Table of ContentsTable of Contents........................................................................................................................................1

List of Figures.............................................................................................................................................2

Chapter 1.....................................................................................................................................................4

Introduction.................................................................................................................................................4

1.1 Nanotechnology and Quantum Technology.................................................................................4

1.2 Basic requirements for QDs in room temperature........................................................................6

1.2.1 Size......................................................................................................................................7

1.2.2 Uniformity..........................................................................................................................7

1.2.3 The material quality..........................................................................................................7

1.3 Quantum Confinement and its consequences...............................................................................8

1.3.1. Confinement effect on density of states...............................................................................8

1.3.2. Lateral confinement.............................................................................................................8

1.3.3. Density of states...................................................................................................................9

1.4 QD implementation...................................................................................................................11

1.4.1 Etching..............................................................................................................................11

1.4.2 Modulated Electric Field.................................................................................................12

1.4.3 Inter-diffusion between the barrier and the quantum well...........................................12

1.4.4 Semiconductor Microcrystals.........................................................................................13

1.4.5 Selective Growth..............................................................................................................13

1.4.6 Self-organized growth......................................................................................................14

1.5 Quantum wells, wires and dots in optics....................................................................................15

Bibliography..............................................................................................................................................22

List of FiguresFigure 1.1 Density of states for confined structures....................................................................................5Figure 1.2 Energy band spectrum: (a) the absorption of photon results in an electron-hole pair, (b) the recombination of an electron-hole pair results in the spontaneous emission of a photon [BO]..................12Figure 1.3: Quantum well Confinement. (a) Geometry of a quantum well structure. (b) Energy level diagram for electrons and holes in a quantum well....................................................................................15Figure 1.5.(b) Self-assembled QDs............................................................................................................22Figure 1.5.(a) Microcrystals......................................................................................................................22

1. Chapter 1

Introduction

1.1Nanotechnology and Quantum Technology

Nanotechnology has become one of the defining technologies for the beginning of the

twenty-first century. Nanotechnology is about: making things small because their

properties change radically as their size is reduced, offering the promise of new artificial

materials made from the nanostructures 2; integration of nanodevices onto a single chip

1; nanodevices functioning down almost to the atomic level 3; and self assembly of

nanostructures into complex devices and structures 5. In the late 1970s and early 1980s,

rapid development of planar growth techniques, such as molecular beam epitaxy (MBE)

and metal organic chemical vapor deposition (MOCVD), made it possible for precise

growth of high-quality, layered, semiconductor heterostructures. Abrupt interfaces could

be made between semiconductors with different bandgaps just by opening and closing

valves to the material cells in the growth system. Exquisite control was achieved with

layers as thin as a few nanometers grown with monolayer precision. Arbitrary sequences

of these layers were grown routinely to form quantum wells and superlattices, as well as

heterojunctions, as shown in Fig. 1.1. At a heterojunction, electrons are confined to move

in a thin two-dimensional (2D) depletion layer just inside the low bandgap region. In a

quantum well (QW), electrons are confined to a thin, low bandgap 2D layer sandwiched

between high bandgap layers2.

These structures provided a quasi-2D world for electrons with properties controlled by

the well size that could be used for novel electronic and optical devices. Electronic

energy and threshold for optical excitation are tailored, via confinement, simply by

varying the thickness of the quantum well layer. Quantum well lasers, Stark-effect

modulators, and the modulation doped field effect transistors were some of the devices

made by exploiting this quasi-2D world. Multilayer structures can be grown specifically

to tailor dielectric response, providing Fabry-Perot cavities that can enhance the

performance of optical devices.

The great successes that is achieved in the exploitation and exploration of the quasi-2D

electronic world motivated research for investigations of nanostructures with even greater

confinement: structures that are confined in two directions to produce quasi-one

dimensional (1D) quantum wires (QWR) with carriers free to move along the wire axis,

and structures that are confined in all three directions to produce quasi-zero dimensional

quantum dots with no free motion. There were predictions of enhanced performance of

optical devices made from dots and wires. There was strong additional motivation to

fabricate dots and wires directly in quantum well systems, so that the dot and the wire

device structures could be routinely inserted into well established technologies. The most

obvious route toward achieving these structure is to: start with a quantum well structure,

with one-dimensional confinement in the vertical direction defined by the well, then etch

away enough material to provide physical confinement laterally to form the thin quantum

wire or quantum dot2.

Figure 1.1 Density of states for confined structures 6.

However, these top-down approaches did not achieve expected success due to their

inability to fabricate enough small useful structures. Quantum wells could be successfully

fabricated because the nanoscale dimension (the well thickness) was controlled by

growth processes. The development of quantum wire and dot structures needs

comparable processes to grow wires and dots from the bottom up, ideally with same

precision and size control as achieved in the growth of quantum wells. Then, dot and wire

structures, that now are promising for providing much-enhanced integrated optical device

structures for lasers and modulators, are realized2.

For semiconductor quantum wells, wires and dots, the radical change of properties occurs

when quantum mechanics begin to govern the properties of the structures. The structures

are referred to as Quantum structures for just this reason. In the quantum limit,

confinement leads to energy shifts and changes in the spatial overlap of conduction and

valence electrons, which determine the strengths of optical transitions. Accordingly, the

size and shape of the structures can be used for engineering their optical response by

tailoring the desired transition energies, transition strengths, and polarization dependence.

1.2Basic requirements for QDs in room temperature

Quantum dots should fulfill the following requirements to be useful for devices at room

temperature 1:

1. sufficiently deep localizing potential and small size is a prerequisite for

observation and utilization of zero-dimensional confinement effects,

2. QD ensembles should show high uniformity and high volume filling factor.

3. And, the material should be coherent without defects like dislocations.

1.2.1 Size

The lower limit of size of QD is given by the condition that at least one energy level of an

electron or a hole or both is present. The critical lateral diameter of QD strongly depends

on the band offset of the corresponding bands in the material system used. There is also

an upper limit for the size of a QD as the thermal population of higher-lying energy is

undesired for some devices (like laser). This upper limit depends on the operating

temperature.

1.2.2 Uniformity

If a single device is based on more than one quantum dot, the issue of uniformity arises.

In principle, all structural parameters of a QD such as shape, size and chemical

composition are exposed to random fluctuation, even in the presence of ordering

mechanisms. In most cases, a dense array of equal-size and equal-shape quantum dots is

desired.

1.2.3 The material quality

The density of defects in a QD material and its interface to the surrounding matrix should

be as low as possible, on the level of the grown quantum wells and interfaces used in

most of the state of the art devices. QD fabrication using self organized growth seems to

be predestined to achieve this goal since all Interfaces are formed in situ during crystal

growth.

1.3Quantum Confinement and its consequences

1.1.1. Confinement effect on density of states

The electronic structure of bulk semiconductors is characterized by delocalized electronic

states and by a quasi continuous spectrum of energies in the conduction and valence

bands3. In semiconductor nanostructures, if the electrons are confined in small regions of

space in the range of a few tens of nanometers or below, the energy spectrum is

particularly affected by this confinement that results in an increase of the width of the

band gap, also the allowed energies become discrete in zero dimensional systems and

form mini bands in 1D and 2D systems.

1.1.2. Lateral confinement

Since quantum dots are created mainly through producing a lateral confinement V (x , y)

restricting the motion of the electrons, which are initially confined in a very narrow

quantum well by the potential V (z ) , they usually have the shape of flat disks, with

transverse dimensions considerably exceeding their thickness5.

The energy of single particle excitations across the disk exceeds other characteristic

energies in the system, and the confined electrons can be considered as two-dimensional

in both growth and lateral directions (Fig. 1.2). The lateral potential can be approximated

modeled depending on the quantum dot implementation method5. Considering the

etching method, the potential of a dot with a considerable radius is fairly close to a

rectangular well with rounded edges. When a dot is small (i.e. when its radius is

comparable to the characteristic length of the variation of the lateral potential near the

edge), a good approximation offers simple smooth potentials, such as a Gaussian well.

1.1.3. Density of states

1.1.3.1. Quantum Well

The considered wells will be square wells, such as those grown by epitaxial techniques.

We consider a well made of semiconductor I (−L/2<z<L/2) sandwiched between barrier

layers of a larger bandgap material II such as the electrons or the holes are confined in the

well. The difference between the potential in the barrier and in the well defines the

confining potential which is commonly approximated by a square potential only

depending on the coordinate z. within this approximation, the calculation of the lowest

electronic states has been done using the envelope function approach, i.e. using K.P

perturbation theory or effective mass approximation. A basic assumption in these

treatments is that the confining potential does not mix the wave functions from different

bands, except those are degenerate. In the simplest case of conduction band states in

direct gap semiconductors like GaAs or InP, it can be shown that the electron wave

function takes the form [3]

ψ=uko

I , II (r ) φ(r), (1.1)

Where uko

I , II is the Bloch wave function at the minimum of the conduction band in the

material I or II, and φ (r )is an envelope function solution of a shrӧdinger-like equation

(−ℏ2

2m¿ ∇2+V conf ( z ))φ (r )=εφ (r ).

(1.2)

If the confining potential is large in the barrier, the problem for the lowest states can be

approximated by an infinite square well. Thus, the above equation can be solved easily to

get the eigen-states of electrons and their energy values they can take. The effect of the

confinement on the electronic structure is evidenced in the density of states

ρ2 D(ε )∝∑n

Θ(ε−εnz). (1.3)

where Θ(x ) is the step function. Thus the density of states in a 2D system is a staircase

function.

Thus the consequences of transition from 3D system to 2D system3 can be concluded as

follows:

1. Density of states is recognized into steps.

2. Width of the band gap is increased compared to bulk.

3. Physical properties are altered by the 2D confinement:

A. It results in blue shift of the optical absorption threshold.

B. It results in blue shift of the photoluminescence threshold.

1.1.3.2. Quantum Wire

The confinement happens in two directions of space, and the carrier motion is free in the

other direction. Following the same procedure as for wells, density of states can be

expressed as follows:

ρ1 D(ε )∝∑nx , ny

(ε−ε nx

x −ε ny

y )−12 .

(1.4)

The density of states is equal to zero when ε<ε1x+ε1

y , thus the quantum confinement leads

to an opening of the band gap like in 2D systems but the 1D density of states is highly

peaked , since it presents singularities at each value of ε nx

x +εn y

y .

1.1.3.3. Cubic quantum dots

In dots, the confinement takes place in the three directions of the space. The main

consequence is that the electronic spectrum consists in series of discrete levels, like in

isolated atoms. Thus, the density of states consists of delta functions at the discrete

energies:

ρ0 D(ε)∝ ∑n x ,n y ,nz

δ (ε−εn x, n y, nz). (1.5)

Since the electronic structure of these quantum dots can be tuned by changing their size

or their shape, the quantum dots are particularly attractive building blocks for the

development of nanotechnologies.

1.4Quantum wells, wires and dots in optics

1.4.1 Basic quantum mechanics

To understand why quantum wells, wires, and dots can provide enhanced optical

performance, it is useful to recall the basic quantum mechanics that governs bulk

semiconductors and to see what changes occur when a semiconductor structures is

reduced to nanometer sizes in one, two, or all three dimensions.

The quantum mechanical energy of a free electron is

E=( kℏ )2

2 m(1.6)

where m is the particle mass and ħk is its quantum mechanical momentum with the

wavevector k related to electron wavelength k=2 π / λ. In a bulk semiconductor, the

electron energy dispersion is more complicated due to the coherent scattering of the

electron off the atoms in the lattice and the mixing of the local electron states on the

atoms into the total electron wavefunction. A typical energy scheme for an electron in a

bulk III-V semiconductor such as GaAs, InAs, or AlGaAs is shown in figure 1.2 for small

k .

1.4.2 Optical response of bulk semiconductor and confinement

modification

The optics of bulk semiconductors is governed by the absorption and emission of

photons. During absorption, an electron is promoted from a filled state to an empty state.

Emission is the reverse process. Typically, emission occurs close to the band-edge, after

the electron and hole have relaxed toward their minimum energies in their respective

bands. Two conservation rules apply. The photon has negligible momentum, so the

momentum of the electron is unchanged during the transition. This means that only

vertical transitions, also referred to as direct transitions are allowed. Semiconductors are

divided into two classes. In direct semiconductors, the conduction and valence band

extreme occur at the same k . In indirect semiconductors, the conduction and valence band

extreme occur at different k . emission is strongly suppressed in indirect semiconductors

because electrons and holes relax to extreme at different k and cannot recombine directly

Figure 1.2 Energy band spectrum: (a) the absorption of photon results in an electron-hole pair, (b) the recombination of an electron-

hole pair results in the spontaneous emission of a photon 6.

to give off a photon. For this reason, bulk semiconductors used for optics are typically

III-V semiconductors, such as GaAs, InAs, and InP which have direct bandgaps, rather

than group IV semiconductors, such as Si, and Ge with indirect gaps.

The second conservation rule is the conservation of energy. The energy of the photon

absorbed or emitted is the same as the energy of the electron and hole involved in the

transition:

ωℏ =Ee+Eh+Ecoul (1.7)

where

Ee, Eh is the electron and hole single-particle energies, respectively, and Ecoul is the

additional energy due to the coulomb interaction between the electron and hole.

Because the hole is the absence of an electron, it behaves as a positive charge. Coulomb

energy results from the direct and exchange interactions between the two oppositely

charged particles, screened by the other electrons in the system. Typically, this energy

binds the electron and hole together. Such a bound pair is referred to as an exciton. More

complicated optical excitations also exist. Two electrons and two holes bound together by

their coulomb interaction are referred to as a biexciton. A single exciton with additional

electrons or additional holes bound to it is referred to as a charged exciton.

The strength of an optical transition is determined by the Fermi Golden Rule. Two factors

are important. The density of states (DOS) is the number of possible electron-hole

transitions that can take place at a given photon energy. For a bulk semiconductor, the

density of states for transitions vanishes for energies below the bandgap. When excitonic

effects are included, a series of discrete transitions involving bound electron-hole pairs is

pulled out of the continuum into the gap. The second factor is the dipole matrix element

⟨ e|r . E|h ⟩. In the dipole approximation, this matrix element couples light to the material

via a transition of an electron between a state in the conduction band and the state of the

hole in the valence band. The transition rate is proportional to|⟨e|r . E|h ⟩|2.

For a larger overlap between the electron and hole state, the transition is stronger. At the

same time, the transition is stronger when the electron and the hole state have larger

extent and a bigger dipole matrix element. Finally, the polarization dependence of the

transition is determined by this matrix element.

To understand how confinement modifies the optical response of a bulk semiconductor,

we first consider a 2D quantum well structure, where a thin layer of a low bandgap

semiconductors, the barriers. The potential profile that describes the confinement is the

profile of the band-edge energy for the conduction band and valence bands as shown in

Figure 1.3.

Figure 1.3: Quantum well Confinement. (a) Geometry of a quantum well structure. (b) Energy level diagram for electrons

and holes in a quantum well.

If the energy bandgap of the low bandgap materials fits entirely inside the energy

bandgap of the higher bandgap material, then the low bandgap region acts as a well that

confines both electrons and holes. In a quantum well structure, the well confines the

electrons and holes in one dimension .they are still free to move in the plane of the layer.

The simplest model for a well is the particle-in-a-box model, where the barrier is

assumed to be infinitely high so that the electron cannot penetrate into the barrier.

In that case, the electron wave motion across the well is constrained to be an integral

number n of half wavelengths. That is d=nλ/2, where d is the well thickness. The

wavevector k for motion perpendicular to the plane of the well is then k=nπ /d . This is

the quantization condition that makes the well a quantum well. The energy for an electron

in the conduction band becomes:

Ee=Ec ,w+¿¿¿ (1.8)

where k ∥ is the wavevector for the motion parallel to the plane of the well.

In terms of the well thickness,

Ee=Ec ,w+¿¿¿. (1.9)

In (1.14) the first term is the band-edge energy; and the second term is the quantized

confinement energy proportional to 1/d2, the third term is the kinetic energy for the

motion in the plane. As a result of quantum well confinement, the bulk density of states is

split into a series of subbands. The band-edge of each subband is shifted to higher energy

due to the confinement energy, but each subband remains a continuum of states due to the

free motion in the plane of the well. Most importantly, there is a pile-up of the DOS near

the subband edges. The density of states remains finite at the subband edge, rather than

vanishes, as happens in the bulk limit. As a consequence, more transitions lie near the

subband-edges and can contribute to band-edge emission. In large part, this is why

confinement of a bulk semiconductor into a quantum well structure enhances optical

performance.

When the quantum well is modeled more realistically with a finite barrier, the effects of

confinement remain the same. The main difference is that the electron can penetrate into

the barriers. This leads to an effective thickness for the well that represents the physical

thickness of the well plus the depth the electron penetrates into the barriers.

When a quantum well structure becomes confined in a second dimension to form a wire

or in all three dimensions to form a dot, the kinetic energy becomes quantized in the

second dimension or in all three dimensions. This additional quantization further piles up

the DOS at the subband edges. This leads to a density of states that is singular at the

subband-edges for a quantum wire.

Each subband is shifted to higher energy due to the additional confinement, but remains a

continuous band. For a quantum dot, the kinetic energy for states localized in the dot is

fully quantized and the subbands for these states become a series of discrete states. A

continuum of states may remain at higher energy, but these correspond to unconfined

states with energies above the barrier.

1.4.3 Advantages of using confined structures for optics

There are several advantages to using confined structures for optics:

1. Extending the confinement to a second dimension or to all three dimensions piles up

the density of states into narrower ranges of energies. This is ideal for optical

performance, because more transitions can contribute to the optical response at the same

energy. In that regard, a quantum dot is ideal because all transitions are concentrated at a

series of discrete energies, rather than distributed over a continuum of energies.

2. Foremost, the subband energies and the splitting between energy levels increase

proportional to 1/d2. Thus, confinement can be used to tailor the transition energies.

3. As a consequence of the level splitting, relaxation processes are slowed. For example,

relaxation via emission of lattice vibrations phonon in quantum dots is slowed because

few phonons have energies that match the level splitting. This can be bad for the use of

dots as light emitters because electrons and holes excited to high energies in the

conduction and valence band will relax slowly to the band edge if relaxation channels

other than phonons are not available.

4. In confined structures, the transitions at a given energy can have a definite

polarization. For example, the lowest energy transition in a quantum well is driven,

typically, by light polarized in the plane of the well. The lowest transition in a quantum

wire responds to light polarized along the wire axis. For quantum dots made from

quantum wells, the polarization is mostly in the plane of the well.

5. The binding between electron and hole is increased by additional confinement because

the confinement localizes the electron and hole to the same region. The enhanced

binding, which scales approximately as 1/d, increase the stability of the exciton to

thermalization.

6. Increased confinement and binding leads to increased electron-hole overlap. This leads

to larger dipole matrix elements and larger transition rates. At the same time, to an

increase of confinement can reduce the extent of the electron and hole states and thereby

reduce the dipole moment.

1.5 QD implementation

1.5.1 Etching

The earliest method for obtaining quantum dots was implemented by Reed et al. 5 who

etched them in a structure containing two-dimensional electron gas. The process can be

summarized in steps:

A) The surface of the sample containing one or two quantum wells is covered by a

polymer mask and then partly exposed to electron/ion beam. The exposed pattern

corresponds to the shape of the created nanostructures.

B) At the exposed areas the mask is removed .Later the entire surface is covered with a

thin metal layer.

C) The polymer film and the protective metal are removed using a special solution except

for the previously exposed areas where the metal layer remains.

D) The areas not protected by the metal mask are chemically etched to get the slim pillars

created.

Thus the electrons, which are initially confined in the plane of the quantum well, are

further confined to a small pillar with a diameter on the order of 10-100 nm.

Figure 1.4 Various quantum dot implementations. (a) metal and metal oxide systems patterned by lithography. (b) metallic dots out of chemical

suspensions. (c) lateral quantum dots through electrical gating of heterostructures. (d) vertical quantum dots through wet etching of quantum well structures. (e) pyramidal quantum dots through self-assembled growth.

(f) trench quantum wire 4.

1.5.2 Modulated Electric Field

At this method, miniature electrodes are created over the surface of a quantum well by

means of lithographic techniques. The application of an appropriate voltage to the

electrodes produces a spatially modulated electric field, which localizes the electron

within a small area. The process of spreading a thin electrode over the surface of a

quantum well may produce both single quantum dots and large arrays of dots. The

advantageous feature of such quantum dots is their smooth lateral confinement, showing

no edge effects. The possibility of controlling certain parameters is also very important.

These quantum dots were created experimentally on gallium arsenide, indium antimonite,

and silicon.

1.5.3 Inter-diffusion between the barrier and the quantum well

A method for obtaining quantum dots based on a quantum –well material by local heating

of a sample with a laser beam. A parent material of a single, 3 nm thick GaAs quantum

well was used, and this was prepared using the molecular beam epitaxy method. It was

then placed between a pair of 20 nm thick AlGaAs barriers. The topmost 10 nm thick

GaAs cap layer was covered with a100nm coating of silicon nitride (ceramic of high

strength) protecting the surface against oxidation or melting by the laser beam. The laser

beam was guided along a rectangular contour surrounding an unilluminated area of

diameter 300-1000 nm. At a temperature of about 1000 C a rapid Inter-diffusion of Al

and Ga atoms occurred between the wall and the barrier, which led to the creation of the

potential barrier which surrounds the unilluminated interior of the rectangle.

1.5.4 Semiconductor Microcrystals

Quantum dots can be created in the form of semiconductor microcrystals immersed in

glass dielectric matrices. Ekimov, who was the first to investigate that idea

experimentally, heated silicate glass with about 1% of the semiconducting phase (CuCl,

CuBr, etc) for several hours at a temperature of several hundred degrees Celsius. This led

to the formation if appropriate microcrystals of almost equal sizes the radii of dots

measured in different samples varied in the range 1.2-38nm.

1.5.5 Selective Growth

Quantum dots can also be created through the selective growth of a semiconductor with a

narrower bandgap (GaAs) on the surface of another semiconductor with a wider band gap

(AlGaAs).The restriction of growth to chosen areas is obtained by covering the sample

with a mask (silicon dioxide) and etching on it miniature triangles. On the surface not

covered with the mask the growth is then carried out with the metal –organic chemical

vapor deposition method (MOCVD).the crystals that are created have the shape of

tetrahedral pyramids, and hence when the first crystallized layers are the layers of the

substrate (AlGaAs) and only the top of the pyramid is created of GaAs, it is possible to

obtain a dot of effective size below 100nm.

1.5.6 Self-organized growth

Quantum dots can be created by the self –crystallization without need for a mask. When

the lattice constants of the substrate and the crystallized material differ considerably, only

the first deposited monolayers crystallize in the form of epitaxial strained layers with the

lattice constant equal to that of the substrate. When the critical thickness is exceeded, a

significant strain occurring in the layer leads to the breakdown of such an ordered

structure and to the spontaneous creation of randomly distributed islets of regular shape

and similar size. The shape and average size of islets depend mainly on the strain

intensity, the growth temperature, and the growth rate. The phase transition from

epitaxial structure to the random arrangement of islets is called the Stranski-Krastanov

transition. The quantum dots formed in the Stranski-Krastanov phase transition are called

self-organized or self-assembled dots (SAD).

The small sizes of the self –assembled quantum dots (~30nm), homogeneity of their

shapes and sizes in a macroscopic sample, perfect crystal structure (without edge effects),

and the fairly convenient growth process without the necessity of the precise deposition

of electrodes or etching –are among their greatest advantages. Thus, this method is very

promising for electronic and optoelectronics applications

Figure 1.5.(a) Microcrystals Figure 1.5.(b) Self-assembled QDs

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