We have numerically investigated parametric variations of transmission peaks of symmetric rectangular double barrier in non-tunneling regime. We have compared the variations with those for tunneling regime. One of the three variations in non-tunneling regime is completely different from that for tunneling regime warranting rapid dissemination. The book contains background on Quantum Mechanics, Microelectronics and Nanostructure Physics to enable readers assimilate the book completely.

Excerpt

Index

Oscillatory transmission through non-tunneling regime of symmetric rectangular

double barrier: parametric variations of transmission peaks

Md. Abidur Rahman and Sujaul Chowdhury

Section

Page number

Index

i-ii

Chapter

Background on Quantum Mechanics

1.1 Wave equation of a free particle: Schrödinger equation

1.2 Schrödinger equation of a particle subject to a conservative

mechanical force

1.3 Conservation of probability and probability current density

1.4 Time-independent Schrödinger equation and stationary state

1.5 Continuous and discontinuous function

1.6 Finite and infinite discontinuity

1.7 Admissibility conditions on wavefunction

1.8 Free particle: eigenfunctions and probability current density

1.9 Single rectangular tunnel barrier

1.9.1 Calculation of transfer matrix and investigation of its properties

1.9.2 Calculation of transmission coefficient

Index

Chapter

Background on Microelectronics

26-

2.1 Insulator and its band model

2.2 Intrinsic semiconductor and its band model

2.3 Elemental and compound semiconductors

2.4 Alloy semiconductors: ternary and quaternary semiconductors

2.5 Bandgap engineering

2.6 Substrate and epitaxial layer

2.7 Semiconductor heterostructure and heterojunction

Chapter

III

Background on Nanostructure Physics

43-

3.1.1 Tunnel barrier: structure and band model

3.1.2 Transport of electron or hole through tunnel barrier

3.2 Quantum

Well

(QW)

3.3 Symmetric

double

barrier

Chapter

Numerical investigation of parametric variations

of transmission peaks of

symmetric rectangular double barrier for non-tunneling regime

67-

130

4.1 Transmission

through

symmetric

rectangular double barrier

4.2 Description of the problem

4.3 Tables and Figures showing parametric variations of energy of

transmission peaks of symmetric rectangular double barrier for

non-tunneling regime

4.4 Conclusions about the parametric variations

130

References 131

Oscillatory transmission through non-tunneling regime of symmetric rectangular double barrier:

Parametric variations of transmission peaks

Md. Abidur Rahman and Sujaul Chowdhury

Chapter I

Background on Quantum Mechanics

Chapter I Background on Quantum Mechanics

1.1 Wave equation of a free particle: Schrödinger equation

If we associate the wave packet

)

(

)

(

)

(

where

a(k) =

)

(

)

(

with a free material particle, we can use de Broglie's equations p =

! k and E =

and rewrite the above two equations as

)

(

)

(

)

(

)

(

or,

)

(

)

(

)

(

where a(p) =

)

(

or, a(p)

)

(

)

(

In three dimensions

)

(

)

(

)

(

)

(

-----------(1.1)

where

Equation (1.1) gives

)

(

)

(

)

(

and

)

(

)

(

)

(

Similarly,

)

(

)

(

)

(

all momentum space

Oscillatory transmission through non-tunneling regime of symmetric rectangular double barrier:

Parametric variations of transmission peaks

Md. Abidur Rahman and Sujaul Chowdhury

and

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

or,

)

(

)

(

-----------(1.2)

Equation (1.1) gives

)

(

)

(

)

(

or, i

= E

-----------(1.3)

For a free particle, E =

. Hence equation (1.2) and (1.3) give

= i

-----------(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

)

(

)

(

as well as a wave packet given by equation (1.1) satisfy it.

Chapter I Background on Quantum Mechanics

1.2 Schrödinger equation of a particle subject to a conservative mechanical

force

1) A comparison of

< x > =

dx and < p > =

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 its position < x > if

the operator

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

By "operator of p is

", we in fact mean, the operator we need to calculate

expectation value of p is

2) Schrödinger equation of a free particle is

= i

. This equation can

be obtained from the classical equation

= E by the operator correspondence of p

and E as

and i

! respectively, and letting the operators operate on the

wavefunction

. Thus operator of E is i

! .

3) If a conservative force acts on a particle, total energy E =

+ V where V is

potential energy and hence V is a function of position only. Hence

)

(

)

(

)

(

)

(

Thus operator of V(x) is V(x).

Oscillatory transmission through non-tunneling regime of symmetric rectangular double barrier:

Parametric variations of transmission peaks

Md. Abidur Rahman and Sujaul Chowdhury

4) Using the operator correspondences of E, p and V,

+ V = E gives

+ V = i

! or,

+ V = i

or,

+ V

= i

or,

= i

This is the Schrödinger equation of a particle moving in a potential V. In three

dimensions, [

+ V]

= i

is the Schrödinger equation of a particle

moving in a potential V. The operator [

+ 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 Schroedinger

equation

+ V = i!

-----------(1.5)

where V is the potential associated with the force. Complex conjugate of equation

(1.5) is

+ V

= - i!

-----------(1.6)

V is real.

Multiplying equation (1.5) by

, we get

+ V

= i!

-----------(1.7)

Multiplying equation (1.6) by

, we get

+ V

- i!

-----------(1.8)

Chapter I Background on Quantum Mechanics

Equation (1.7) (1.8)

[

] = i! ¨

or,

· (

) = i!

(

)

or,

· [

(

)] = -

-----------(1.9)

or,

· S

where

and S

(

)

----------(1.10)

Equation (1.9)

over a volume v.

or,

· A

----------(1.11)

Using

Gauss's

divergence

theorem

The surface integral is over the closed surface that encloses the volume v. In

equation (1.11),

d is probability of presence of a particle in the volume v.

d is time rate of decrease of probability of presence of the particle in the

volume v. Since this rate is equal to

· A

, we can say that

is time rate of flow-

out of the probability per unit area through the surface enclosing the volume v. The

probability changes or decreases because of change of

with time. Hence

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.

Oscillatory transmission through non-tunneling regime of symmetric rectangular double barrier:

Parametric variations of transmission peaks

Md. Abidur Rahman and Sujaul Chowdhury

1.4 Time-independent Schrödinger equation and stationary state

Schrödinger equation for a particle under conservative force is

)

(

+ V( r

)

(

= i

)

(

---------(1.12)

which is called time-dependent Schroedinger equation. V( r

) is independent of time.

The Hamiltonian is also time independent. Equation (1.12) simplifies considerably.

Let us try

)

(

= 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)

[u( r& ) f(t)] + V( r& ) u( r& ) f(t) = i!

[u( r

) f(t)]

or,

f(t)

u( r& ) + V( r& ) u( r& ) f(t) = i! u( r& )

f(t)

or,

( )

+ V( r

) =

)

(

)

(

---------(1.13)

Dividing by u( r

) f(t)

LHS of equation (1.13) is a function of position only and 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

( )

+ V( r

) = C

---------(1.14)

and

)

(

)

(

---------(1.15)

Equation (1.14)

u( r& ) + V( r& ) u( r& ) = C u( r& )

or, [

+ V( r& )] u( r& ) = C u( r& )

--------(1.16a)

or,

u( r

) = C u( r

)

--------(1.16b)

Chapter I Background on Quantum Mechanics

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)

[

+ V( r& )] u( r& ) = E u( r& )

--------(1.17a)

or,

u( r

) = E u( r

)

--------(1.17b)

Equation (1.17) is called time-independent Schroedinger equation.

Equation (1.15)

f(t) = E f(t)

or,

f(t) =

)

(

Let f(t) =

= E

or, (n

- E

)

= 0

n - E

= 0

or, n = E

- i

f(t) =

( r& , t) = u( r& ) f(t) = u( r& )

= u( r

)

---------(1.18)

)

(

( r& , t) ( r& , t) = u*( r& )

u( r

)

= u*( r

) u( r

)

E is real,

E* = E.

)

(

---------(1.19)

From equation (1.19), we find that

)

(

is independent of time. Thus the states

given by equation (1.18) are stationary states.

Expectation value of any observable A is

< A > =

)

(

)

(

]

)

(

[

)

(

using

equation

(1.18)

)

(

)

(

E is real.

does not explicitly contain the variable t (time).

Oscillatory transmission through non-tunneling regime of symmetric rectangular double barrier:

Parametric variations of transmission peaks

Md. Abidur Rahman and Sujaul Chowdhury

)

(

)

(

For stationary states, expectation value of any observable A is independent of time,

provided the operator

itself does not depend explicitly on t.

In hydrogen atom, e.g., the potential V(r) =

is a function of position

only. Thus solution of equation (1.12) will give stationary state. Thus

for the

electron and expectation values of all observables (e.g. energy) of the electron remain

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 single valued at every value of x.

is also continuous and single valued.

f(x)

Chapter I Background on Quantum Mechanics

f(x) above is a discontinuous function of x. The discontinuity is at x =

where the function is many valued. f(

x ) is unspecified.

f < f(

x ) <

f .

) =

x +

) =

f .

at x =

- is tan

f(x)

Oscillatory transmission through non-tunneling regime of symmetric rectangular double barrier:

Parametric variations of transmission peaks

Md. Abidur Rahman and Sujaul Chowdhury

at x =

x +

is tan

. Here 0.

and

may or may not be equal, depending on the nature of f(x). Thus the

derivative

may or may not be continuous.

At x =

x ,

= tan 90

1.6 Finite and infinite discontinuity

The discontinuity considered above is finite discontinuity, because

and

are finite. If a < f(

x ) < b where either a or b (or both) is +

, the

discontinuity is infinite discontinuity.

f(x)

Chapter I Background on Quantum Mechanics

x +

) = a

- ) = +

a < f(

x ) <

1.7 Admissibility conditions on wavefunction

(x , t)

0 as x

± , because

= 1 is finite.

(x , t) must be finite, single valued and continuous function of x for all time t;

this is because of probability interpretation of

is a continuous function of x, because otherwise

= c where

c < c <

c at

say x =

x .

f(x)

all space

Oscillatory transmission through non-tunneling regime of symmetric rectangular double barrier:

Parametric variations of transmission peaks

Md. Abidur Rahman and Sujaul Chowdhury

This means

= c t which is many valued at x =

x . But

must be single valued

according to condition (2).

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.

is also continuous (condition (3))

function of x. Hence Schroedinger equation

(x,

(x , t) +V(x , t) (x , t)

gives

is continuous. Thus

is continuous function of x; otherwise

becomes infinite at points (x) where

is discontinuous.

must be continuous function of x if V does not have infinite discontinuity. In

Schroedinger equation i

+ V(x, t) (x, t),

is always

Chapter I Background on Quantum Mechanics

continuous function of x (condition 3). If

has any discontinuity say at x = x

, the

versus x curve becomes vertical at x =

x and hence its slope

becomes

infinite at x =

x . Thus

. This forces V to become + to keep

the Schroedinger equation valid. Since

is finite, continuous and single valued

everywhere, V is forced to be +

at x =

x . Thus unless V has an infinite

discontinuity,

must be a continuous function of x.

Any finite discontinuity of V makes V

finite discontinuous. This is adjusted

by a finite discontinuity of

because

is always continuous [see

Schroedinger equation]. Finite discontinuity of

means

is continuous,

otherwise

gets infinite discontinuity there.

1.8 Free particle: eigenfunctions and probability current density

If the potential is constant, V(x) =

V , force acting on the particle F(x) =

= 0; so the particle is free.

V can be taken to be zero, without any loss of generality.

Time-independent Schroedinger equation is

)

(

+ 0 = E u(x)

or,

u = 0

or,

u = 0.

0, k =

)

(

u(x) = A

ikx

(x , t) = A

)

(

= A

)

(

Oscillatory transmission through non-tunneling regime of symmetric rectangular double barrier:

Parametric variations of transmission peaks

Md. Abidur Rahman and Sujaul Chowdhury

where k =

, E =

Probability current density corresponding to the free particle eigenfunction

)

(

S =

[

]

[

)

(

)

(

)

- A

)

(

)

(

)]

A [

p (

p)] =

A 2

p =

A =

A = v

and

A .

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

one-dimensional, 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

. The

barrier is defined as

V(x) = 0 for

= V

for

V(x)

(0, 0)

Region I

Region II

Region III

Chapter I Background on Quantum Mechanics

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

) 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.

1.9.1 Calculation of transfer matrix and investigation of its properties (E < V

)

- a

+a x

Solutions of time-independent Schrödinger equation

[

+ V(x)] u(x) = E u(x)

or,

))

(

in the three regions are u

, u

and u

given by

ikx

)

(

where

)

(

where

)

(

ikx

)

(

The expressions for u

and

imply that we are considering free electrons of kinetic

energy less than V

impinging on the barrier from the left.

V(x)

(0, 0)

Region I

Region II

Region III

Oscillatory transmission through non-tunneling regime of symmetric rectangular double barrier:

Parametric variations of transmission peaks

Md. Abidur Rahman and Sujaul Chowdhury

Using the boundary condition

= u

at x =

-a, we get

ika

---------(1.20)

Again, the boundary condition

at x =

-a gives

ikx

ikBe

at x =

-a

or, ik

ika

ikBe

or,

ika

---------(1.21)

The boundary condition u

= u

at x = +a gives

ika

---------(1.22)

And

at x = +a gives

ikx

ikHe

ikGe

at x = +a

ika

ikHe

ikGe

ika

---------(1.23)

From equation (1.20) + (1.21), we have

ika

)

(

)

(

ika

)

(

)

(

From equation (1.20)

- (1.21), we have

ika

)

(

)

(

ika

)

(

)

(

Now in the form of matrix we can write

Chapter I Background on Quantum Mechanics

¸¸

¨¨

¸¸

¨¨

¸¸

¨¨

)

(

)

(

)

(

)

(

ika

From equation (1.22) + (1.23), we have

From equation (1.22)

- (1.23), we have

ika

)

(

)

(

)

(

)

(

Now in the form of matrix we can write

¸¸

¨¨

¸¸

¨¨

)

(

)

(

)

(

)

(

ika

¸¸

¨¨

¸¸

¨¨

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

ika

or,

¸¸¹

¨¨©

¸¸¹

¨¨©

¸¸¹

¨¨©

---------(1.24)

ika

)

(

)

(

)

(

)

(

Oscillatory transmission through non-tunneling regime of symmetric rectangular double barrier:

Parametric variations of transmission peaks

Md. Abidur Rahman and Sujaul Chowdhury

The M matrix is called transfer matrix. The elements of the matrix are calculated in

the following.

ika

)

(

)

(

ika

)

(

)

(

ika

]

)

(

)

[(

]

)

)(

(

)

)(

[(

]

)

)(

(

)

(

[

]

sinh

)

(

[cosh

----------(1.25)

ika

)

(

)

(

ika

)

(

)

(

ika

]

)

)(

(

)

)(

[(

]

)

)(

(

)

(

[

]

)

(

)

[(

]

sinh

)

(

[cosh

]

sinh

)

(

[cosh

----------(1.26)

Chapter I Background on Quantum Mechanics

ika

)

(

)

(

]

)

)(

(

)

)(

[(

)

(

)

(

ika

]

)

(

)

[(

)

)(

(

sinh

)

(

sinh

)

(

----------(1.27)

ika

)

(

)

(

]

)

)(

(

)

)(

[(

)

(

)

(

ika

sinh

)

(

)

)(

(

]

)

(

)

[(

sinh

)

(

---------(1.28)

We find that the elements of the transfer matrix M obey the following properties.

----------(1.29)

or,

----------(1.30)

Oscillatory transmission through non-tunneling regime of symmetric rectangular double barrier:

Parametric variations of transmission peaks

Md. Abidur Rahman and Sujaul Chowdhury

We now evaluate the determinant of the M matrix.

]

sinh

)

(

[cosh

]

sinh

)

(

[cosh

]

sinh

)

(

[

]

sinh

)

(

[

sinh

cosh

sinh

]

[

cosh

sinh

]

)

(

)

[(

cosh

sinh

)

(

sinh

)

(

cosh

Thus

----------(1.31)

i.e. the determinant of the transfer matrix is unity.

1.9.2 Calculation of transmission coefficient

- a

+a x

V(x)

(0 , 0)

Region I

Region II

Region III

Chapter I Background on Quantum Mechanics

We have solutions of time-independent Schrödinger equation

[

+ V(x)] u(x) = E u(x) or,

))

(

in the three regions given by u

, u

and u

ikx

)

(

where

)

(

where

)

(

ikx

)

(

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.

The matrix equation is equivalent to the following two equations.

---------(1.32)

and B

---------(1.33)

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

---------(1.34)

which we can put in equation (1.32) to get

---------(1.35)

using

equation

(1.31)

Oscillatory transmission through non-tunneling regime of symmetric rectangular double barrier:

Parametric variations of transmission peaks

Md. Abidur Rahman and Sujaul Chowdhury

Now transmission coefficient of the single barrier is given by

. With the aid

of equation (1.35), T

reduces to

----------(1.36)

Reflection coefficient of the single barrier is given by

With the aid of equation (1.34), R

reduces to

---------(1.37)

---------(1.38)

using equation (1.36).

We now use equation (1.36) to obtain an analytic expression for T

for E < V

The transmission coefficient is given by

]

sinh

)

(

[cosh

]

sinh

)

(

[cosh

using

equation

(1.26)

sinh

)

(

cosh

sinh

)

(

sinh

]

)

(

[

sinh

]

[

sinh

)

(

--------(1.39a)

Chapter I Background on Quantum Mechanics

sinh

¸¸¹

¨¨©

)

(

sinh

¸¸¹

¨¨©

)

(

sinh

)

(

))

(

¸¸¹

¨¨©

)

(

sinh

)

(

--------(1.39b)

Equation (1.39) is for E < V

For E > V

)

(

Using x

tan

tanh

cos

cosh

sin

sinh

in equation (1.39a),

we get

sinh

)

(

)

sin

(

)

(

sin

)

(

---------(1.40a)

sin

¸¸¹

¨¨©

)

(

sin

Excerpt out of 133 pages - scroll top

Details

Title: Oscillatory transmission through non-tunneling regime of symmetric rectangular double barrier
Subtitle: Parametric variations of transmission peaks
College: Shahjalal University of Science and Technology (Department of Physics)
Course: Nanostructure Physics
Authors: Dr Sujaul Chowdhury (Author), Abidur Rahman (Author)
Publication Year: 2013
Pages: 133
Catalog Number: V230804
ISBN (eBook): 9783656461142
ISBN (Book): 9783656461777
Language: English
Tags: Numerical investigation Nanostructure Physics GaAs-AlGaAs symmetric rectangular double barrier non-tunneling regime (E > V0) oscillatory transmission parametric variations of transmission peaks unbound states virtual bound states above barrier states semiconductor nanostructure

Quote paper: Dr Sujaul Chowdhury (Author), Abidur Rahman (Author), 2013, Oscillatory transmission through non-tunneling regime of symmetric rectangular double barrier, Munich, GRIN Verlag, https://www.grin.com/document/230804