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Project report
On
QUANTUM MECHANICAL STUDY OF DOUBLE
HETEROSTRUCTURE LED
This project report is submitted
towards partial fulfillment of the requirement for the award of degree of
“B.E. Electronics and Communication”
Submitted by
Kalyani Yeotikar Shikha Paliwal
Urvashi Dhoot Vipul Hada
Under guidance of
Dr. Rajesh Raut
Co guide
Dr. D.K. Bopardikar
Electronics and Communication Department
Shri Ramdeobaba College of Engineering and Management
Nagpur-440013
(An Autonomous College of Rashtrasant Tukadoji Maharaj Nagpur University)
2014-2015

CERTIFICATE
Shri Ramdeobaba College of Engineering and Management, Nagpur
This is to certify that project titled
“QUANTUM MECHANICAL STUDY OF
DOUBLE HETEROSTRUCTURE LED”
has been successfully completed by following students in recognition of
partial fulfillment for Final Year B.E. E&C Engineering
Shri Ramdeobaba College of Engineering and Management
Nagpur (2014-2015)
Submitted by
Dr. Rajesh Raut Dr. D.K. Bopardikar
Associate Professor Professor
Department of E&C Department of Physics
(Guide) (Co-guide)
Dr. S.B. Pokle
(HOD, E&C Department)
Dr. R.S. Pande
(Principal, RCOEM)

ANKNOWLEGEMENTS
Success is manifestation of perseverance inspiration and motivation.
We, the projectees take this opportunity to express our profound
gratitude and deep regards to our guide Dr. Rajesh Raut for his
exemplary guidance, monitoring and constant encouragement
throughout the course of this thesis. The blessing, help and guidance
given by him time to time shall carry us a long way in the journey of
life on which we are about to embark.
We would like to express our sincere thanks to our co guide
Dr.D.K. Bopardikar. He has guided and supported us in all our
endeavours. We are deeply indebted to him for giving us clarity of
vision and thought which enabled us to complete this project.
In our journey we have always been guided and supported by our
respected Head of Department Dr.S.B. Pokle. He has always instilled
confidence in us and has inculcated many skills in us which we will
carry with us all our lives.
We would like to thank our principal Dr.R.S. Pande and all the staff of
the Electronics and Communication department for extending us the
facilities without which our project would not have been a success.
Submitted by

Contents
1 Quantum Mechanical Study of Double Heterostructure LED 2
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Introduction to Semiconductor . . . . . . . . . . . . . . . 4
1.1.2 Intrinsic and Extrinsic Semiconductor . . . . . . . . . . . 4
1.1.3 Semiconductor Materials: . . . . . . . . . . . . . . . . . 6
1.1.4 Why Direct Bandgap and Why Not Indirect Bandgap Semi-
conductors !? . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.5 Direct Band Gap Light Emitting Diode . . . . . . . . . . 10
1.1.6 LED Characteristics . . . . . . . . . . . . . . . . . . . . 13
1.1.7 Drawbacks of Homojunction LED . . . . . . . . . . . . . 15
1.1.8 Heterojunction Structure . . . . . . . . . . . . . . . . . . 15
2 The Schrodinger Equation 18
2.1 QUANTUM MECHANICAL STUDY:- . . . . . . . . . . . . . . 18
2.1.1 So which is it - a particle or a wave? . . . . . . . . . . . . 19
2.1.2 WHAT IS QUANTUM-MECHANICS? . . . . . . . . . . 19
2.1.3 WHY IS QUANTUM -MECHANICS IMPORTANT? . . 20
2.1.4 WAVE FUNCTION . . . . . . . . . . . . . . . . . . . . 21
2.1.5 SCHRODINGER EQUATION . . . . . . . . . . . . . . . 21
3 Mathematica 24
3.1 Introduction to Mathematica . . . . . . . . . . . . . . . . . . . . 24
3.1.1 What is Mathematica . . . . . . . . . . . . . . . . . . . . 24
3.1.2 Features of Mathematica . . . . . . . . . . . . . . . . . . 24
3.1.3 Our purpose of using Mathematica . . . . . . . . . . . . . 25
4 Application and future scope 47
4.1 Application and Future Scope . . . . . . . . . . . . . . . . . . . 47
4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1.2 RESULT AND CONCLUSION . . . . . . . . . . . . . . 48
4.1.3 FUTURE SCOPE . . . . . . . . . . . . . . . . . . . . . 48
1

Chapter 1
Quantum Mechanical Study of
Double Heterostructure LED
1.1 Introduction
The engineering students generally study light emitting diodes (LED) as an ele-
ment in an electronic circuit. The light emitting diodes are generally used as a
basic unit of a big optical display or as a source in optical communication. In fact
the fiber optic communication forms the basis of the current technology which
has revolutionized the telecommunication industry. It has enabled telecommuni-
cations links to be made over much greater distances and with much lower level
of loss in its medium besides high rate of data transmission.
The present project work is aimed at study of LED from entirely different point
of view. In the present work an attempt has been made to study LED from the
point of view of semiconductor physics. A LED is essentially a p-n junction diode
typically made from direct bandgap semiconductor, for example GaAs in which
the electron hole pair recombination results in the emission of photon. The emitted
energy photon is therefor approximately equal to the bandgap energy h=Eg. In its
simpest form the semiconductor material used on either side of a p-n junction, has
same bandwidth(Eg), and hence termed as homojunction LED.
The homojunction LED suffers from certain drawbacks. The radiationless
recombination of electron and re-absorption of photon due to long electron dif-
fusion length causes substantial reduction in the intensity of the output radiation.
These drawbacks have been taken care in the double heterostructure(DH) device
based on two junctions between different semiconductor materials with different
bandgap. In this case the semiconductors are AlGaAs with Eg = 1.4eV. The het-
erostructure is shown in figure 1.
In a forward bias conditions electrons are injected in narrow confined region
2

1.1. INTRODUCTION
Figure 1.1: Double heterostructure LED
of GaAs of 0.2 micro m. These electrons suffer recombination the holes in central
p-region of GaAs to produce a photons. The wavelength of the output radiation
obviously depend on the energy levels of electron and holes in the confined region
of GaAs. In the present work this confined region has been modelled as Quantum
well as shown in figure 2.
Figure 1.2: (a) The energy band structure of an intrinsic semiconductor at an
temperature above absolute zero. (b) The Fermi Dirac Probability Distribution
corresponding to (a)
The attempt has been made in the present project to calculate energy levels
in central confined region of the quantum well. This has been done by solving
Schrodinger equation for finite potential well.
To understand the topic we need to study the basics about the semiconduc-
tor and how we reached to our problem statement. For which further we will be
gathering information about semiconductors, LED and how we reached to het-
3

1.1. INTRODUCTION
erostructure.
1.1.1 Introduction to Semiconductor
A semiconductor material has an electrical conductivity value falling between
conductor and an insulator. A semiconductor is a substance, usually a solid chem-
ical element or compound, that can conduct electricity under some conditions but
not others, making it a good medium for the control of electrical current. Further
it is subdivided into two categories:
Intrinsic Semiconductor
Extrinsic Semiconductor
1.1.2 Intrinsic and Extrinsic Semiconductor
Intrinsic Semiconductor: A perfect semiconductor containing no impurities or
lattice defects is said to be an intrinsic semiconductor. An intrinsic semiconductor
in which the valence and conduction bands separated by a forbidden energy gap or
bandgap ’Eg’. In the semiconductor at a temperature above absolute zero where
thermal excitation raises some electrons from the valence band into the conduc-
tion band, leaving empty hole states in the valence band. These excited electrons
in the conduction band and the holes left in the valence band allow conduction
through the material, called as carriers. For a semiconductor in thermal equilib-
rium the energy-level occupation is described by the FermiDirac distribution func-
tion. Consequently, the probability P(E) that an electron gains sufﬁcient thermal
energy at an absolute temperature T, such that it will be found occupying a partic-
ular energy level E, is given by the FermiDirac distribution: P(E)= 1
1+exp(E−Ef )KT
The energy band structure of an intrinsic semiconductor is shown in ﬁgure
3. The Fermi level gives an indication of the distribution of carriers within the
material. For the intrinsic semiconductor where the Fermi level is at the center
of the bandgap, indicating that there is a small probability of electrons occupying
energy levels at the bottom of the conduction band and a corresponding number
of holes occupying energy levels at the top of the valence band.
Extrinsic Semiconductor: To create an extrinsic semiconductor, the material is
doped with impurity atoms which create either more free electrons (donor impu-
rity) or holes (acceptor impurity). These two situations where the donor impurities
form energy levels just below the conduction band while acceptor impurities form
energy levels just above the valence band.
When donor impurities are added, the excited electrons from the donor levels
are moved into the conduction band to create an excess of negative charge carri-
4

1.1. INTRODUCTION
Figure 1.3: (a) The energy band structure of an intrinsic semiconductor at an
temperature above absolute zero. (b) The Fermi Dirac Probability Distribution
corresponding to (a)
Figure 1.4: Energy band diagrams: (a) n-type semiconductor; (b) p-type semicon-
ductor
ers and the semiconductor is said to be n-type, with the majority carriers being
electrons. The Fermi level is above the center of the bandgap. When acceptor
impurities are added, the excited electrons goes from the valence band to the ac-
ceptor impurity levels leaving an excess of positive charge carriers in the valence
band and creating a p-type semiconductor where the majority carriers are holes.
In this case Fermi level is lowered below the center of the bandgap.
P-N Junction Diode: The pn junction diode is formed by joining p-type and
n-type semiconductor layers. A thin depletion region or layer is formed at the
junction through carrier recombination. This establishes a potential barrier be-
tween the p-type and n-type regions which restricts diffusion of majority carriers
from their respective regions. In the absence of an externally applied voltage no
current ﬂows as the potential barrier prevents the ﬂow of carriers from one region
to another. When the junction is in this equilibrium state the Fermi level for the
5

1.1. INTRODUCTION
p- and n-type semiconductor is the same.
Figure 1.5: (a) The impurities and charge carriers at a pn junction. (b) The energy
band diagram corresponding to (a)
The width of the depletion region and thus the magnitude of the potential
barrier is dependent upon the doping in the p-type and n-type regions and any
external applied voltage. When an external positive voltage is applied to the p-type
region, the depletion region width and the resulting potential barrier are reduced
and the diode is said to be forward biased. Electrons from the n-type region and
holes from the p-type region can flow across the junction into the opposite type
region. By the application of the external voltage, these minority carriers are
injected across the junction and form a current flow through the device. However,
this situation in suitable semiconductor materials allows carrier recombination
with the emission of light.
1.1.3 Semiconductor Materials:
The semiconductor material need to possess-
1. pn junction formation. The materials must lend themselves to the formation
of pn junctions with suitable characteristics for carrier injection. 2. Efficient elec-
troluminescence. The devices fabricated must have a high probability of radiative
transitions and therefore a high internal quantum efficiency. Hence the materials
6

1.1. INTRODUCTION
utilized must be either direct bandgap semiconductors or indirect bandgap semi-
conductors with appropriate impurity centers. 3. Useful emission wavelength.
The materials must emit light at a suitable wavelength to be utilized with current
optical fibers and detectors 0.8 to 1.7 um. Ideally, they should allow bandgap vari-
ation with appropriate doping and fabrication in order that emission at a desired
specific wavelength may be achieved.
The electroluminescent materials for LEDs in the early 1960s centered around
the direct bandgap IIIV alloy semiconductors including the gallium arsenide (GaAs)
and gallium phosphide (GaP) and the ternary gallium arsenide phosphide (GaAsxP1−x).
Gallium arsenide gives efficient electroluminescence over an appropriate wave-
length band (0.88 to 0.91 um). It was quickly realized that improved devices
could be fabricated with heterojunction structures which through carrier and radia-
tion confinement would give enhanced light output. These heterostructure devices
were first fabricated using liquid-phase epitaxy (LPE) to produce GaAs/AlxGa1−xAs
single heterojunction lasers. This process involves the precipitation of material
from a cooling solution onto an underlying substrate. When the substrate con-
sists of a single crystal and the lattice constant or parameter of the precipitating
material is the same or very similar to that of the substrate (i.e. the unit cells
within the two crystalline structures are of a similar dimension), the precipitat-
ing material forms an epitaxial layer on the substrate surface. Subsequently,
the same technique was used to produce double heterojunctions consisting of
AlxGa1−xAs/GaAs/AlxGa1−xAs epitaxial layers, which gave continuous wave op-
eration at room temperature. Some of the common material systems now utilized
for DH device fabrication, together with their useful wavelength ranges, are shown
in Table:
Active Layer/Confinement Layer Wavelength Range(um) Substrate
GaAs/AlxGa1−xAs 0.8-0.9 GaAs
GaAs/InxGa1−xP 0.9 GaAs
AlyGa1−yAs/AlxGa1−xAs 0.65-0.9 GaAs
InyGa1−yAs/InxGa1−xP 0.85-1.1 GaAs
Ga1−yAlyAs1−xSbx 0.9-1.1 GaAs
Ga1−yAlyAs1−xSbx 1.0-1.7 GaSb
In1−xGaxAsyP1−y 0.92-1.7 InP
InxGa1−xAs 1.3 InGaAs
In1−xGaNyAs1−y 1.3-1.55 GaAs
In1−xGaxN1−yAsySb 1.31 GaAs
Table 1.1: Some common material systems used in the fabrication of electro-
luminescent sources
The GaAs/AlGaAs Double Heterojuntion system is the best developed and is
used for fabricating LEDs for the shorter wavelength region. There is very little
7

1.1. INTRODUCTION
lattice mismatch (0.017layer and the GaAs substrate which gives good internal
quantum efficiency.
1.1.4 Why Direct Bandgap and Why Not Indirect Bandgap Semi-
conductors !?
To get proper electroluminescence, it is must to get an appropriate semiconductor
material. The best device for this purpose is the direct bandgap semiconductor in
which electrons and holes on either side of the forbidden energy gap have the same
value of crystal momentum and thus direct recombination can be obtained. It may
be observed in the figure 1.4 the energy maximum of the valence band occurs at
the same (or very nearly the same) value of electron crystal momentum as the en-
ergy minimum of the conduction band. Hence when electronhole recombination
occurs the momentum of the electron remains constant and the energy released,
which corresponds to the bandgap energy Eg, may be emitted as light. This direct
transition of an electron across the energy gap provides an efficient mechanism
for photon emission and the average time that the minority carrier remains in a
free state before recombination is short (108 to 10−10 s).
Semiconductor Material Energy Bandgap (eV)
GaAs Direct: 1.43
CaSb Direct: 0.73
InAs Direct: 0.35
InSb Direct: 0.18
Si Indirect: 1.12
Ge Indirect: 0.67
GaP Indirect: 2.26
Table 1.2 Some direct and indirect bandgap semiconductors with calculated
recombination coefficients
In indirect bandgap semiconductors, however, the maximum and minimum
energies occur at different value of crystal momentum. For electronhole recombi-
nation to take place it is necessary that the electron loses momentum such that it
has a value of momentum corresponding to the maximum energy of the valence
band. The conservation of momentum requires the emission or absorption of a
third particle, a phonon. The Figure 6 illustrates the carrier recombination giving
spontaneous emission of light in a pn junction diode.
This three-particle recombination process is far less probable than the two-
particle process exhibited by direct bandgap semiconductors. Hence, the recombi-
nation in indirect bandgap semiconductors is relatively slow (10−2 to 104 s). This
is reflected by a much longer minority carrier lifetime, together with a greater
probability of nonradiative transitions. The competing nonradiative recombina-
8

1.1. INTRODUCTION
Figure 1.6: Energymomentum diagrams showing the types of transition: (a) direct
bandgap semiconductor; (b) indirect bandgap semiconductor
tion processes which involve lattice defects and impurities become more likely as
they allow carrier recombination in a relatively short time in most materials.
Thus the indirect bandgap emitters such as silicon and germanium shown in
Table 1.2 give insignificant levels of electroluminescence. This disparity is further
illustrated in Table 1.2 by the values of the recombination coefficient Br given for
both the direct and indirect bandgap recombination semiconductors shown.
The recombination coefficient is obtained from the measured absorption coef-
ficient of the semiconductor, and for low injected minority carrier density relative
to the majority carriers it is related approximately to the radiative minority carrier
lifetime* τ r by τr = [Br(N + P)]−1 where N and P are the respective majority
carrier concentrations in the n-type and p-type regions. The significant differ-
ence between the recombination coefficients for the direct and indirect bandgap
semiconductors shown underlines the importance of the use of direct bandgap ma-
terials for electroluminescent sources. Direct bandgap semiconductor devices in
general have a much higher internal quantum efficiency. This is the ratio of the
number of radiative recombinations (photons produced within the structure) to the
number of injected carriers which is often expressed as a percentage.
9

1.1. INTRODUCTION
1.1.5 Direct Band Gap Light Emitting Diode
Spontaneous Emission
The interaction of light with matter takes place in discrete packets of energy or
quanta, called photons. Furthermore, the quantum theory suggests that atoms
exist only in certain discrete energy states such that absorption and emission of
light causes them to make a transition from one discrete energy state to another.
The frequency of the absorbed or emitted radiation f is related to the difference in
energy E between the higher energy state E2 and the lower energy state E1 by the
expression:
E = E2 - E1 = hf
where h = 6.626 * 10−34 J s is Plancks constant. These discrete energy states
for the atom may be considered to correspond to electrons occurring in particular
energy levels relative to the nucleus.
Figure 1.7: Energy state diagram showing: (a) absorption; (b) spontaneous emis-
sion; The black dot indicates the state of the atom before and after a transition
takes place
Hence, different energy states for the atom correspond to different electron
10

1.1. INTRODUCTION
configurations, and a single electron transition between two energy levels within
the atom will provide a change in energy suitable for the absorption or emission
of a photon. It must be noted, however, that modern quantum theory gives a
probabilistic description which specifies the energy levels in which electrons are
most likely to be found. Nevertheless, the concept of stable atomic energy states
and electron transitions between energy levels is still valid.
The given figure 7 represents a two energy state or level atomic system where
an atom is initially in the lower energy state E1. Alternatively, when the atom is
initially in the higher energy state E2 it can make a transition to the lower energy
state E1 providing the emission of a photon at a frequency corresponding to Equa-
tion given above. This emission process can occur by spontaneous emission in
which the atom returns to the lower energy state in an entirely random manner.
The random nature of the spontaneous emission process where light is emitted
by electronic transitions from a large number of atoms gives incoherent radiation.
A similar emission process in semiconductors provides the basic mechanism for
light generation within the LED
How exactly An LED Works !!
Figure 1.8: The pn junction with forward bias giving spontaneous emission of
photons
The increased concentration of minority carriers in the opposite type region
11

1.1. INTRODUCTION
in the forward-biased pn diode leads to the recombination of carriers across the
bandgap. This process is shown in Figure 8 for a direct bandgap semiconductor
material where the normally empty electron states in the conduction band of the p-
type material and the normally empty hole states in the valence band of the n-type
material are populated by injected carriers which recombine across the bandgap.
The energy released by this electronhole recombination is approximately equal to
the bandgap energy Eg
Figure 1.9: An illustration of carrier recombination giving spontaneous emission
of light in a pn junction diode
Excess carrier population is therefore decreased by recombination which may
be radiative or nonradiative. In nonradiative recombination the energy released is
dissipated in the form of lattice vibrations and thus heat. However, in band-to-
band radiative recombination the energy is released with the creation of a pho-
ton with a frequency following where the energy is approximately equal to the
bandgap energy Eg and therefore:
Eg = hf = hc
λ
where c is the velocity of light in a vacuum and is the optical wavelength.
Substituting the appropriate values for h and c in Equation and rearranging gives:
λ = 1.24
Eg
where is written in m and Eg in eV.
12

1.1. INTRODUCTION
1.1.6 LED Characteristics
Optical output power:
The graph of an ideal light output power vs current characteristics of an LED is
given below:
Figure 1.10: An ideal light output against current characteristic for an LED
It is linear corresponding to the linear part of the injection laser optical power
output characteristic before lasing occurs. Intrinsically the LED is a very linear
device in comparison with the majority of injection lasers and hence it tends to
be more suitable for analog transmission where severe constraints are put on the
linearity. LEDs do exhibit significant nonlinearities which depend upon the con-
figuration utilized. It is therefore often necessary to use some form of linearizing
circuit technique in order to ensure the linear performance of the device to allow
its use in high-quality analog transmission systems.
With an increase in the temperature, the internal quantum efficiency of an
LED decreases exponentially. Hence as the p-n junction increases the light emit-
ted from these devices decreases. resonant cavity LEDs have shown a similar
reduction in output power when operated at higher temperatures. When operating
at room temperature, however, RC-LEDs can provide high levels of optical output
power.
Output spectrum:
The spectral linewidth of an LED operating at room temperature in the 0.8 to
0.9 m wavelength band is usually between 25 and 40 nm at the half maximum
13

1.1. INTRODUCTION
intensity points.
Figure 1.11: Output spectrum for an AlGaAs with doped active region.
The output spectra also tend to broaden at a rate of between 0.1 and 0.3 nm
with increase in temperature due to the greater energy spread in carrier distribu-
tions at higher temperatures. Increases in temperature of the junction affect the
peak emission wavelength as well, and it is shifted by +0.3 to 0.4 nm for AlGaAs
devices
Modulation bandwidth:
The modulation bandwidth of LEDs is depended on- These are:(a) the doping
level in the active layer; (b) the reduction in radiative lifetime due to the injected
carriers; (c) the parasitic capacitance of the device.
The carrier lifetime is dependent on the doping concentration, the number of
injected carriers into the active region, the surface recombination velocity and
the thickness of the active layer. All these parameters tend to be interdependent
and are adjustable within limits in present-day technology. In general, the car-
rier lifetime may be shortened by either increasing the active layer doping or by
decreasing the thickness of the active layer.
LEDs have a very thin, virtually undoped active layer and the carrier lifetime is
controlled only by the injected carrier density. At high current densities the carrier
lifetime decreases with injection level because of a bimolecular recombination
14

1.1. INTRODUCTION
process This bimolecular recombination process allows edge-emitting LEDs with
narrow recombination regions to have short recombination times, and therefore
relatively high modulation capabilities at reasonable operating current densities
Reliability:
LEDs are not generally affected by the catastrophic degradation mechanisms that
is their life is not shortened by its usage. Rapid degradation in LED is due to
both the growth of dislocations and precipitate-type defects in the active region
giving rise to dark line defects (DLDs) and dark spot defects (DSDs), respectively,
under device aging. DLDs tend to be the dominant cause of rapid degradation in
GaAs-based LEDs. The growth of these defects does not depend upon substrate
orientation but on the injection current density, the temperature and the impurity
concentration in the active region.
Good GaAs substrates have dislocation densities around 5 104 cm2 . Hence,
there is less probability of dislocations in devices with small active regions.
It is clear, that with the long-term LED degradation process there is no absolute
end-of-life power level
1.1.7 Drawbacks of Homojunction LED
:
The homojunction LED has two main drawbacks- The p-region in the LED
must be narrow so as to allow the emission of photons without getting reabsorbed.
As the electrons in the valence band can absorb the emitted photon to gain the en-
ergy and jump into conduction band. To avoid the reabsorption the p-region must
be narrow. But when the p-region is narrowed, some of the injected electrons in
the p-side reach the surface by diffusion and re-combine through crystal defects
of the surface. This gives radiationless recombination of electron and hole pair
which reduces the light output. This is the major disadvantage of homojunction
LED. The another one being that if the recombination of e-h pairs takes place
over a large volume then chances of reabsorption of photons becomes higher
wherein the amount of reabsorption of photons increases with material volume.
Heterostrucutre
1.1.8 Heterojunction Structure
A junction between two differently doped semiconductors that are of the same
material is known as a homojunction. A junction between two bandgap semicon-
ductors is called as a heterojunction. A semiconductor device structure that has
junctions between two different bandgap materials is known as a heterostructure
15

1.1. INTRODUCTION
device (HD). The refractive index of a semiconductor depends on its bandgap.
A wider bandgap semiconductor has lower refractive index. This means by con-
structing LEDs from heterostructures, we engineer a dielectric waveguide within
the device and hence photons out from the recombination region. LED con-
structions for increasing the intensity of the output light make use of the double
heterostructure. Figure 12 shows a double heterostructure device based on two
junctions between different semiconductor materials with different bandgaps. In
this case, semiconductors are AlGaAs with Eg=2eV and GaAs with Eg=1.4eV.
The double heterostructure has an n+p heterojunction between n+ AlGaAs and
p-GaAs. The p-GaAs is a thin layer, typically of the fraction of a micron and it is
lightly doped.
Figure 1.12: Double heterostructure LED
The simplified energy band diagram for the whole device in the absence of
an applied voltage is shown below. The fermi level Ef is continuous through
the whole structure. There is a potentil energy barrier eVo for electrons in the
CB of n+-AlGaAs against diffusion into p-GaAs. There is a band gap change at
the junction between p-GaAs and p-AlGaAs that results in a step change, δEc in
Ec, between the two bands of p-GaAs and p-AlGaAs. This, δEc is effectively a
potential energy barrier that prevents any electrons in the CB in p-GaAs passing
to the CB of p-AlGaAs.
When a forward bias is applied, majority of this voltage drops between the
n+-AlGaAs and p-GaAs and reduces the potential energy barrier eVo, just a in the
normal p-n junction diode. This allows electrons in the CB of n+- AlGaAs to
be injected into p-GaAs. This electrons however are confined to the conduction
band of p-GaAs since there is a barrier of δEc between p-GaAs and p-AlGaAs.
The wide band-gap AlGaAs layers therefore act as confining layers that restrict
injected electrons to the p-AlGaAs layer results in spontaneous photon emission.
Since the bandgap Eg of AlGaAs is greater than GaAs, the emitted photons do not
16

1.1. INTRODUCTION
Figure 1.13: The quantum well structure formed by different energy bandgaps
get reabsorbed as they escape the active region and can reach the surface of the
device. Since light is also not absorbed in p-AlGaAs it can be reﬂected to increase
the light output.
Therefore a quantum well structure is found to be present in DH-LED struc-
ture. The study of this structure is done with the help of Quantum-Mechanics
which has been explained in the subsequent chapters.
17

Chapter 2
The Schrodinger Equation
2.1 QUANTUM MECHANICAL STUDY:-
In our everyday life we come across a number of instances where we do not ob-
serve many things which actually might occur behind the scene. For an example,
we drop a glass and it will smash on the ﬂoor. Walk to a wall and we cannot
walk through it. These are various simple physics related examples going around
us, but, do we actually ponder why these things are happening? There are very
basic laws of physics going on all around us that we instinctively grasp: gravity
makes things fall to the ground, pushing something makes it move, two things
can’t occupy the same place at the same time.
At the turn of the century, scientists thought that all the basic rules like this
should apply to everything in nature – but then they began to study the world of
the ultra-small. Atoms, electrons, light waves, none of these things followed the
normal rules. As physicists like Niels Bohr and Albert Einstein began to study
particles, they discovered new physics laws that were downright quirky. These
were the laws of quantum mechanics, and they got their name from the work of
Max Planck. QUANTA- CONCEPT!
The idea that particles could only contain lumps of energy in certain sizes
moved into various areas of physics. Over the next decade, Niels Bohr pulled
it into his description of how an atom worked. He said that electrons traveling
around a nucleus couldn’t have arbitrarily small or arbitrarily large amounts of
energy; they could only have multiples of a standard ”quantum” of energy.
Eventually scientists realized this explained why some materials are conduc-
tors of electricity and some aren’t – since atoms with differing energy electron
orbits conduct electricity differently. This understanding was crucial to building
a transistor, since the crystal at its core is made by mixing materials with vary-
ing amounts of conductivity. Interestingly, the fact that light was thought of as
18

2.1. QUANTUM MECHANICAL STUDY:-
being constituted of quanta of energy didnt mean that it could not be thought of
to be continuous wave. Infact, in most cases light works as a wave and exhibits
wave properties. This wave nature produces some interesting effects. For exam-
ple, if an electron traveling around a nucleus behaves like a wave, then its position
at any one time becomes fuzzy. Instead of being in a concrete point, the elec-
tron is smeared out in space. This smearing means that electrons don’t always
travel quite the way one would expect. Unlike water flowing along in one direc-
tion through a hose, electrons traveling along as electrical current can sometimes
follow weird paths, especially if they’re moving near the surface of a material.
Moreover, electrons acting like a wave can sometimes burrow right through a bar-
rier. Understanding this odd behavior of electrons was necessary as scientists tried
to control how current flowed through the first transistors.
2.1.1 So which is it - a particle or a wave?
Scientists interpret quantum mechanics to mean that a tiny piece of material like
a photon or electron is both a particle and a wave. It can be either, depending on
how one looks at it or what kind of an experiment one is doing. In fact, it might
be more accurate to say that photons and electrons are neither a particle or a wave
– they’re undefined up until the very moment someone looks at them or performs
an experiment, thus forcing them to be either a particle or a wave.
This comes with other side effects: namely that a number of qualities for par-
ticles aren’t well-defined. For example, there is a theory by Werner Heisenberg
called the Uncertainty Principle. It states that if a researcher wants to measure the
speed and position of a particle, he can’t do both very accurately. If he measures
the speed carefully, then he can’t measure the position nearly as well. This doesn’t
just mean he doesn’t have good enough measurement tools – it’s more fundamen-
tal than that. If the speed is well-established then there simply does not exist a
well-established position (the electron is smeared out like a wave) and vice versa.
Albert Einstein disliked this idea. When confronted with the notion that the
laws of physics left room for such vagueness he announced: ”God does not play
dice with the universe.” Nevertheless, most physicists today accept the laws of
quantum mechanics as an accurate description of the subatomic world. And cer-
tainly it was a thorough understanding of these new laws which helped Bardeen,
Brattain, and Shockley invent the transistor.
2.1.2 WHAT IS QUANTUM-MECHANICS?
It is the science of materials on the nano-level. It deals with the wave particle
duality of the matter. Quantum mechanics is the body of scientific principles that
19

explains the behavior of matter and its interactions with energy on the scale of
atoms and subatomic particles.
There are many phenomenon which the classical physics fails to explain. The
quantum mechanics comes into picture when dealing with the matter on micro and
nano level. It provides a mathematical picture of much of the dual particle-like
and wave-like behavior and interaction of energy and matter.
Quantum mechanics provides a substantially useful framework for many fea-
tures of the modern periodic table of elements including the behavior of atoms dur-
ing chemical bonding and has played a significant role in the development of many
modern technologies. In the context of quantum mechanics, the waveparticle du-
ality of energy and matter and the uncertainty principle provide a unified view
of the behavior of photons, electrons, and other atomic-scale objects. Other dis-
ciplines including quantum chemistry, quantum electronics, quantum optics, and
quantum information science. Much 19th-century physics has been re-evaluated
as the ”classical limit” of quantum mechanics and its more advanced develop-
ments in terms of quantum field theory, string theory, and speculative quantum
gravity theories. The name quantum mechanics derives from the observation that
some physical quantities can change only in discrete amounts (Latin quanta), and
not in a continuous (analog) way.
2.1.3 WHY IS QUANTUM -MECHANICS IMPORTANT?
Quantum mechanics is essential to understanding the behavior of systems at atomic
length scales and smaller. If the physical nature of an atom was solely described
by classical mechanics electrons would not ”orbit” the nucleus since orbiting elec-
trons emit radiation (due to circular motion) and would eventually collide with
the nucleus due to this loss of energy. This framework was unable to explain the
stability of atoms. Instead, electrons remain in an uncertain, non-deterministic,
”smeared”, probabilistic, waveparticle orbital about the nucleus, defying the tra-
ditional assumptions of classical mechanics and electromagnetism.
In short, the quantum-mechanical atomic model has succeeded spectacularly
in the realm where classical mechanics and electromagnetism falter.
Broadly speaking, quantum mechanics incorporates four classes of phenom-
ena for which classical physics cannot account:
Quantization of certain Physical Properties
WaveParticle Duality
Principle of Uncertainty
Quantum Entanglement
20

2.1.4 WAVE FUNCTION
An isolated systems information can be well defined using the wave function
which in quantum mechanics describes the quantum state of a system of one or
more particles. Quantities associated with measurements, such as the average
momentum of a particle, are derived from the wave function by mathematical op-
erations that describe its interaction with observational devices. Thus it is of a
great importance in quantum mechanics. The most common symbols for a wave
function are the Greek letters ψ (lower-case and capital psi). The Schrdinger
equation determines how the wave function evolves over time, that is, the wave
function is the solution of the Schrdinger equation. The wave function behaves
qualitatively like other waves, such as water waves or waves on a string, because
the Schrdinger equation is mathematically a type of wave equation. This explains
the name ”wave function”, and gives rise to waveparticle duality. The wave of
the wave function, however, is not a wave in physical space; it is a wave in an
abstract mathematical ”space”, and in this respect it differs fundamentally from
water waves or waves on a string.
However, the wave function ψ alone does not give much of information.
Therefore, the squared function |ψ|2 comes into picture. It represents the proba-
bility density of measuring a particle as being at a given place at a given time.
2.1.5 SCHRODINGER EQUATION
In quantum mechanics a Schrodinger equation which is basically a second-order
differential equation in terms of a wave function Y which represents a state of
a bound electron in the molecular world. It is a function of space coordinates
and time. The wave function y is a mathematical function representing a state
of electron in an atomic world and does not possess any physical significance.
However,it has a definite physical significance. It is a probability density of an
electron in the atomic world. In such situations electron is regarded as a particle
confined in potential well which is ideally a infinite well with width of the order
of nanometer. However in most of the practical situations it is a finite potential
well. The Schrodinger equation for a particle like electron located at any general
point P(r) is given by
The solution of the Schrodinger equation gives a state function y(r,t) and dis-
crete energy levels. For this one has to invoke a mathematical model of an electron
confined in potential well. The corresponding boundary conditions may be writ-
ten down for a one dimensional potential well normally represented by a potential
energy function U(x)
The time-independent Schrodinger equation for this one-dimensional potential
21

ı¯h
∂
∂t
ψ(r,t) = ˆH ψ(r,t)
The Schrdinger equation is the fundamental equation of physics for describing
quantum mechanical behavior. It is also often called the Schrdinger wave equa-
tion, and is a partial differential equation that describes how the wave function of
a physical system evolves over time.
Time-dependent equation The form of the Schrdinger equation depends on the
physical situation. The most general form is the time-dependent Schrdinger equa-
tion, which gives a description of a system evolving with time Time-dependent
Schrdinger equation (general)
ı¯h
∂
∂t
ψ = ˆH ψ
where ı is the imaginary unit, ¯h is the Planck constant divided by 2, the symbol
∂
∂t indicates a partial derivative with respect to time t, ψ (the Greek letter Psi) is
the function of the quantum system, and ˆH is the Hamiltonian operator (which
characterizes the total energy of any given wave function and takes different forms
depending on the situation).
A wave function that satisfies the non-relativistic Schrdinger equation with
V = 0. In other words, this corresponds to a particle traveling freely through
empty space. The real part of the wave function is plotted here. The most famous
example is the non-relativistic Schrdinger equation for a single particle moving in
an electric field (but not a magnetic field; see the Pauli equation): Time-dependent
Schrdinger equation (single non-relativistic particle)
ı¯h
∂
∂t
ψ(r,t) = [−
[¯h2
]
2µ
∇2
+V(r,t)]ψ(r,t)
where µ is the particle’s ”reduced mass”, V is its potential energy, ∇2 is the
Laplacian, and ψ is the wave function (more precisely, in this context, it is called
the ”position-space wave function”). In plain language, it means ”total energy
equals kinetic energy plus potential energy”, but the terms take unfamiliar forms
for reasons explained below.
Given the particular differential operators involved, this is a linear partial dif-
ferential equation. It is also a diffusion equation, but unlike theheat equation, this
one is also a wave equation given the imaginary unit present in the transient term.
The term ”Schrdinger equation” can refer to both the general equation, or the
specific nonrelativistic version. The general equation is indeed quite general, used
throughout quantum mechanics, for everything from the Dirac equation to quan-
tum field theory, by plugging in various complicated expressions for the Hamilto-
nian. The specific nonrelativistic version is a simplified approximation to reality,
22

which is quite accurate in many situations, but very inaccurate in others (see rela-
tivistic quantum mechanics and relativistic quantum field theory).
To apply the Schrdinger equation, the Hamiltonian operator is set up for the
system, accounting for the kinetic and potential energy of the particles constitut-
ing the system, then inserted into the Schrdinger equation. The resulting partial
differential equation is solved for the wave function, which contains information
about the system. Time-independent equation
Eψ = ˆH ψ
The time-independent Schrdinger equation predicts that wave functions can
form standing waves, called stationary states(also called ”orbitals”, as in atomic
orbitals or molecular orbitals). These states are important in their own right, and
if the stationary states are classified and understood, then it becomes easier to
solve the time-dependent Schrdinger equation forany state. The time-independent
Schrdinger equation is the equation describing stationary states. (It is only used
when theHamiltonian itself is not dependent on time. In general, the wave func-
tion still has a time dependency.) Time-independent Schrdinger equation (general)
In words, the equation states: When the Hamiltonian operator acts on a certain
wave function ψ, and the result is proportional to the same wave function ψ, then
ψ is a stationary state, and the proportionality constant, E, is the energy of the
state ψ. The time-independent Schrdinger equation is discussed further below. In
linear algebra terminology, this equation is aneigenvalue equation. As before, the
most famous manifestation is the non-relativistic Schrdinger equation for a single
particle moving in an electric field (but not a magnetic field): Time-independent
Schrdinger equation (single non-relativistic particle)
ı¯h
∂
∂t
ψ(r) = [−
[¯h2
]
2µ
∇2
+V(r)]ψ(r)
with definitions as above.
23

Chapter 3
Mathematica
3.1 Introduction to Mathematica
Mathematica software can be used for solving Schrodinger equation with the fol-
lowing types of potential: infinite double rectangular well and double rectangular
well. The package outputs are the energy eigen values and plots of their corre-
sponding eigen functions. The single square limit is beautifully reproduced in
each case by the software and quantization of energy is demonstrated for large
barrier limit for the double well cases.
3.1.1 What is Mathematica
Mathematica is a computational software program used in many scientific, engi-
neering, mathematical and computing fields, based on symbolic mathematics. It
was conceived by Stephen Wolfram and is developed by Wolfram Research of
Champaign, Illinois. The Wolfram Language is the programming language used
in Mathematica .
3.1.2 Features of Mathematica
Elementary and special mathematical function library.
Matrix and data manipulation tools including support for spare arrays
Support for complex number, arbitrary precision, interval arithmetic and sym-
bolic computation.
2D and 3D data, function and geo visualization and animation tools
24

3.1. INTRODUCTION TO MATHEMATICA
3.1.3 Our purpose of using Mathematica
As we have mentioned earlier Mathematica is a very powerful calculations soft-
ware which is capable of handling complex differential and intergral equations.
In our project we have solved Schrodingers differential equation and also we have
plotted the graphs using this software.
For example, If we want to solve an equation over the postitve integers
x2
+2y3
= 3681wherex > 0andy > 0 (3.1)
Solve
[x∧
2+2y∧
3==3681&&x > 0&&y > 0,{x,y} (3.2)
The output which we obtain is
{x → 15,y → 12},{x → 41,y → 10},{x → 57,y → 6}Integers (3.3)
Also Mathematica can solve integral of a given equation using proper commands-
Example- If we want the answer to an improper integral -
∞
0
e−xdx
(3.4)
In Mathematica, the command which is used to solve integration is -
Integrate
[E(
−x2
),x,0,] (3.5)
The output obtained in mathematica is
√
π
2
k = 8π2mE
h2k = 8π2mE
h2k = 8π2mE
h2
25

Figure 3.1: Particle in a ﬁnite potential well
2
√
2e m
h2 π
eq : Ψ”[x]+k∧2Ψ[x] == 0eq : Ψ”[x]+k∧2Ψ[x] == 0eq : Ψ”[x]+k∧2Ψ[x] == 0
eq : 8emπ2Ψ[x]
h2 +Ψ [x] == 0
DSolve[Ψ”[x]+k∧2Ψ[x] == 0,Ψ[x],x]DSolve[Ψ”[x]+k∧2Ψ[x] == 0,Ψ[x],x]DSolve[Ψ”[x]+k∧2Ψ[x] == 0,Ψ[x],x]
Ψ[x] → C[1]Cos 2
√
2e
√
mπx
h +C[2]Sin 2
√
2e
√
mπx
h
sol1 = DSolve[{Ψ”[x]+k∧2Ψ[x] == 0,Ψ[0] == 0},Ψ[x],x]sol1 = DSolve[{Ψ”[x]+k∧2Ψ[x] == 0,Ψ[0] == 0},Ψ[x],x]sol1 = DSolve[{Ψ”[x]+k∧2Ψ[x] == 0,Ψ[0] == 0},Ψ[x],x]
Ψ[x] → C[2]Sin 2
√
2e
√
mπx
h
sol2 = sol1[[1,1]]sol2 = sol1[[1,1]]sol2 = sol1[[1,1]]
Ψ[x] → C[2]Sin 2
√
2e
√
mπx
h
sol3 = Ψ[x]==C[2]Sin 2
√
2e
√
mπx
hsol3 = Ψ[x]==C[2]Sin 2
√
2e
√
mπx
hsol3 = Ψ[x]==C[2]Sin 2
√
2e
√
mπx
h
Ψ[x] == C[2]Sin 2
√
2e
√
mπx
h
Ψ[x] == C[2]Sin 2
√
2e
√
mπx
hΨ[x] == C[2]Sin 2
√
2e
√
mπx
hΨ[x] == C[2]Sin 2
√
2e
√
mπx
h
26

Ψ[x] == C[2]Sin 2
√
2e
√
mπx
h
Whenweapplythesecondboundaryconditionwherewhenx = L(whichisthewidthofthepotentialwell),Ψ[x]Whenweapplythesecondboundaryconditionwherewhenx = L(whichisthewidthofthepotentialwell),Ψ[x]Whenweapplythesecondboundaryconditionwherewhenx = L(whichisthewidthofthepotentialwell),Ψ[x]
so,inorderthatΨ[x]isequaltozerothesinetermintheaboveequationshouldbezeroasc[2]cannotbezero.therefoso,inorderthatΨ[x]isequaltozerothesinetermintheaboveequationshouldbezeroasc[2]cannotbezero.therefoso,inorderthatΨ[x]isequaltozerothesinetermintheaboveequationshouldbezeroasc[2]cannotbezero.therefo
thereforetheterm2
√
2e
√
mπx
h shouldbeanintegralmultipleofπ.kL = nπthereforetheterm2
√
2e
√
mπx
h shouldbeanintegralmultipleofπ.kL = nπthereforetheterm2
√
2e
√
mπx
h shouldbeanintegralmultipleofπ.kL = nπ
asaresultenergyE = h2n2
8mL2asaresultenergyE = h2n2
8mL2asaresultenergyE = h2n2
8mL2
where,h– → planck sconstantwhere,h– → planck sconstantwhere,h– → planck sconstant
m– → massofelectronm– → massofelectronm– → massofelectron
L– → lengthofpotentialwellL– → lengthofpotentialwellL– → lengthofpotentialwell
n– → quantumstaten– → quantumstaten– → quantumstate
nowassumingstandardvaluesnowassumingstandardvaluesnowassumingstandardvalues
h = 6.626∗10−34h = 6.626∗10−34
h = 6.626∗10−34
m = 9.1∗10−31m = 9.1∗10−31
m = 9.1∗10−31
letlengthofpotentialwellL = 5∗10−9mletlengthofpotentialwellL = 5∗10−9mletlengthofpotentialwellL = 5∗10−9m
E[n] = 0.0150n2E[n] = 0.0150n2
E[n] = 0.0150n2
4.55`*∧-39
0.015n2
Plot 0.015b2,{b,0,4}Plot 0.015b2,{b,0,4}Plot 0.015b2,{b,0,4}
Plot “0.3737”
L2 , L,0, 1
20000000Plot “0.3737”
L2 , L,0, 1
20000000Plot “0.3737”
L2 , L,0, 1
20000000
NowfindingvalueofC[2]NowfindingvalueofC[2]NowfindingvalueofC[2]
considernormalizationconditionforthewavefunction Ψ[x]considernormalizationconditionforthewavefunction Ψ[x]considernormalizationconditionforthewavefunction Ψ[x]
27

Integrate c2(Sin[k ∗x])∧2,{x,0,L}Integrate c2(Sin[k ∗x])∧2,{x,0,L}Integrate c2(Sin[k ∗x])∧2,{x,0,L}
−c2(−2kL+Sin[2kL])
4k
sinceSin[2kL] = 0 (provedinitiallywithsecondboundarycondition)sinceSin[2kL] = 0 (provedinitiallywithsecondboundarycondition)sinceSin[2kL] = 0 (provedinitiallywithsecondboundarycondition)
thustheoutput = c2 L
2thustheoutput = c2 L
2thustheoutput = c2 L
2
sincetheintegrationwasnormalizedandhenceoutput = 1sincetheintegrationwasnormalizedandhenceoutput = 1sincetheintegrationwasnormalizedandhenceoutput = 1
therefore,therefore,therefore,
c2L
2 = 1c2L
2 = 1c2L
2 = 1
C[2] = c = 2
LC[2] = c = 2
LC[2] = c = 2
L
soourfinalwavefunctionsoourfinalwavefunctionsoourfinalwavefunction
28

Ψ[x] = 2
LSin 2
√
2e
√
mπx
hΨ[x] = 2
LSin 2
√
2e
√
mπx
hΨ[x] = 2
LSin 2
√
2e
√
mπx
h
ﬁnaloursowavefunction
√
2 1
LSin[2.10917×1019x]
NowplottingthewaveequationΨ[x]andplotofcorrespondingprobablitydistributionfunctiony2[x]forapotentiNowplottingthewaveequationΨ[x]andplotofcorrespondingprobablitydistributionfunctiony2[x]forapotentiNowplottingthewaveequationΨ[x]andplotofcorrespondingprobablitydistributionfunctiony2[x]forapotenti
for n = 1for n = 1for n = 1
Plot[0.632Sin[π ∗x/5],{x,0,5}]Plot[0.632Sin[π ∗x/5],{x,0,5}]Plot[0.632Sin[π ∗x/5],{x,0,5}]
Plot (0.632Sin[π ∗x/5])2,{x,0,5}Plot (0.632Sin[π ∗x/5])2,{x,0,5}Plot (0.632Sin[π ∗x/5])2,{x,0,5}
for n = 2for n = 2for n = 2
Plot[0.632Sin[2π ∗x/5],{x,0,5}]Plot[0.632Sin[2π ∗x/5],{x,0,5}]Plot[0.632Sin[2π ∗x/5],{x,0,5}]
29

Plot (0.632Sin[2π ∗x/5])2,{x,0,5}Plot (0.632Sin[2π ∗x/5])2,{x,0,5}Plot (0.632Sin[2π ∗x/5])2,{x,0,5}
forn = 3forn = 3forn = 3
Plot[0.632Sin[3π ∗x/5],{x,0,5}]Plot[0.632Sin[3π ∗x/5],{x,0,5}]Plot[0.632Sin[3π ∗x/5],{x,0,5}]
Plot (0.632Sin[3π ∗x/5])2,{x,0,5}Plot (0.632Sin[3π ∗x/5])2,{x,0,5}Plot (0.632Sin[3π ∗x/5])2,{x,0,5}
1DFinite Potential Well1DFinite Potential Well1DFinite Potential Well
ConsiderpotentialwellsuchthatpotentialenergyConsiderpotentialwellsuchthatpotentialenergyConsiderpotentialwellsuchthatpotentialenergy
U[x] = U ,x ≤ 0U[x] = U ,x ≤ 0U[x] = U ,x ≤ 0
0 ,0 < x < L0 ,0 < x < L0 ,0 < x < L
U ,x ≥ LU ,x ≥ LU ,x ≥ L
30

Intherange0 < x < L ,kineticenergyE ≤ UIntherange0 < x < L ,kineticenergyE ≤ UIntherange0 < x < L ,kineticenergyE ≤ U
Therfore,dividetheregioninthreeregionsTherfore,dividetheregioninthreeregionsTherfore,dividetheregioninthreeregions
region1,forx ≤ 0 whereU = Uregion1,forx ≤ 0 whereU = Uregion1,forx ≤ 0 whereU = U
anditsschrodingerequationisanditsschrodingerequationisanditsschrodingerequationis
Ψ”[x]−l2Ψ[x] = 0Ψ”[x]−l2Ψ[x] = 0Ψ”[x]−l2Ψ[x] = 0
region2,whereU = 0region2,whereU = 0region2,whereU = 0
schrodingerequationforcorrespondingregionschrodingerequationforcorrespondingregionschrodingerequationforcorrespondingregion
Ψ”[x]+k2Ψ[x] = 0Ψ”[x]+k2Ψ[x] = 0Ψ”[x]+k2Ψ[x] = 0
region3,forx ≥ L whereU = Uregion3,forx ≥ L whereU = Uregion3,forx ≥ L whereU = U
Ψ”[x]−l2Ψ[x] = 0Ψ”[x]−l2Ψ[x] = 0Ψ”[x]−l2Ψ[x] = 0
k = 8π2mE
h2k = 8π2mE
h2k = 8π2mE
h2
31

l = 8π2m(U−E)
h2l = 8π2m(U−E)
h2l = 8π2m(U−E)
h2
FromtheprevioustwosolutionsweknowtheequationsofwavefunctionofinsidepotentialwellwherepotentialeFromtheprevioustwosolutionsweknowtheequationsofwavefunctionofinsidepotentialwellwherepotentialeFromtheprevioustwosolutionsweknowtheequationsofwavefunctionofinsidepotentialwellwherepotentiale
Forregion1,Forregion1,Forregion1,
Ψ1[x] = A∗elx ,x ≤ 0Ψ1[x] = A∗elx ,x ≤ 0Ψ1[x] = A∗elx ,x ≤ 0
Ψ2[x] = B∗Sin[kx+θ] ,0 < x < LΨ2[x] = B∗Sin[kx+θ] ,0 < x < LΨ2[x] = B∗Sin[kx+θ] ,0 < x < L
Ψ3[x] = C ∗e−lx ,x ≥ LΨ3[x] = C ∗e−lx ,x ≥ LΨ3[x] = C ∗e−lx ,x ≥ L
InordertofindthevaluesofA,B,andC.Letusconsidercontinuityequationatx = 0andx = L.InordertofindthevaluesofA,B,andC.Letusconsidercontinuityequationatx = 0andx = L.InordertofindthevaluesofA,B,andC.Letusconsidercontinuityequationatx = 0andx = L.
At x = 0,At x = 0,At x = 0,
Ψ1[0] = AΨ1[0] = AΨ1[0] = A
Ψ2[0] = B∗Sin[θ]Ψ2[0] = B∗Sin[θ]Ψ2[0] = B∗Sin[θ]
32

and Ψ1[0] = Ψ2[0]and Ψ1[0] = Ψ2[0]and Ψ1[0] = Ψ2[0]
A = B∗Sin[θ]A = B∗Sin[θ]A = B∗Sin[θ]
BSin[θ]
Atx = L,Atx = L,Atx = L,
Ψ2[L] = B∗Sin[lL+θ]Ψ2[L] = B∗Sin[lL+θ]Ψ2[L] = B∗Sin[lL+θ]
Ψ3[L] = C ∗e−lLΨ3[L] = C ∗e−lL
Ψ3[L] = C ∗e−lL
and Ψ2[L] = Ψ3[L]and Ψ2[L] = Ψ3[L]and Ψ2[L] = Ψ3[L]
B∗Sin[lL+θ] = C ∗e−lLB∗Sin[lL+θ] = C ∗e−lL
B∗Sin[lL+θ] = C ∗e−lL
33

C = BSin[lL+θ]∗elLC = BSin[lL+θ]∗elL
C = BSin[lL+θ]∗elL
A = h∗k
2π
√
2∗m∗U
∗BA = h∗k
2π
√
2∗m∗U
∗BA = h∗k
2π
√
2∗m∗U
∗B
C = e−kl ∗ h∗k
2π
√
2∗m∗U
∗BC = e−kl ∗ h∗k
2π
√
2∗m∗U
∗BC = e−kl ∗ h∗k
2π
√
2∗m∗U
∗B
ConsiderpotentialwellsuchthatpotentialenergyConsiderpotentialwellsuchthatpotentialenergyConsiderpotentialwellsuchthatpotentialenergy
U[x] = U ,x ≤ 0U[x] = U ,x ≤ 0U[x] = U ,x ≤ 0
0 ,0 < x < L0 ,0 < x < L0 ,0 < x < L
U ,x ≥ LU ,x ≥ LU ,x ≥ L
Intherange0 < x < L ,kineticenergyE ≤ UIntherange0 < x < L ,kineticenergyE ≤ UIntherange0 < x < L ,kineticenergyE ≤ U
Therfore,dividetheregioninthreeregionsTherfore,dividetheregioninthreeregionsTherfore,dividetheregioninthreeregions
region1,forx ≤ 0 whereU = Uregion1,forx ≤ 0 whereU = Uregion1,forx ≤ 0 whereU = U
34

Ψ”[x]+l2Ψ[x] = 0Ψ”[x]+l2Ψ[x] = 0Ψ”[x]+l2Ψ[x] = 0
region2,whereU = 0region2,whereU = 0region2,whereU = 0
schrodingerequationforcorrespondingregionschrodingerequationforcorrespondingregionschrodingerequationforcorrespondingregion
Ψ”[x]+k2Ψ[x] = 0Ψ”[x]+k2Ψ[x] = 0Ψ”[x]+k2Ψ[x] = 0
region3,forx ≥ L whereU = Uregion3,forx ≥ L whereU = Uregion3,forx ≥ L whereU = U
Ψ”[x]+l2Ψ[x] = 0Ψ”[x]+l2Ψ[x] = 0Ψ”[x]+l2Ψ[x] = 0
k = 8π2mE
h2k = 8π2mE
h2k = 8π2mE
h2
2
√
2e m
h2 π
35

l = 8π2m(U−E)
h2l = 8π2m(U−E)
h2l = 8π2m(U−E)
h2
2
√
2π m(−e+U)
h2
FromtheprevioustwosolutionsweknowtheequationsofwavefunctionofinsidepotentialwellwherepotentialeFromtheprevioustwosolutionsweknowtheequationsofwavefunctionofinsidepotentialwellwherepotentialeFromtheprevioustwosolutionsweknowtheequationsofwavefunctionofinsidepotentialwellwherepotentiale
Ψ1[x] = A∗elx ,x ≤ 0Ψ1[x] = A∗elx ,x ≤ 0Ψ1[x] = A∗elx ,x ≤ 0
Ψ2[x] = B∗Sin[kx+θ] ,0 < x < LΨ2[x] = B∗Sin[kx+θ] ,0 < x < LΨ2[x] = B∗Sin[kx+θ] ,0 < x < L
Ψ3[x] = C ∗e−lx ,x ≥ LΨ3[x] = C ∗e−lx ,x ≥ LΨ3[x] = C ∗e−lx ,x ≥ L
Thefollowingwavefunctionwillbecontinousforthecompleterangeifitsatisfiescontinuityequationsat x = 0aThefollowingwavefunctionwillbecontinousforthecompleterangeifitsatisfiescontinuityequationsat x = 0aThefollowingwavefunctionwillbecontinousforthecompleterangeifitsatisfiescontinuityequationsat x = 0a
Ψ1[0] = Ψ2[0],Ψ1[0] = Ψ2[0],Ψ1[0] = Ψ2[0],
Ψ1 [0] = Ψ2 [0]andΨ1 [0] = Ψ2 [0]andΨ1 [0] = Ψ2 [0]and
Ψ2[L] = Ψ3[L]Ψ2[L] = Ψ3[L]Ψ2[L] = Ψ3[L]
Ψ1 [L] = Ψ2 [L]Ψ1 [L] = Ψ2 [L]Ψ1 [L] = Ψ2 [L]
BOUNDARY1BOUNDARY1BOUNDARY1
InordertofindthevaluesofA,B,C.Letusconsidercontinuityequationatx = 0andx = L.InordertofindthevaluesofA,B,C.Letusconsidercontinuityequationatx = 0andx = L.InordertofindthevaluesofA,B,C.Letusconsidercontinuityequationatx = 0andx = L.
At x = 0,At x = 0,At x = 0,
Ψ1[0] = AΨ1[0] = AΨ1[0] = A
Ψ2[0] = B∗Sin[θ]Ψ2[0] = B∗Sin[θ]Ψ2[0] = B∗Sin[θ]
and Ψ1[0] = Ψ2[0]and Ψ1[0] = Ψ2[0]and Ψ1[0] = Ψ2[0]
36

A = B∗Sin[θ]A = B∗Sin[θ]A = B∗Sin[θ]
BSin[θ]
Also,Also,Also,
Ψ1 [x] = Al∗elxΨ1 [x] = Al∗elx
Ψ1 [x] = Al∗elx
Ψ2 [x] = Bk∗Cos[kx+θ]Ψ2 [x] = Bk∗Cos[kx+θ]Ψ2 [x] = Bk∗Cos[kx+θ]
and Ψ1 [x] = Ψ2 [x]and Ψ1 [x] = Ψ2 [x]and Ψ1 [x] = Ψ2 [x]
B = A∗l
k∗Cos[θ]
B = A∗l
k∗Cos[θ]B = A∗l
k∗Cos[θ]
Thereforedividingbothcontinuityequations,Thereforedividingbothcontinuityequations,Thereforedividingbothcontinuityequations,
Ψ1[0]
Ψ1 [0]
= Ψ2[0]
Ψ2 [0]
orsubstitutingthevaluesof BinA
Ψ1[0]
Ψ1 [0]
= Ψ2[0]
Ψ2 [0]
orsubstitutingthevaluesof BinAΨ1[0]
Ψ1 [0]
= Ψ2[0]
Ψ2 [0]
orsubstitutingthevaluesof BinA
Tan[θ] = k
lTan[θ] = k
lTan[θ] = k
l
Sin[θ] = Tan[θ]
(1+Tan2[θ])
Sin[θ] = Tan[θ]
(1+Tan2[θ])
Sin[θ] = Tan[θ]
(1+Tan2[θ])
Sin[θ] = k√
k2+l2
Sin[θ] = k√
k2+l2Sin[θ] = k√
k2+l2
k2 +l2 = 8π2∗m∗U
h2k2 +l2 = 8π2∗m∗U
h2k2 +l2 = 8π2∗m∗U
h2
Therefore,Therefore,Therefore,
Sin[θ] = k
b whereb2 = 8π2∗m∗U
h2 ............(A)Sin[θ] = k
h2 ............(A)Sin[θ] = k
h2 ............(A)
BOUNDARY2BOUNDARY2BOUNDARY2
37

Atx = L,Atx = L,Atx = L,
Ψ2[L] = B∗Sin[kL+θ]Ψ2[L] = B∗Sin[kL+θ]Ψ2[L] = B∗Sin[kL+θ]
Ψ3[L] = C ∗e−lLΨ3[L] = C ∗e−lL
Ψ3[L] = C ∗e−lL
and Ψ2[L] = Ψ3[L]and Ψ2[L] = Ψ3[L]and Ψ2[L] = Ψ3[L]
B∗Sin[kL+θ] = C ∗e−lLB∗Sin[kL+θ] = C ∗e−lL
B∗Sin[kL+θ] = C ∗e−lL
B = C∗e−lL
Sin[kL+θ]
B = C∗e−lL
Sin[kL+θ]B = C∗e−lL
Sin[kL+θ]
Also,Also,Also,
Ψ2 [x] = Bk∗Cos[kx+θ]Ψ2 [x] = Bk∗Cos[kx+θ]Ψ2 [x] = Bk∗Cos[kx+θ]
Ψ3 [x] = −lC∗e−lxΨ3 [x] = −lC∗e−lx
Ψ3 [x] = −lC∗e−lx
and Ψ2 [x] = Ψ3 [x]and Ψ2 [x] = Ψ3 [x]and Ψ2 [x] = Ψ3 [x]
Ψ2 [0] = Ψ3 [0]Ψ2 [0] = Ψ3 [0]Ψ2 [0] = Ψ3 [0]
C = B∗k∗Cos[kL+θ]
−l∗e−lLC = B∗k∗Cos[kL+θ]
−l∗e−lLC = B∗k∗Cos[kL+θ]
−l∗e−lL
NowsubstitutionthevalueofCweget,NowsubstitutionthevalueofCweget,NowsubstitutionthevalueofCweget,
Tan[kL+θ] = −k
lTan[kL+θ] = −k
lTan[kL+θ] = −k
l
Sin[kL+θ] = Tan[kL+θ]
(1+Tan2[kL+θ])
(1+Tan2[kL+θ])
(1+Tan2[kL+θ])
Sin[kL+θ] = −k√
k2+l2
Sin[kL+θ] = −k√
k2+l2Sin[kL+θ] = −k√
k2+l2
k2 +l2 = 8π2∗m∗U
h2k2 +l2 = 8π2∗m∗U
h2k2 +l2 = 8π2∗m∗U
h2
38

Sin[kL+θ] = −k
h2 ............(B)Sin[kL+θ] = −k
h2 ............(B)Sin[kL+θ] = −k
h2 ............(B)
Here Sin[kL] = ±γkLHere Sin[kL] = ±γkLHere Sin[kL] = ±γkL
where γ = h
2π∗
√
2mL2U
where γ = h
2π∗
√
2mL2U
where γ = h
2π∗
√
2mL2U
Graphicalsolution :Graphicalsolution :Graphicalsolution :
ApplicationofboundaryconditionstothesolutionofschrodingerequationleadstoApplicationofboundaryconditionstothesolutionofschrodingerequationleadstoApplicationofboundaryconditionstothesolutionofschrodingerequationleadsto
Sin(kL) = ±γ(kL),whereγ = h
2π
√
2meL2Uo
,if L = 200nmandUo = 0.6eVthenγ = 0.004867Sin(kL) = ±γ(kL),whereγ = h
2π
√
2meL2Uo
,if L = 200nmandUo = 0.6eVthenγ = 0.004867Sin(kL) = ±γ(kL),whereγ = h
2π
√
2meL2Uo
,if L = 200nmandUo = 0.6eVthenγ = 0.004867
substituting kL = x,wegetsubstituting kL = x,wegetsubstituting kL = x,weget
Sin(x) = ±γxSin(x) = ±γxSin(x) = ±γx
Plot[{Sin[x],0.004867x},{x,0,70π}]Plot[{Sin[x],0.004867x},{x,0,70π}]Plot[{Sin[x],0.004867x},{x,0,70π}]
FindRoot[Sin[x] == 0.004867x,{x,1}]FindRoot[Sin[x] == 0.004867x,{x,1}]FindRoot[Sin[x] == 0.004867x,{x,1}]
{x → 5.048709793414476`*∧-29}
{x → 3.12638}
39

{x → 6.31392}
{x → 9.37911}
{x → 12.6279}
{x → “15.6318”}{x → “15.6318”}{x → “15.6318”}
...
...
...
...
40

{x → 196.644}
{x → 202.794}
{x → 202.462}
{x → 208.911}
41

Plot[{Sin[x],−0.0044x},{x,0,80π}]Plot[{Sin[x],−0.0044x},{x,0,80π}]Plot[{Sin[x],−0.0044x},{x,0,80π}]
FindRoot[Sin[x] == −0.0044x,{x,0}]FindRoot[Sin[x] == −0.0044x,{x,0}]FindRoot[Sin[x] == −0.0044x,{x,0}]
{x → 3.15548}
{x → 6.25566}
{x → 9.46644}
{x → 12.5113}
42

{x → “15.7774”}{x → “15.7774”}{x → “15.7774”}
...
...
...
...
{x → 218.055}
{x → 218.618}
{x → 224.467}
{x → 224.772}
43

By getting the values of x we can calculate Energy Levels in the conduction
band for electrons and valence band for holes.
x for electrons x for holes E Levels Electron E levels (eV) Holes E levels (eV)
5.04871x10−29 3.15 E1 1.39x10−4 1.19x10−4
3.126 6.25 E2 5.671x10−4 4.45x10−4
6.313 9.46 E3 1.249x10−3 1.03x10−3
9.379 12.41 E4 2.268x10−3 1.81x10−3
12.627 15.77 E5 3.477x10−3 2.88x10−3
15.631 18.76 E6 5.105x10−3 4.08x10−3
18.9419 22.088 E7 6.81x10−3 5.61x10−3
21.8814 25.022 E8 9.07x10−3 5.61x10−3
25.25 28.39 E9 0.0112 9.34x10−3
28.13 31.27 E10 0.0141 0.011
31.57 34.71 E11 0.0168 0.013
34.38 37.53 E12 0.0204 0.016
37.88 41.02 E13 0.0235 0.0195
40.64 43.78 E14 0.0277 0.022
44.19 47.33 E15 0.03129 0.0259
46.89 50.04 E16 0.0363 0.029
50.51 53.64 E17 0.04019 0.032
53.14 56.29 E18 0.0495 0.036
59.39 59.95 E19 0.0502 0.0416
63.14 62.55 E20 0.05673 0.0453
65.64 66.26 E21 0.06132 0.05092
69.45 68.80 E22 0.06865 0.054
71.89 72.58 E23 0.07356 0.0611
75.77 75.6 E24 0.0817 0.06535
78.14 78.89 E25 0.0869 0.072
82.09 81.31 E26 0.0958 0.076
84.39 85.20 E27 0.1013 0.084
88.40 87.56 E28 0.1112 0.088
90.64 91.52 E29 0.1109 0.096
94.72 93.82 E30 0.1276 0.102
96.89 97.83 E31 0.1336 0.111
101.045 100.07 E32 0.1452 0.1161
103.147 104.14 E33 0.15139 0.125
44

x for electrons x for holes E Levels Electron E levels (eV) Holes E levels (eV)
107.36 106.32 E34 0.164 0.1311
109.39 110.36 E35 0.17029 0.1415
113.68 112.59 E36 0.1839 0.146
115.64 116.77 E37 0.1902 0.156
120.00 118.83 E38 0.2049 0.1637
121.88 123.09 E39 0.2114 0.175
126.32 125.08 E40 0.227 0.1814
128.13 129.41 E41 0.233 0.194
132.64 131.33 E42 0.2503 0.20007
134.37 135.72 E43 0.2569 0.236
138.97 137.58 E44 0.2748 0.2195
140.61 142.04 E45 0.2813 0.2343
145.29 148.36 E46 0.3003 0.2533
146.85 150.07 E47 0.3069 0.2612
151.62 154.68 E48 0.32715 0.277
153.09 156.32 E49 0.3355 0.2834
157.95 161.00 E50 0.355 0.30007
159.33 162.56 E51 0.3612 0.30656
164.28 167.33 E52 0.384 0.324
165.56 168.8 E53 0.39 0.33055
170.62 173.65 E54 0.4142 0.35
171.97 175.05 E55 0.4199 0.3554
176.96 179.98 E56 0.4456 0.3757
178.02 181.28 E57 0.4509 0.381
184.24 186.31 E58 0.484 0.4026
189.67 187.52 E59 0.5119 0.4077
190.45 192.75 E60 0.5161 0.4304
196.046 198.98 E61 0.5469 0.4354
202.74 199.98 E62 0.5502 0.4592
208.91 205.31 E63 0.5852 0.463
206.20 E64 0.489
211.68 E65 0.4932
212.42 E66 0.519
218.05 E67 0.5234
218.47 E68 0.5513
220.77 E69 0.5543
224.46 E70 0.5844
224.77 E71 0.586
45

Using values of energy levels given as E we can calculate wavelength for a
given DH LED.
This can be found using following formula
λ = hc
E
When we are calculating value of ,we get to know that the we need to ﬁnd
allowed transition of E.
Whether a particular electronic transition is allowed or not is decided by the
selection rule. According to the selection rule only those electronic transitions are
allowed for which the difference in the quantum number is even number.
The number of energy levels in the quantum well is more than 50. Nearly
equal number of energy levels for holes. Then one has to apply the selection rule to
remove forbidden transtions. This is quite cumbersome and tedious. Therefore we
have limited our task to estimate minimum and maximum possible wavelengths
in the output radiation of DH LED. After getting values of E we have calculated
values of Emin and Emax.
Emin value gives us limiting case value and Emax value gives us maximum
allowed difference between two well structure.
Therefore the considered values are 1.39 ∗ 104 and 0.5852 for electrons and
for holes considered values are 1.19∗104 and 0.5860.
Now consider a quantum well structure, the value of band gap between the
two wells gives us the value of Emin and maximum value of E is calculated by
taking the difference between two quantum well.
We get values of Emin = 1.4eV and Emax = 1.98eV.
Therefore we are considering values of energy levels for electrons as well as
for holes as Emin=1.4eV and Emax=1.98eV, we get values of wavelength as
λmax = 8.88∗107 m or 888 nm.
λmin= 6.26∗107 m or 626 nm.
46

Chapter 4
Application and future scope
4.1 Application and Future Scope
4.1.1 Introduction
Modern technological capabilities of epitaxial growth allow us to fabricate types
of nanostructures even with more complicated potential profiles. For example, it
is possible to fabricate nanostructures that contain two or more coupled low di-
mensional Nano objects (a structure with size less than or about 10 nm is called a
Nano object or a nanostructure). It becomes possible to control the energy spec-
trum of electrons in such structures not only by changing the form of an individual
Nano object, but also by changing the distance and barrier height between neigh-
boring Nano objects. If individual Nano objects are separated by low and narrow
potential barriers, electrons can easily tunnel from one Nano object to another.
This significantly affects the character of the electron energy spectrum in such a
structure.
Further, because of their quasi-two dimensional nature, electrons in quantum
wells have a density of states as a function of energy that has distinct steps, versus
a smooth square root dependence that is found in bulk materials. Additionally, the
effective mass of holes in the valence band is changed to more closely match that
of electrons in the conduction band. These two factors, together with the reduced
amount of active material in quantum wells, leads to better performance in optical
devices such as laser diodes. As a result quantum wells are in wide use in diode
lasers, including red lasers for DVDs and laser pointers, infra-red lasers in fiber
optic transmitters, or in blue lasers.
47

4.1. APPLICATION AND FUTURE SCOPE
4.1.2 RESULT AND CONCLUSION
Quantum mechanics paved way for the effective and in depth study of the dou-
ble heterojunction LED structure. We could understand the very idea behind the
light which a LED emits. The Schrodinger equation has been solved for potential
well created in double hetero-junction LED structure and various energy levels
have been calculated. We have used MATHEMATICA software for optimized
mathematical solution and representation of our problem.
Various energy bands thus calculated give an idea of the possible transition in
LED structure. Photons are emitted when any kind of transition from conduction
band to valence band occur. The minimum wavelength results when transition
takes place between levels whose energy difference is maximum. The maximum
wavelength of the emitted photon is achieved when transition takes place between
levels whose energy difference is minimum. The calculated wavelength is found
to be less than or equal to the maximum permitted/observed experimental wave-
length in LED.
4.1.3 FUTURE SCOPE
Different colors of LED have different power consumptions and therefore there
costs vary. Hence, we strive hard to innovate and develop LEDs of different col-
ors. In the year 2015, blue LED was a revolutionary innovation in the field on
Nanotechnology. Blue was the last and most difficult – advance required to create
white LED light. And with white LED light, companies are able to create smart-
phone and computer screens, as well as light bulbs that last longer and use less
electricity than any bulb invented before.
LEDs are basically semiconductors that have been built so they emit light
when they’re activated. Different chemicals give different LEDs their colors. En-
gineers made the first LEDs in the 1950s and 60s. Early iterations included laser-
emitting devices that worked only when bathed in liquid nitrogen. At the time,
scientists developed LEDs that emitted everything from infrared light to green
light. But they couldn’t quite get to blue. That required chemicals, including
carefully-created crystals that they weren’t yet able to make in the lab.
Once researchers did figure it out, however, the results were remarkable. Upon
studying various nanostructures and finding solutions using schrodingers’ equa-
tion realization and fabrication of white LEDs by combining red, green and blue
ones, could be possible with tunable colors. The various permutation and combi-
nation will give various possible energy level transitions. Based on the wavelength
of the emitted photon and/or materials used for fabrication the color of the LED
can be manipulated. Detailed calculations of the energy levels of holes and subse-
quent application of the selection rule has not been carried out in the present work.
48

4.1. APPLICATION AND FUTURE SCOPE
Therefore authentic prediction of wavelengths present in the output radiation of
the double hetero-structure is beyond the scope of the current work. This work
can deﬁnitely be carried out in future.
49

Bibliography
[1] Quantum Mechanics for Nanostructures (2010) (Malestrom)
[2] Introduction to Quantum Mechanics by David Grifﬁths
[3] Principles of Electronic Materials and Devices by Kasap and Islam
[4] Design parameters of frequency response of GaAs(Ga, Al) As double het-
erostructure LED’s for optical communications from IEEE
[5] Schrdinger Equations and Diffusion Theory (Modern Birkhuser Classics)
[6] Getting Started with LATEX by David R. Wilkins
[7] Differential equations with Mathematica by Martha L. Abell and James
P.Barselton
[8] Mathematica help document
[9] LATEX help document
[10] Wikipedia
50

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