Excerpt
i
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha*
Department of Physics, Shahjalal University of Science and Technology,
Sylhet 3114, Bangladesh, http://www.sust.edu
*Scientific Officer, Bangladesh Atomic Energy Commission: http://www.baec.org.bd
Chapter
I
Derivation of WKB solution of Schroedinger equation
and introduction to WKB connection formulae
110
1.1 WKB
approximation
2
1.2 WKB solution of (1 dimensional) Schroedinger equation
2
1.3 Classical turning point
6
1.4 WKB connection formulae: case I
8
1.5 WKB connection formulae: case II
9
Chapter
II
Construction of classical Hamiltonian function
of a charged particle in an electric and a magnetic field
11
18
2.1 Lagrange's equation, Lagrangian function and generalized potential
12
2.2 Construction of Lagrangian function for a charged particle in an
electric field and a magnetic field
14
2.3 Construction of Hamiltonian function for a charged particle in an
electric field and a magnetic field
16
Index
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
ii
Chapter
III
Reducing 3D problem to 1D problem:
using one Landau gauge
19
31
3.1 Description of the problem
20
3.2 The general eigenvalue equation of energy
22
3.3 Simplification of the general eigenvalue equation of energy
22
3.4 Adapting the general eigenvalue equation of energy to our problem
23
3.5
op
y
p
commutes with Hamiltonian operator of our problem
24
3.6 Reducing 3D eigenvalue equation to 1D eigenvalue equation and
obtaining eigenvalue spectrum and identification of different parts of
eigenvalue spectrum
26
3.7 Landau level index n is a constant of motion
29
Chapter
IV
Reducing 3D problem to 1D problem:
using the other Landau gauge
32
44
4.1 Description of the problem
33
4.2 The general eigenvalue equation of energy
35
4.3 Simplification of the general eigenvalue equation of energy
35
4.4 Adapting the general eigenvalue equation of energy to our problem
36
4.5
op
z
p
commutes with Hamiltonian operator of our problem
37
4.6 Reducing 3D eigenvalue equation to 1D eigenvalue equation and
obtaining eigenvalue spectrum and identification of different parts of
eigenvalue spectrum
39
4.7 Landau level index n is a constant of motion
42
Index
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
iii
Chapter
V
Obtaining analytical expressions for tunneling regime of
longitudinal magnetic field dependent transmission coefficient
of single and symmetric double barriers of general shape
and of many different shapes we encounter
in studying Nanostructure Physics
using WKB method
45
68
5.1 Results of 1D problem at zero magnetic field
46
5.2 Single and symmetric double barriers of general shape
48
5.3 Single rectangular tunnel barrier
50
5.4 Symmetric rectangular double barrier
52
5.5 Single rectangular barrier biased to Fowler Nordheim tunneling regime 55
5.6 Moderately biased single rectangular tunnel barrier
57
5.7 Single parabolic tunnel barrier
59
5.8 Schottky
barrier
60
5.9 Single triangular tunnel barrier
62
5.10 Two identical triangular tunnel barriers separated by a triangular
Quantum Well
64
5.11 Symmetric double barrier obtained by biasing asymmetric rectangular
double barrier
66
Index
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
iv
Chapter
VI
Numerical investigation of
longitudinal magnetic field dependent transmission coefficient
of different types of single and symmetric double barriers
encountered in Nanostructure Physics
using WKB method
69
175
6.1 Single rectangular tunnel barrier
70
6.2 Symmetric rectangular double barrier
88
6.3 Single rectangular barrier biased to Fowler Nordheim tunneling regime 103
6.4 Moderately biased single rectangular tunnel barrier
115
6.5 Single
parabolic tunnel barrier
128
6.6 Schottky
barrier
136
6.7 Single triangular tunnel barrier
145
6.8 Two identical triangular tunnel barriers separated by a triangular
Quantum well
152
6.9 Symmetric double barrier obtained by biasing asymmetric rectangular
double barrier
163
References 176
177
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
1
Chapter I
Derivation of WKB solution of Schroedinger equation
and introduction to WKB connection formulae
Chapter I: Derivation of WKB solution of Schroedinger equation
and introduction to WKB connection formulae
2
1.1 WKB approximation
WentzelKramersBrillouin (WKB) approximation gives direct and approximate
solution of Schroedinger equation if
1) potential V is such that 3D Schroedinger equation can be reduced to 1D
differential equation,
2) potential V(x) is a slowly varying function of position x.
As such
!
!
)
x
(
p
))
x
(
V
E
(
m
2
)
x
(
k
2
=

=
(1.1)
is local wavenumber. Since k(x) is slowly varying function of x,
k
dx
dk
k
1
<<
(1.2)
i.e. fractional change of k is much smaller than k. Equation (1.2) is called condition
of validity of WKB approximation.
If we use
)
x
(
2
)
x
(
k
=
(1.3)
where )
x
(
is local de Broglie wavelength, equation (1.2) becomes
)
x
(
2
2
dx
d
2
<<
¸
¹
·
¨
©
§
or,
<<
¸¸¹
·
¨¨©
§

2
dx
d
1
2
or,
<<
2
dx
d
or,
<<
2
dx
d
or,
<<
dx
d
(1.4)
i.e. change of de Broglie wavelength over a wavelength is much smaller than de
Broglie wavelength. Equation (1.4) is also called condition of validity of WKB
approximation.
1.2 WKB solution of (1 dimensional) Schroedinger equation
0
))
x
(
V
E
(
m
2
dx
d
2
2
2
=

+
!
or,
0
)
x
(
k
dx
d
2
2
2
=
+
(1.5)
where
))
x
(
V
E
(
m
2
)
x
(
k
2
2

=
!
(1.6)
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
3
If k were independent of x, solution of equation (1.5) would become
.
e
)
x
(
ikx
±
=
This suggests that if k(x) is slowly varying function of x, we may try a solution of
equation (1.5) of the form
)
x
(
iu
e
)
x
(
=
where u(x) is a function of x. Equation (1.5)
then gives
0
e
k
]
e
[
dx
d
)
x
(
iu
2
)
x
(
iu
2
2
=
+
or,
0
e
k
dx
du
ie
dx
d
)
x
(
iu
2
)
x
(
iu
=
+
»¼
º
«¬
ª
or,
0
e
k
dx
du
e
i
dx
u
d
ie
)
x
(
iu
2
2
)
x
(
iu
2
2
2
)
x
(
iu
=
+
¸
¹
·
¨
©
§
+
or,
0
k
dx
du
dx
u
d
i
2
2
2
2
=
+
¸
¹
·
¨
©
§

or,
2
2
k
dx
du
=
¸
¹
·
¨
©
§
0
e
)
x
(
iu
in general
If k were independent of x, we would get u = kx by comparing
ikx
e
and
and
)
x
(
iu
e
. Then
k
)
kx
(
dx
d
dx
du
=
=
(constant) and
0
)
k
(
dx
d
dx
u
d
2
2
=
=
.
Since k(x) is not constant but is slowly varying,
2
2
dx
u
d
is not zero but is
negligibly small.
or,
k
dx
du
±
=
or,
³
±
=
dx
)
x
(
k
u
+ arbitrary constant
³
±
=
x
a
dx
)
x
(
k
where a is arbitrary
constant, we are free to choose it
³
±
x
a
dx
)
x
(
k
i
)
x
(
iu
0
e
e
)
x
(
(1.7)
is an approximate solution of equation (1.5). Let a more accurate solution of equation
(1.5) be
³
±
=
=
x
a
dx
)
x
(
k
i
0
e
)
x
(
F
)
x
(
)
x
(
F
)
x
(
Here F(x) is still unknown (to be determined). Equation (1.5) gives
0
F
k
]
F
[
dx
d
0
2
0
2
2
+
or,
0
F
k
dx
dF
dx
d
F
dx
d
0
2
0
0
+
»¼
º
«¬
ª
+
or,
0
F
k
dx
d
dx
dF
dx
F
d
dx
d
dx
dF
dx
d
F
0
2
0
0
2
2
0
2
0
2
+
+
+
+
Chapter I: Derivation of WKB solution of Schroedinger equation
and introduction to WKB connection formulae
4
or,
0
F
k
e
dx
d
F
e
dx
d
dx
dF
2
dx
F
d
0
2
x
a
dx
)
x
(
k
i
2
2
x
a
dx
)
x
(
k
i
2
2
0
+
¸
¸
¸
¸
¹
·
¨
¨
¨
¨
©
§
+
¸
¸
¸
¸
¹
·
¨
¨
¨
¨
©
§
+
³
³
±
±
or,
(
)
0
F
k
dx
d
ik
dx
dk
i
F
)
x
(
ik
dx
dF
2
dx
F
d
0
2
0
0
0
2
2
0
+
¸
¹
·
¨
©
§
±
±
+
±
+
or,
0
F
k
)
ik
(
ikF
F
dx
dk
i
dx
dF
ik
2
dx
F
d
0
2
0
0
0
2
2
0
+
±
±
±
±
or,
0
F
k
F
k
F
dx
dk
i
dx
dF
ik
2
dx
F
d
0
2
0
2
0
0
2
2
0
+

±
±
or,
0
F
dx
dk
i
dx
dF
ik
2
0
0
±
±
If k were constant,
ikx
Ae
±
=
with amplitude A = constant.
Since k is slowly varying, we have
0
)
x
(
F
=
in place of
ikx
Ae
±
=
.
As such F(x) is slowly varying. Hence
2
2
dx
F
d
is negligibly small.
or,
dx
dk
k
2
1
dx
dF
F
1

or,
k
dk
2
1
F
dF

dividing by i
0
kF
or,
C
k
ln
2
1
F
ln
+

where C
is an arbitrary constant of integration.
or,
C
ln
C
)
k
F
ln(
or,
C
)
x
(
k
F
or,
)
x
(
k
1
F
(1.8)
except for a multiplicative constant C.
Thus WKB solution of equation (1.5) is
0
WKB
F
)
x
(
=
or,
³
±
=
x
a
dx
)
x
(
k
i
WKB
e
)
x
(
k
1
)
x
(
(1.9)
This is WKB solution because WKB approximation (equation (1.2)) has been used in
obtaining it. WKB solution of equation (1.5) is also written as
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
5
³
³

+
+
=
x
a
dx
)
x
(
k
i
2
x
a
dx
)
x
(
k
i
1
WKB
e
)
x
(
k
C
e
)
x
(
k
C
)
x
(
(1.10)
This is also written as
¸¸¹
·
¨¨©
§
³
+
¸¸¹
·
¨¨©
§
³
=
x
a
x
a
WKB
dx
)
x
(
k
Cos
)
x
(
k
B
dx
)
x
(
k
Sin
)
x
(
k
A
)
x
(
(1.11)
For E < V(x), equation (1.5) becomes
0
)
E
)
x
(
V
(
m
2
dx
d
2
2
2
=


!
(1.12)
or,
0
)
x
(
k
dx
d
2
2
2
=

(1.13)
where
2
2
2
k
)
E
)
x
(
V
(
m
2
)
x
(
k

=

=
!
(1.14)
see equation (1.6)
WKB solution of equation (1.13) is
or,
³
±
=
x
a
dx
)
x
(
k
WKB
e
)
x
(
k
1
)
x
(
(1.15)
which can be obtained using calculations as for equation (1.5). General solution is
³
³

+
+
=
x
a
dx
)
x
(
k
2
x
a
dx
)
x
(
k
1
WKB
e
)
x
(
k
D
e
)
x
(
k
D
)
x
(
(1.16)
For E close to V(x), k and k
become very small ( 0) and hence the factor
)
x
(
k
1
or
)
x
(
k
1
of the WKB solution(s) become very large (
) and hence the
solutions become nonadmissible or invalid. WKB solutions are valid for E
appreciably different from V(x).
Chapter I: Derivation of WKB solution of Schroedinger equation
and introduction to WKB connection formulae
6
1.3 Classical turning point
Figure 1.1 Diagram to help describe classical turning point if V is an increasing
function of x.
For a particle having total energy E (kinetic plus potential), in Figure 1.1, x >
x
1
is classically forbidden region, because kinetic energy of the particle becomes
negative for x > x
1
. x < x
1
is classically allowed region. x = x
1
is called classical
turning point. A particle travelling from x < x
1
to x = x
1
turns back at x = x
1
as it
cannot classically enter the region x > x
1
. Near x = x
1
, E is close to V and hence
WKB solutions are invalid near x = x
1
.
Figure 1.2 Diagram to help describe classical turning point if V is a decreasing
function of x. For a particle of total energy E, x = x
2
is classical turning point in
Figure 1.2. WKB solutions are invalid near x = x
2
for which E is close to V.
V
E
(0, 0)
x
1
Classically forbidden region
Classically allowed region
V
E
(0, 0)
x
2
Classically
allowed region
Classically
forbidden region
x
x
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
7
Figure 1.3 Diagram showing classical turning points (x = a and b) for a particle of
total energy E encountering a single potential barrier of general shape. WKB
solutions are valid in the three regions because the regions are away from classical
turning points.
Figure 1.4 Diagram showing classical turning points (a and b) for a particle of total
energy E residing in the quantum well of general shape. WKB solutions are valid in
the three regions because the regions are away from the classical turning points.
V
(0, 0)
a
b
E
Region I
Region II
Region III
V
(0, 0)
a
b
E
Region I
Region II
Region III
x
x
Chapter I: Derivation of WKB solution of Schroedinger equation
and introduction to WKB connection formulae
8
1.4 WKB connection formulae: case I
Figure 1.5 Diagram to help describe WKB connection formulae. We consider V(x)
as an increasing function of x as case I. x = x
1
is classical turning point for a particle
of total energy E approaching the potential from left.
Let
I
and
II
be WKB solutions of Schroedinger equation in region I and II
respectively. WKB solutions are valid in region I and II which are sufficiently away
from classical turning point x = x
1
. WKB solution of Schroedinger equation is invalid
in region III which is close to classical turning point x = x
1
.
Connections between WKB solutions
I
and
II
are given by
)
4
dx
)
x
(
k
(
Sin
)
x
(
k
2
e
)
x
(
k
1
1
x
x
II
I
x
1
x
dx
)
x
(
k
+
³

³

(1.17)
and
)
4
dx
)
x
(
k
(
Cos
)
x
(
k
1
e
)
x
(
k
1
1
x
x
II
I
x
1
x
dx
)
x
(
k
+
³

³
+
(1.18)
V
E
(0, 0)
x
1
Region II
Region III
Region I
x
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
9
Relation (1.17) and (1.18) are called WKB connection formulae. As we go from
region I to region II,
I
becomes
II
, i.e. LHS of (1.17) becomes RHS of (1.17),
LHS of (1.18) becomes RHS of (1.18).
In fact, e.g., if we have LHS of (1.17) as a multiplicative factor in expression
of
I
, we can replace LHS of (1.17) by RHS of (1.17) in the expression of
I
and
get expression of
II
valid / correct within WKB approximation. Such is the game
while using the WKB connection formulae.
1.5 WKB connection formulae: case II
Figure 1.6 Diagram to help describe WKB connection formulae. We consider V(x)
as a decreasing function of x as case II. x = x
2
is classical turning point for a particle
of total energy E approaching the potential from right.
V
x
(0, 0)
x
2
Region II
Region I
Region III
E
Chapter I: Derivation of WKB solution of Schroedinger equation
and introduction to WKB connection formulae
10
Let
I
and
II
be WKB solutions of Schroedinger equation in region I and II
respectively. WKB solutions are valid in region I and II which are sufficiently away
from classical turning point x = x
2
. WKB solution of Schroedinger equation is invalid
in region III which is close to classical turning point x = x
2
.
Connections, in fact ratio, between WKB solutions
I
and
II
are given by
)
4
dx
)
x
(
k
(
Sin
)
x
(
k
2
e
)
x
(
k
1
x
2
x
II
I
2
x
x
dx
)
x
(
k
+
³

³

(1.19)
and
)
4
dx
)
x
(
k
(
Cos
)
x
(
k
1
e
)
x
(
k
1
x
2
x
II
I
2
x
x
dx
)
x
(
k
+
³

³
+
(1.20)
Relation (1.19) and (1.20) are WKB connection formulae. As we go from region I to
region II,
I
becomes
II
i.e. LHS of (1.19) becomes RHS of (1.19), and LHS of
(1.20) becomes RHS of (1.20). In fact, e.g., if we have LHS of (1.19) as a
multiplicative factor in expression of
I
, we can replace LHS of (1.19) by RHS of
(1.19) in the expression of
I
and get expression of
II
valid / correct within WKB
approximation. Such is the game while using the WKB connection formulae.
Detailed derivation of WKB connection formulae in the two cases can be
found in reference [3] which contains an extensive account of WKB method and its
use in Nanostructure Physics at zero magnetic field.
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
11
Chapter II
Construction of classical Hamiltonian function
of a charged particle in an electric and a magnetic field
Chapter II Construction of classical Hamiltonian function of a charged particle in an electric and a magnetic field
12
2.1 Lagrange's equation, Lagrangian function and generalized potential
Let
q
1
, q
2
, q
3
, ..., q
n
be n = 3N k generalised coordinates for an N particle
system having k holonomic constraints that depend on time. Let external force acting
on ith particle be
L
which is different from resultant force acting on ith particle in
that vector sum of
L
and force of constraint (if any) acting on the ith particle gives
resultant force on ith particle. Let
i
r
&
be location of ith particle. Change of value of q
j
as a function of time is determined by solution of the differential equation
M
M
M
where
M
is called generalized force corresponding to the generalised coordinate q
j
,
and is given by
M
L
L
L
M
To solve equation (2.1), amazingly, we do not at all need to know force of constraint
if virtual work done by force of constraint is zero as is fortunately the case with most
dynamical systems or problems of practical interest.
If
L
is a conservative force derivable from a scalar potential V which is a
function of
M
(not of
M
),
L
where
equation
(2.1) modifies to
M
M
called Lagrange's equation. Here
L = T V
(2.4)
is called Lagrangian or Lagrangian function. T is kinetic energy of the system of
particles given by
L
L L
where v
i
is speed of ith particle given by
L
L
M
M
M
L
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
13
The term
L
is absent (zero) from equation (2.6) if constraints do not depend on time.
Use of equation (2.6) in (2.5) gives
L
L
L
L
L
M
M
M
L
L
N
N
N
L
(2.7)
where T
0
does not contain
M
, T
1
is linear in
M
and T
2
is quadratic in
M
i.e. T
2
M N
and / or
M
given by
L
L
L
L
L
M
M
L
M
L
M
L
N
L
M N
N
M
If constraints are independent of time,
L
and as such, T
0
= T
1
= 0 and hence T =
T
2
i.e. T contains only term(s) quadratic in
M
.
Apart from the scalar potential V which depends on coordinates q
j
's only,
there is great importance of a velocity dependent potential
M
M
also called
generalised potential given by
M
M
M
as will be evident later in this chapter. If such a potential is available, equation (2.1)
modifies to
M
M
M
M
M
M
which has the form of Lagrange's equation
Chapter II Construction of classical Hamiltonian function of a charged particle in an electric and a magnetic field
14
M
M
but with the Lagrangian function L = T U
(2.10)
containing U instead of V.
If U contains terms U
0
, U
1
and U
2
such that U = U
0
+ U
1
+ U
2
where U
0
does
not contain
M
, U
1
is linear in
M
and U
2
is quadratic in
M
i.e. U
2
M N
and / or
M
,
we can collect similar terms and write L as
where, evidently, L
0
does not contain
M
, L
1
is linear in
M
and L
2
is quadratic in
M
i.e.
L
2
M N
and / or
M
. As is evident,
,
and
(2.12)
2.2 Construction of Lagrangian function for a charged particle in an electric
field and a magnetic field
A magnetic field is represented by a vector potential such that
(2.13)
Again, an electric field is represented, in general, by a combination of scalar
potential V and the vector potential such that
In static case
Total force acting on a charged particle in an electric field
and a magnetic field is
(2.15)
Using equation (2.13) and (2.14) in equation (2.15), we find
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
15
Now,
(2.17)
Hence
We can write
Since can, in general, depend on x, y, z and t, can also depend on x, y, z and t.
Hence
Use of equation (2.19) and (2.20) in equation (2.18) gives
Chapter II Construction of classical Hamiltonian function of a charged particle in an electric and a magnetic field
16
Use of equation (2.21) in equation (2.16) gives
We can write
Equation (2.22) and (2.23) give
where
In view of equation (2.8) and (2.24), U can be considered as generalised potential.
Hence according to equation (2.10), Lagrangian L is given by
L = T U or,
using equation (2.25)
Equation (2.26) is Lagrangian of a charged particle in an electric field and a magnetic
field.
2.3 Construction of Hamiltonian function for a charged particle in an electric
field and a magnetic field
Before constructing the Hamiltonian function, we introduce Euler's theorem.
If f is a homogeneous function of degree n of the variables x
1
, x
1
, x
3
, ..., x
N
, Euler's
theorem is
L
L
L
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
17
We now verify this theorem using
as an example. f is a homogeneous
function of degree (n =) 11 of the variables x
1
and x
2
. (N = 2). As such
L
L
L
. Indeed
L
L
L
We now decompose T as in equation (2.7): T = T
0
+T
1
+T
2
to write equation
(2.26)
as
We now equate this L to L
0
+L
1
+L
2
in light of equation (2.11) to get
which gives
,
,
(2.27)
We have
. This implies that
As such
equation (2.27) gives
,
,
(2.28)
We now construct Hamiltonian function H using Legendre transformation of
the Lagrangian as
M
M
M
M
using Euler's theorem
. Thus
(2.29)
Equation (2.28) and (2.29) give
or,
(2.30)
Chapter II Construction of classical Hamiltonian function of a charged particle in an electric and a magnetic field
18
We have momentum conjugate to x, y, z given by
respectivevly. Use of equation (2.26) in (2.31) gives
Again,
and
Equation (2.32), (2.33), (2.34) in equation (2.30) give
Equation (2.35) gives expression of Hamiltonian function of a charged particle in an
electric and a magnetic field. In view of equation (2.30), we find that H is total
energy, i.e. sum of kinetic energy and potential energy of the particle. Hence if we
replace in equation (2.35) by operator
, we get Hamiltonian operator H
op
to be
used in eigenvalue equation of (total) energy solved in the following chapters.
Operators of V and of are V and themselves respectively because operator of any
function of space and time is the function itself.
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
19
Chapter III
Reducing 3D problem to 1D problem:
using one Landau gauge
Chapter III Reducing 3D problem to 1D problem: using one Landau gauge
20
3.1 Description of the problem
Space
(Y)
(a)
Space (X)
Space (Y)
(b)
Space (X)
Figure 3.1 (a) Single tunnel barrier of general shape V(x) in applied magnetic field
called longitudinal magnetic field, (b) symmetric double barrier of general shape
V(x) in applied magnetic field called longitudinal magnetic field.
V
(0 , 0)
x
V
(0, 0)
x
Z
Z
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
21
We consider nanostructures containing in their band model single tunnel
barrier of general shape V(x) or symmetric double barrier of general shape V(x) in
uniform magnetic field called longitudinal magnetic field applied along x
direction. Figure 3.1(a) and (b) show the situations. In Figure 3.1(b), we have
potential profile of two identical tunnel barriers separated by a Quantum Well.
Individual barriers need not be symmetric in shape. An electron traversing structures
containing these potential profiles in their band models can feel the potential profile
for motion of electron along x direction only. In y and z directions, potential
variations can be taken constant or zero without any loss of generality.
We consider electrons free in 3D (not 2D) impinging in general obliquely on
the (left hand) tunnel barrier in presence of nonzero magnetic field, and we wish to
obtain analytical expressions of transmission coefficient of electron to find out
effects of the magnetic field on transmission coefficient. We shall restrict us to
tunneling regime only.
In this chapter, we shall carry out thorough and complete analytical
calculations to reduce 3D eigenvalue equation of energy of the problem to 1D
equation. In Chapter IV, we repeat calculations of this chapter using another Landau
gauge ending up with same results.
In Chapter V, we shall use the 1D equation(s) to obtain general analytical
expressions of longitudinal magnetic field dependent transmission coefficient of
single tunnel barrier of general shape and of symmetric double barrier of general
shape for tunneling regime. We shall apply these general analytical expressions and
obtain analytical expressions of longitudinal magnetic field dependent transmission
coefficient of single and symmetric double barriers of many different shapes we
encounter in studying Nanostructure Physics.
In Chapter VI, we shall carry out extensive numerical investigations to bring
out effects of the longitudinal magnetic field on transmission coefficient of single
and symmetric double barriers of the many different shapes we encounter in studying
Nanostructure Physics.
Chapter III Reducing 3D problem to 1D problem: using one Landau gauge
22
3.2 The general eigenvalue equation of energy
The general eigenvalue equation of energy of a particle of mass m and charge
Q in presence of electric field and magnetic field is given by
(3.1)
or,
(3.2)
with Hamiltonian operator
obtained by replacing by
in the classical
Hamiltonian function
(3.3)
Operator of V and are V and respectively, because they are functions of position
and time (in general). V and are scalar potential and vector potential respectively
and are given by
and
in general, although the time
derivative is absent in our static case.
3.3 Simplification of the general eigenvalue equation of energy
We now simplify the factor
of equation (3.1).
=
=
.
=
=
=
(3.4)
Thus equation (3.1) becomes
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
23
(3.5)
Equation (3.5) is the simplified form of general eigenvalue equation of energy.
3.4 Adapting the general eigenvalue equation of energy to our problem
In our problem,
. We have a magnetic field along x direction.
Vector potential must be such that
. Hence x component of
must
be equal to B.
gives
(3.6)
There are two renowned, obvious choices of in this regard. These are called
Landau gauges: (
)
(3.7)
and
(
)
(3.8)
which when put in equation (3.6) gives
. We can use vector potential
given by equation (3.7) or equation (3.8) in equation (3.5) to get eigenvalue equation
of energy of our problem. Use of equation (3.7) in equation (3.5) gives
Chapter III Reducing 3D problem to 1D problem: using one Landau gauge
24
( )
=
+
»
»
¼
º
«
«
¬
ª


+
+

E
x
V
y
z
i
2
z
1
z
y
x
m
2
2
2
4
2
2
2
2
2
2
2
"
"
!
(3.9)
or,
=
E
H
op
(3.10)
Here
QB
!
" =
or,
QB
2
!
" =
or,
2
1
QB
"
!
=
or,
4
2
1
QB
"
!
=
¸
¹
·
¨
©
§
(3.11)
Equation (3.9) is eigenvalue equation of energy and
op
H
of equation (3.10) is
Hamiltonian operator of our problem.
3.5
commutes with Hamiltonian operator of our problem
We now show that
y
i
p
op
y
= !
commutes with
op
H
of equation (3.10) which
is Hamiltonian operator of our problem. We start with
»
»
¼
º
«
«
¬
ª
op
op
y
H
,
p
( )
»
»
¼
º
«
«
¬
ª
+
¸
¸
¹
·
¨
¨
©
§


+
+

=
x
V
y
z
i
2
z
1
z
y
x
m
2
,
y
i
2
2
4
2
2
2
2
2
2
2
"
"
!
!
=
( )
»
»
¼
º
«
«
¬
ª
+
+
+



x
V
y
z
i
2
m
2
z
1
m
2
z
m
2
y
m
2
x
m
2
,
y
i
2
2
2
4
2
2
2
2
2
2
2
2
2
2
"
!
"
!
!
!
!
!
=
+
»
»
¼
º
«
«
¬
ª

2
2
2
x
m
2
,
y
i
!
!
+
»
»
¼
º
«
«
¬
ª

2
2
2
y
m
2
,
y
i
!
!
»
»
¼
º
«
«
¬
ª

2
2
2
z
m
2
,
y
i
!
!
»
»
¼
º
«
«
¬
ª
+
2
4
2
z
1
m
2
,
y
i
"
!
!
+
»
»
¼
º
«
«
¬
ª
+
y
z
i
2
m
2
,
y
i
2
2
"
!
!
( )
»
»
¼
º
«
«
¬
ª
x
V
,
y
i
!
[
] [ ] [ ] [
] [ ]
E
,
A
D
,
A
C
,
A
B
,
A
E
D
C
B
,
A
Because
+
+
+
=
+
+
+
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
25
=
¸¸¹
·
¨¨©
§
¸
¸
¹
·
¨
¨
©
§


¸
¸
¹
·
¨
¨
©
§

¸¸¹
·
¨¨©
§
y
i
x
m
2
x
m
2
y
i
2
2
2
2
2
2
!
!
!
!
¸¸¹
·
¨¨©
§
¸
¸
¹
·
¨
¨
©
§


¸
¸
¹
·
¨
¨
©
§

¸¸¹
·
¨¨©
§
+
y
i
y
m
2
y
m
2
y
i
2
2
2
2
2
2
!
!
!
!
¸¸¹
·
¨¨©
§
¸
¸
¹
·
¨
¨
©
§


¸
¸
¹
·
¨
¨
©
§

¸¸¹
·
¨¨©
§
+
y
i
z
m
2
z
m
2
y
i
2
2
2
2
2
2
!
!
!
!
¸¸¹
·
¨¨©
§
¸
¸
¹
·
¨
¨
©
§

¸
¸
¹
·
¨
¨
©
§
¸¸¹
·
¨¨©
§
+
y
i
z
1
m
2
z
1
m
2
y
i
2
4
2
2
4
2
!
"
!
"
!
!
¸¸¹
·
¨¨©
§
¸
¸
¹
·
¨
¨
©
§

¸
¸
¹
·
¨
¨
©
§
¸¸¹
·
¨¨©
§
+
y
i
y
z
i
2
m
2
y
z
i
2
m
2
y
i
2
2
2
2
!
"
!
"
!
!
( )
(
) ( )
¸¸¹
·
¨¨©
§

¸¸¹
·
¨¨©
§
+
y
i
x
V
x
V
y
i
!
!
[ ]
.
BA
AB
B
,
A
Because

=
=
y
x
m
2
i
x
y
m
2
i
2
2
2
2
2
2
+

!
!
!
!
3
2
2
3
3
2
y
m
2
i
y
m
2
i
+

!
!
!
!
y
z
m
2
i
z
y
m
2
i
2
2
2
2
2
2
+

!
!
!
!
y
z
1
m
2
i
y
z
1
m
2
i
2
4
2
2
4
2

+
"
!
!
"
!
!
+

+
2
2
2
2
2
2
2
2
y
z
i
2
m
2
i
y
z
i
2
m
2
i
"
!
!
"
!
!
( )
( )
y
x
V
i
y
x
V
i

!
!
= 0
Thus indeed
.
0
H
,
p
op
op
y
=
»
»
¼
º
«
«
¬
ª
In partial derivatives of
with respect to x and y for example, order of
differentiation whether w.r.t. x first or y first is immaterial.
Chapter III Reducing 3D problem to 1D problem: using one Landau gauge
26
3.6 Reducing 3D eigenvalue equation to 1D eigenvalue equation and obtaining
eigenvalue spectrum and identification of different parts of eigenvalue spectrum
We have eigenvalue equation of energy of electron given by equation (3.9):
( )
=
+
»
»
¼
º
«
«
¬
ª


+
+

E
x
V
y
z
i
2
z
1
z
y
x
m
2
2
2
4
2
2
2
2
2
2
2
"
"
!
(3.12)
or,
=
E
H
op
(3.13)
op
y
p
commutes with
op
H
of equation (3.13). Moreover,
y
i
p
op
y
= !
does not
contain t (time) explicitly. Hence
y
p is a constant of motion. Hence y part of
can
be taken as plane wave as for a particle free for motion in y direction. Hence
can
be written as
(
)
( )
z
Z
e
)
x
(
X
z
,
y
,
x
y
y
p
i
!
=
(3.14)
Using equation (3.14) in equation (3.12), we get
( ) ( )
( ) ( )
2
2
y
y
p
i
y
y
p
i
2
y
y
y
p
i
2
2
2
dz
Z
d
e
x
X
z
Z
e
p
i
x
X
z
Z
e
dx
X
d
m
2
!
!
!
!
!
+
«
«
¬
ª
¸
¹
·
¨
©
§
+

( )
( )
»
»
¼
º
¸
¹
·
¨
©
§


z
Z
e
p
i
x
zX
i
2
z
1
y
y
p
i
y
2
2
4
!
!
"
"
+
( ) (
)
z
,
y
,
x
x
V
=
(
)
z
,
y
,
x
E
or,
( )
E
x
V
z
p
2
z
1
dz
Z
d
Z
1
p
dx
X
d
X
1
m
2
2
y
2
4
2
2
2
2
y
2
2
2
=
+
»
»
¼
º
«
«
¬
ª


+


!"
"
!
!
or,
m
2
p
dx
X
d
X
1
m
2
2
y
2
2
2
+
 !
2
2
2
dz
Z
d
Z
1
m
2
!

2
4
2
z
1
m
2
"
!
+
( )
E
x
V
z
m
p
2
y
=
+
+
"
!
or,
2
2
2
dz
Z
d
Z
1
m
2
!

2
4
2
z
1
m
2
"
!
+
z
m
p
2
y
"
!
+
=

E
m
2
p
2
y
( )
2
2
2
dx
X
d
X
1
m
2
x
V
!
+

(3.15)
L.H.S. of equation (3.15) contains z only while R.H.S. of equation (3.15) contains x
only. Since x and z are independent, we can equate both sides of equation (3.15) to a
constant, say C. Thus
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
27
2
2
2
dz
Z
d
Z
1
m
2
!

2
4
2
z
1
m
2
"
!
+
z
m
p
2
y
"
!
+
=

E
m
2
p
2
y
( )
2
2
2
dx
X
d
X
1
m
2
x
V
!
+

= C (3.16)
Now,
2
2
2
dz
Z
d
Z
1
m
2
!

2
4
2
z
1
m
2
"
!
+
z
m
p
2
y
"
!
+
= C
or,
2
2
2
dz
Z
d
m
2
!

Z
z
1
m
2
2
4
2
"
!
+
zZ
m
p
2
y
"
!
+
= CZ
or,
2
2
2
dz
Z
d
m
2
!

Z
z
B
Q
m
2
2
2
2
2
2
!
!
+
zZ
QB
m
p
y
!
!
+
= CZ
QB
!
" =
or,
2
2
2
dz
Z
d
m
2
!

Z
z
m
2
1
2
2
c
+
zZ
p
c
y
+
= CZ
m
QB
c
=
or,
2
2
2
dz
Z
d
m
2
!

Z
z
p
m
2
z
m
2
1
y
c
2
2
c
¸¸¹
·
¨¨©
§
+
+
= CZ
or,
2
2
2
dz
Z
d
m
2
!

Z
m
p
m
p
z
2
z
m
2
1
2
c
y
c
y
2
2
c
¸¸
¸
¹
·
¨¨
¨
©
§
¸¸¹
·
¨¨©
§
+
+
+
=
¸
¸
¹
·
¨
¨
©
§
+
2
c
2
2
y
2
c
m
p
m
2
1
C
Z
or,
2
2
2
dz
Z
d
m
2
!

Z
m
p
z
m
2
1
2
c
y
2
c
¸¸¹
·
¨¨©
§
+
+
=
¸
¸
¹
·
¨
¨
©
§
+
m
2
p
C
2
y
Z
or,
(
)
2
0
2
2
z
z
d
Z
d
m
2
+
 !
(
)
Z
z
z
m
2
1
2
0
2
c
+
+
=
¸
¸
¹
·
¨
¨
©
§
+
m
2
p
C
2
y
Z
(3.17)
Equation (3.17) has the form of eigenvalue equation of simple harmonic oscillator.
Hence
c
2
y
2
1
n
m
2
p
C
¸
¹
·
¨
©
§ +
=
+
!
where
...
,
3
,
2
,
1
,
0
n
=
or,
c
2
y
2
1
n
m
2
p
C
¸
¹
·
¨
©
§ +
+

=
!
(3.18)
Chapter III Reducing 3D problem to 1D problem: using one Landau gauge
28
Equations (3.16) and (3.18) yield

E
m
2
p
2
y
( )
2
2
2
dx
X
d
X
1
m
2
x
V
!
+

=
c
2
y
2
1
n
m
2
p
¸
¹
·
¨
©
§ +
+

!
or,
2
2
2
dx
X
d
X
1
m
2
!

( )
x
V
+
=
c
2
1
n
E
¸
¹
·
¨
©
§ +

!
or,
2
2
2
dx
X
d
m
2
!

( )
X
x
V
+
=
X
2
1
n
E
c
»¼
º
«¬
ª
¸
¹
·
¨
©
§ +

!
or,
( )
X
x
V
dx
d
m
2
2
2
2
»
»
¼
º
«
«
¬
ª
+
 !
=
X
2
1
n
E
c
»¼
º
«¬
ª
¸
¹
·
¨
©
§ +

!
(3.19)
Equation (3.19) is eigenvalue equation of the operator
2
2
2
dx
d
m
2
!

( )
x
V
+
which is
operator of total energy (kinetic plus potential) of electron for motion along x
direction. Hence eigenvalue
c
2
1
n
E
¸
¹
·
¨
©
§ +

! is eigenvalue of total energy (kinetic
plus potential) of electron for motion along x direction. Let us denote it by
x
E . Hence
=
x
E
c
2
1
n
E
¸
¹
·
¨
©
§ +

!
(3.20)
Here
E
is grand total energy which is a constant of motion. Hence
=

x
E
E
c
2
1
n
¸
¹
·
¨
©
§ + !
(3.21)
is total energy (which is kinetic energy only) of electron for motion in y and z
directions taken together. Discrete energy levels given by equation (3.21) are called
Landau levels, and these are allowed values of energy of electron for motion in YZ
plane i.e. in the plane normal to magnetic field (which is along x direction). Equation
(3.20) or (3.21) gives energy spectrum of electron.
Equation (3.19) can be written as
( )
X
x
V
dx
d
m
2
2
2
2
»
»
¼
º
«
«
¬
ª
+
 !
=
X
E
x
(3.22)
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
29
This can be written as
( )
(
)
0
X
x
V
E
m
2
dx
X
d
x
2
2
2
=

+
!
for
( )
x
V
E
x
>
(3.23)
and
( )
(
)
0
X
E
x
V
m
2
dx
X
d
x
2
2
2
=


!
for
( )
x
V
E
x
<
(3.24)
or,
0
X
k
dx
X
d
2
2
2
=
+
for
( )
x
V
E
x
>
(3.25)
and
0
X
k
dx
X
d
2
2
2
=

for
( )
x
V
E
x
<
(3.26)
where
=
2
k
( )
(
)
x
V
E
m
2
x
2

!
(3.27)
and
=
2
k
( )
(
)
x
2
E
x
V
m
2

!
(3.28)
where
c
x
2
1
n
E
E
¸
¹
·
¨
©
§ +

=
!
(3.29)
Equations (3.19) or (3.22), or (3.23) and (3.24), or (3.25) and (3.26) are differential
equations for motion of electron along x direction through the barrier
( )
x
V
in
presence of longitudinal magnetic field B (along x direction).
3.7 Landau level index n is a constant of motion
We now show that value of n associated with an electron remains unchanged
as the electron passes through the barrier(s) in presence of the magnetic field.
Equation (3.19) gives
Chapter III Reducing 3D problem to 1D problem: using one Landau gauge
30
c
op
op
2
1
n
E
¸
¹
·
¨
©
§
+

! =
2
2
2
dx
d
m
2
!

( )
x
V
+
or,
=
op
E
c
op
2
1
n
¸
¹
·
¨
©
§
+
!
2
2
2
dx
d
m
2
!

( )
x
V
+
(3.30)
Equation (3.9) gives
( )
x
V
y
z
i
2
z
1
z
y
x
m
2
E
2
2
4
2
2
2
2
2
2
2
op
+
»
»
¼
º
«
«
¬
ª


+
+

=
"
"
!
(3.31)
Equations (3.30) and (3.31) yield
c
op
2
1
n
¸
¹
·
¨
©
§
+
!
2
2
2
dx
d
m
2
!

( )
x
V
+


=
2
2
2
x
m
2
!

2
2
2
y
m
2
!
+
2
2
2
z
m
2
!
+
2
4
2
z
1
m
2
"
!
( )
x
V
y
z
i
2
m
2
2
2
+
"
!
or,
c
op
1
n
=
!
¨
¨
©
§


2
2
2
y
m
2
!
+
2
2
2
z
m
2
!
+
2
4
2
z
1
m
2
"
!
¸
¸
¹
·
y
z
i
2
m
2
2
2
"
!
2
1

or,
2
1
y
z
i
2
z
1
z
y
m
2
n
2
2
4
2
2
2
2
c
op

¸
¸
¹
·
¨
¨
©
§


+

=
"
"
!
(3.32a)
or,
2
1
A
m
2
n
op
c
op


=
!
(3.32b)
Equation (3.9) gives
( )
x
V
y
z
i
2
z
1
z
y
x
m
2
H
2
2
4
2
2
2
2
2
2
2
op
+
»
»
¼
º
«
«
¬
ª


+
+

=
"
"
!
(3.33a)
=
op
op
2
B
A
m
2
+
 !
(3.33b)
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
31
Here
y
z
i
2
z
1
z
y
A
2
2
4
2
2
2
2
op


+
=
"
"
(3.34)
and
( )
x
V
x
m
2
B
2
2
2
op
+

= !
(3.35)
Equations (3.32) and (3.33) give
[
]
»
»
¼
º
«
«
¬
ª
+



=
op
op
2
op
c
op
op
B
A
m
2
,
2
1
A
m
2
H
,
n
!
!
=
+
»
»
¼
º
«
«
¬
ª
+


op
op
2
op
c
B
A
m
2
,
A
m
2
!
!
»
»
¼
º
«
«
¬
ª
+


op
op
2
B
A
m
2
,
2
1
!
=
+
»
»
¼
º
«
«
¬
ª


op
2
op
c
A
m
2
,
A
m
2
!
!
0
B
,
A
m
2
op
op
c
+
»
¼
º
«
¬
ª
 !
=
¸
¸
¹
·
¨
¨
©
§


m
2
m
2
2
c
!
!
»¼
º
«¬
ª
op
op
A
,
A
¸¸¹
·
¨¨©
§

+
c
m
2
!
»¼
º
«¬
ª
op
op
B
,
A
=
c
m
2
0
 !
»¼
º
«¬
ª
op
op
B
,
A
= 0
This is because
op
A
contains y and z, not x, and
op
B
contains x only, not y and z.
Hence
op
A
and
op
B
commute. Since
[
]
op
op
H
,
n
= 0, and n
op
does not contain t (time)
explicitly (see equation (3.32a)), we note that n is a constant of motion. That is, value
of n associated with an electron does not change as the electron passes through the
barrier in presence of the magnetic field.
Longitudinal magnetic field in WKB method in Nanostructure Physics:
effects on transmission coefficient of single and symmetric double barriers by 3DEG
Md. Mahbub Alam, Sujaul Chowdhury, Sudipta Saha
32
Chapter IV
Reducing 3D problem to 1D problem:
using the other Landau gauge
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 Dr Sujaul Chowdhury (Author)Mahbub Alam (Author)Sudipta Saha (Author), 2011, Longitudinal Magnetic Field in WKB Method in Nanostructure Physics, Munich, GRIN Verlag, https://www.grin.com/document/231376
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