chapter 1 introduction
TRANSCRIPT
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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