Nanostructure Physics of non-isolated Quantum Well of symmetric double barrier

Chowdhury, Sujaul; Samad, Abdus; Dash, Dipak

Blick ins Buch

Nanostructure Physics of non-isolated Quantum Well of symmetric double barrier

analytical calculation of equation obeyed by quasi-bound energy levels of the Quantum Well

Masterarbeit, 2011
137 Seiten

Physik - Sonstiges

Leseprobe

Index

Nanostructure Physics of non-isolated Quantum Well of symmetric double barrier:

analytical calculation of equation obeyed by quasi-bound 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

Background on Quantum Mechanics

1-24

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

(E < V

)

1.9.2 Calculation of transmission coefficient

1.10 Further topics on Quantum Mechanics

Index

Chapter

Background on Microelectronics

25-

2.1 Intrinsic

semiconductor 26

2.2 Semiconductors: elemental and binary

2.3 Alloy semiconductors (ternary and quaternary)

2.4 Bandgap

engineering

2.5 Semiconductor heterojunction and heterostructure

2.6 Effective

mass

2.7 Further topics on Microelectronics

Chapter

III

Background on Nanostructure Physics

36-

3.1 Single rectangular tunnel barrier

3.2 Transmission coefficient of a single rectangular tunnel barrier

3.3 Quantum

well

(QW)

3.4 Double

potential

barrier

3.5 Transmission coefficient of double potential barrier

3.6 Transmission coefficient of double barrier if two barriers are identical

3.7 Profile of transmission peak

3.8 T versus E curve of symmetric rectangular double barrier

3.9 Further study of double barrier structures

Index

iii

Chapter

Analytical calculation of transcendental equation

obeyed by quasi-bound energy levels of the non-isolated Quantum Well

of symmetric rectangular double barrier

59-

4.1 Description of the problem

4.2 Methodology and Physics of calculation of the transcendental equation

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

4.5 Calculation of transcendental equation obeyed by quasi-bound energy

levels

4.6 Resonant transmission peaks obey the same condition

Chapter

Taking effective mass inequality into account:

analytical calculation of transcendental equation

obeyed by quasi-bound energy levels

of the non-isolated Quantum Well

of symmetric rectangular double barrier:

75-

5.1 Introduction

5.2 Calculation of transfer matrix of the left hand rectangular tunnel

barrier: taking effective mass inequality into account

5.3 Calculation of inverse transfer matrix of the right hand rectangular

tunnel barrier: taking effective mass inequality into account

5.4 Calculation of transcendental equation obeyed by quasi-bound energy

levels: taking effective mass inequality into account

5.5 Resonant transmission peaks obey the same condition:

taking effective mass inequality into account

Index

Chapter

Derivation of WKB solution of Schroedinger equation

and introduction to WKB connection formulae

91-

100

6.1 WKB

approximation

6.2 WKB solution of (1 dimensional) Schroedinger equation

6.3 Classical turning point

6.4 WKB connection formula: case I

6.5 WKB connection formula: case II

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 quasi-bound energy levels of non-isolated 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 quasi-bound 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 non-isolated Quantum Well of symmetric double barrier:

analytical calculation of equation obeyed by quasi-bound energy levels of the Quantum Well

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 write

)

(

)

(

)

(

where a(p) =

)

(

)

(

using de Broglie's equations p =

! k and E =

! . In three dimensions, we have

)

(

)

(

)

(

)

(

__________ (1.1)

where r

. Equation (1.1) gives

)

(

)

(

)

(

and

)

(

)

(

)

(

Similarly,

)

(

)

(

)

(

and

)

(

)

(

)

(

all momentum space

Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:

Nanostructure Physics of non-isolated Quantum Well of symmetric double barrier:

analytical calculation of equation obeyed by quasi-bound energy levels of the Quantum Well

)

(

)

(

)

(

)

(

)

(

)

(

)

(

or,

)

(

)

(

_________(1.2)

Again, 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.

1.2 Schrödinger equation of a particle subject to a conservative mechanical

force

1) A comparison of

< x > =

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 position < x > if the

Chapter I Background on Quantum Mechanics

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 the

expectation value of p is

2) The 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 the operator of E is i

! .

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

+ V where V is

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

)

(

)

(

)

(

)

(

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

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

+ V = E gives

+ V = i

or,

+ V = i

or,

+ V

= i

or,

= i

Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:

Nanostructure Physics of non-isolated Quantum Well of symmetric double barrier:

analytical calculation of equation obeyed by quasi-bound energy levels of the Quantum Well

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 Schrödinger

equation

+ V = i!

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

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

(1.5) is

+ V

= - i!

V is real.

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

Multiplying equation (1.5) by

, we get

+ V

= i!

_______(1.7)

Multiplying equation (1.6) by

, we get

+ V

= - i!

_______(1.8)

Equation (1.7) (1.8)

[

] = i! ¨

or,

· (

) = i!

(

)

or,

· [

(

)] = -

_______(1.9)

Chapter I Background on Quantum Mechanics

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 the probability of the presence of a particle in the volume 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

· A

, we can say that

is the 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 the change of

with time.

Hence

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 Time-independent Schrödinger equation and stationary state

The Schrödinger equation for a particle under conservative force is

)

(

+ V

)

(

= i

)

(

______(1.12)

which is called time-dependent Schrödinger equation. If V( r

) is independent of time,

the Hamiltonian is also time-independent and equation (1.12) simplifies

considerably.

Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:

Nanostructure Physics of non-isolated Quantum Well of symmetric double barrier:

analytical calculation of equation obeyed by quasi-bound energy levels of the Quantum Well

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)

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

Thus

( )

+ V( r

) = C

______(1.14)

and

)

(

)

(

= C

______(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)

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

[

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

______(1.17a)

or,

u( r

) = E u( r

)

______(1.17b)

Equation (1.17) is called time-independent Schrödinger 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.

The expectation value of any observable A is

< A > =

)

(

)

(

]

)

(

[

)

(

using equation (1.18)

)

(

)

(

E is real.

does not explicitly contain the variable t (time).

)

(

)

(

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

time, provided the operator

itself does not depend explicitly on t.

Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:

Nanostructure Physics of non-isolated Quantum Well of symmetric double barrier:

analytical calculation of equation obeyed by quasi-bound energy levels of the Quantum Well

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

is a function of position

only. Thus the solutions of equation (1.12) will give stationary states. Thus

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 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 = x

where the

function is many valued. f(x

) is unspecified.

f < f(x

) <

f .

f(x

) =

f(x

) =

f .

at x = x

- is tan

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

°=

f(x)

Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:

Nanostructure Physics of non-isolated Quantum Well of symmetric double barrier:

analytical calculation of equation obeyed by quasi-bound energy levels of the Quantum Well

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

) = a

f(x

- ) = +

a < f(x

) <

f(x)

Chapter I Background on Quantum Mechanics

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 <

say x = x

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 Schrödinger equation

(x,

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

all space

Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:

Nanostructure Physics of non-isolated Quantum Well of symmetric double barrier:

analytical calculation of equation obeyed by quasi-bound energy levels of the Quantum Well

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

Schrödinger equation i

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

is always

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

Schrödinger 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 Schrödinger

equation]. Finite discontinuity of

means

is continuous, otherwise

gets infinite discontinuity there.

Chapter I Background on Quantum Mechanics

1.8 Free particle: eigenfunctions and probability current density

If the potential is constant, V(x) =

V , the 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.

The time-independent Schrödinger equation becomes

)

(

+ 0 = E u(x)

or,

u = 0

or,

u = 0.

0, k

)

(

u(x) = A

ikx

(x , t) = A

)

(

= A

)

(

k =

, E =

The probability current density corresponding to the free particle eigenfunction

)

(

S =

[

]

[

)

(

)

(

)

- A

)

(

)

(

)]

A [

p (

p)] =

A 2

Thus S = v

A and

A .

Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:

Nanostructure Physics of non-isolated Quantum Well of symmetric double barrier:

analytical calculation of equation obeyed by quasi-bound energy levels of the Quantum Well

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

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.

V(x)

(0, 0)

Region I

Region II

Region III

Chapter I Background on Quantum Mechanics

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.

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

V(x)

(0, 0)

Region I

Region II

Region III

Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:

Nanostructure Physics of non-isolated Quantum Well of symmetric double barrier:

analytical calculation of equation obeyed by quasi-bound energy levels of the Quantum Well

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

¸¸

¨¨

¸¸

¨¨

¸¸

¨¨

)

(

)

(

)

(

)

(

ika

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

ika

)

(

)

(

)

(

)

(

Chapter I Background on Quantum Mechanics

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)

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

the following.

ika

)

(

)

(

ika

)

(

)

(

ika

]

)

(

)

[(

]

)

)(

(

)

)(

[(

Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:

Nanostructure Physics of non-isolated Quantum Well of symmetric double barrier:

analytical calculation of equation obeyed by quasi-bound energy levels of the Quantum Well

]

)

)(

(

)

(

[

]

sinh

)

(

[cosh

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

ika

)

(

)

(

ika

)

(

)

(

ika

]

)

)(

(

)

)(

[(

]

)

)(

(

)

(

[

]

)

(

)

[(

]

sinh

)

(

[cosh

]

sinh

)

(

[cosh

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

ika

)

(

)

(

]

)

)(

(

)

)(

[(

)

(

)

(

ika

]

)

(

)

[(

)

)(

(

sinh

)

(

sinh

)

(

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

Chapter I Background on Quantum Mechanics

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)

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.

Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:

Nanostructure Physics of non-isolated Quantum Well of symmetric double barrier:

analytical calculation of equation obeyed by quasi-bound energy levels of the Quantum Well

1.9.2 Calculation of transmission coefficient

-a

+a x

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)

V(x)

(0, 0)

Region I

Region II

Region III

Chapter I Background on Quantum Mechanics

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)

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

)

(

sinh

)

(

cosh

Md. Abdus Samad, Sujaul Chowdhury, Dipak Dash:

Nanostructure Physics of non-isolated Quantum Well of symmetric double barrier:

analytical calculation of equation obeyed by quasi-bound energy levels of the Quantum Well

sinh

]

[

sinh

]

)

(

[

sinh

)

(

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

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 (1.39),

we get:

sinh

)

(

)

sin

(

)

(

sin

)

(

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

Ende der Leseprobe aus 137 Seiten - nach oben

Details

Titel: Nanostructure Physics of non-isolated Quantum Well of symmetric double barrier
Untertitel: analytical calculation of equation obeyed by quasi-bound energy levels of the Quantum Well
Hochschule: Shahjalal University of Science and Technology (Department of Physics)
Veranstaltung: Nanostructure Physics
Autoren: Sujaul Chowdhury (Autor:in), Abdus Samad (Autor:in), Dipak Dash (Autor:in)
Jahr: 2011
Seiten: 137
Katalognummer: V211440
ISBN (eBook): 9783656395065
ISBN (Buch): 9783656395966
Dateigröße: 1067 KB
Sprache: Englisch
Anmerkungen: KEYWORDS Nanostructure Physics, symmetric rectangular double barrier, symmetric double barrier of general shape, non-isolated Quantum Well, quasi-bound energy levels, WKB method, analytical calculation, semiconductor nanostructures
Schlagworte: nanostructure physics quantum well
Preis (Ebook): US$ 36,99
Preis (Book): US$ 47,99

Arbeit zitieren: Sujaul Chowdhury (Autor:in), Abdus Samad (Autor:in), Dipak Dash (Autor:in), 2011, Nanostructure Physics of non-isolated Quantum Well of symmetric double barrier, München, Page::Imprint:: GRINVerlagOHG, https://www.diplomarbeiten24.de/document/211440

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Nanostructure Physics of non-isolated Quantum Well of symmetric double barrier

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