
[
1
1
2
2
2
2
2
2
2

+
+
+
=

+
+
=
a
2
sinh
)
k
k
(
4
1
1
1
2
2
+
+
=
(1.39a)
a
2
sinh
2
k
k
4
1
1
1
2
2
2
»
»
¼
º
«
«
¬
ª
+
¸
¹
·
¨
©
§
+
¸
¹
·
¨
©
§
+
=
¸¸¹
·
¨¨©
§

»
¼
º
«
¬
ª
+

+

+
=
2
0
2
0
0
)
E
V
(
m
2
a
2
sinh
2
E
E
V
E
V
E
4
1
1
1
!
¸¸¹
·
¨¨©
§



+
+
=
2
0
2
0
2
0
)
E
V
(
m
2
a
2
sinh
)
E
V
(
E
))
E
V
(
E
(
4
1
1
1
!
¸¸¹
·
¨¨©
§


+
=
2
0
2
0
2
0
)
E
V
(
m
2
a
2
sinh
)
E
V
(
E
V
4
1
1
1
!
(1.39b)
Equation (1.39) is for E < V
0
. For E > V
0
,
±
=

=


=
i
)
V
E
(
m
2
2
0
2
2
!
Using x
tan
i
ix
tanh
,
x
cos
ix
cosh
,
x
sin
i
ix
sinh
,
i
=
=
=
=
in equation (1.39),
we get:
a
2
i
sinh
)
k
i
i
k
(
4
1
1
1
T
2
2
1
+
+
=
)
a
2
sin
(
)
k
k
(
i
4
1
1
1
2
2
2


+
=
a
2
sin
)
k
k
(
4
1
1
1
2
2

+
=
(1.40a)
Excerpt out of 137 pages
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 Sujaul Chowdhury (Author)Abdus Samad (Author)Dipak Dash (Author), 2011, Nanostructure Physics of nonisolated Quantum Well of symmetric double barrier, Munich, GRIN Verlag, https://www.grin.com/document/211440
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