quantum wells, dots, Quantum Limit of Conductance, Quantum Capacitance & Quantum HALL effect R. John Bosco Livro

Prévia do material em texto

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Quantum Wells, Quantum Wires, Quantum 
Dots, Quantum Limit of Conductance, 
Quantum Capacitance & Quantum HALL 
Effect 
 
 
 
 
 
 
 
 
R. John Bosco Balaguru 
Professor 
School of Electrical & Electronics Engineering 
SASTRA University 
 
B. G. Jeyaprakash 
Assistant Professor 
School of Electrical & Electronics Engineering 
SASTRA University 
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1 INTRODUCTION………………………………………………………………3 
Table of contents 
 
1.1 WHAT ARE LOW-DIMENSIONAL STRUCTURES?............................................................3 
1.2 CLASSIFICATION OF LOW-DIMENSIONAL MATERIALS……………………………...3 
1.3 WHY WE NEED QUANTUM MECHANICS?........................................................................4 
 
2 INTRODUCTION ABOUT QUANTUM WELLS, 
QUANTUM WIRES AND QUANTUM DOTS……………………………….6 
 
2.1 QUANTUM WIRE…………………………………………………………………………...11 
2.1.1 How to prepare Quantum Wire?.....................................................................................11 
2.2QUANTUM DOT…………………………………………………………………………....12 
2.3 TWO-DIMENSIONAL STRUCTURES: QUANTUM WELLS…………………….…..….13 
2.3.1 Zero-Point Energy……………………………………………………………………...15 
 
3 ONE-DIMENSIONAL STRUCTURES: 
QUANTUM WIRES AND NANOWIRES…………………………………...15 
 
4 ZERO-DIMENSIONAL STRUCTURES: 
QUANTUM DOTS AND NANODOTS………………………………………18 
 
5 QUANTUM CONDUCTANCE……………………………………………....19 
 
6 QUANTUM CAPACITNECE………………………………………………..23 
 
6.1 QUANTUM CONDUCTANCE……………………………………………………………..23 
6.2 EFFECT OF QUANTUM CONDUCTANCE……………………………………………....24 
 
7 QUANTUM HALLEFFECT…………………………………………..……..24 
 
7.1 
7.2 QUANTUM HALL EFFECT………………………………………………………..……….27 
INTEGER QUANTUM HALL EFFECT – LANDAU LEVELS………………...………….25 
 
8 REFERENCES…………………………………………….…………………..29 
 
 
 
 
 
 
 
 
 
 
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1. INTRODUCTION 
 
1.1 What are Low-Dimensional Structures? 
 
When one or more of the dimensions of a solid are reduced sufficiently, its 
physicochemical characteristics notably depart from those of the bulk solid. With 
reduction in size, novel electrical, mechanical, chemical, magnetic, and optical properties 
can be introduced. The resulting structure is then called a low-dimensional structure (or 
system). The confin emen t o f p articles, u su ally electro n s o r h o les, to a lo w - 
dimensional structure leads to a dramatic change in their behaviour and to the 
manifestation of size effects that usually fall into the category of quantum-size 
effects. 
The low dimensional materials exhibit new physicochemical properties not shown 
by the corresponding large-scale structures of the same composition. Nanostructures 
constitute a bridge between molecules and bulk materials. Suitable control of the 
properties and responses of nanostructures can lead to new devices and technologies. 
 
1.2 Classification of Low-dimensional Materials 
 
Low-dimensional structures are usually classified according to the number of 
reduced dimensions they have. More precisely, the dimensionality refers to the number 
of degrees of freedom in the particle momentum. Accordingly, depending on the 
dimensionality, the following classification is made: 
 
Three-dimensional (3D) structure or bulk structure: No quantization of the particle 
motion occurs, i.e., the particle is free. 
 
Two-dimensional (2D) structure or quantum well: Quantization of the particle motion 
occurs in one direction, while the particle is free to move in the other two directions. 
 
One-dimensional (1D) structure or quantum wire: Quantization occurs in two 
directions, leading to free movement along only one direction. 
 
Zero-dimensional (0D) structure or quantum dot (sometimes called “quantum 
box”): Quantization occurs in all three directions. 
 
 
 
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Table 1. Nanostructures and their typical nanoscale dimensions 
 
 
 
 
 
 
 
 
Tradition has determined that reduced-dimensionality structures are labeled by the 
remaining degrees of freedom in the particle motion, rather than by the number of 
directions with confinement. 
 
1.3 Why we need Quantum Mechanics? 
 
As a spatial dimension approaches the atomic scale, a transition occurs from the 
classical laws to the quantum-mechanical laws of physics. Phenomena that occur on the 
atomic or subatomic scale cannot be explained outside the framework of quantum-
mechanical laws. 
 
For example, the existence and properties of atoms, the chemical bond, and the 
motion of an electron in a crystal cannot be understood in terms of classical laws. 
Moreover, many phenomena exhibited on a macroscopic scale reveal underlying 
quantum phenomena. It is in this reductionist sense that quantum mechanics is 
proclaimed as the basis of our present understanding of all natural phenomena studied 
and exploited in chemistry, biology, physics, materials science, engineering, etc. Physical 
behaviour at the nanoscale is accurately predicted by quantum mechanics, as 
represented by the Schrödinger equation, which therefore provides a quantitative 
understanding of the properties of low-dimensional structures. 
 
 In quantum mechanics, the trajectory of a moving particle loses its meaning when 
the distance over which potential energy varies is on the order of the de Broglie 
wavelength: 
 
2
2me
πλ = ĥ (1) 
 
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where ĥ is the reduced Planck constant, m is the mass of the particle, and E is its energy. 
In other words, a basic characteristic of all matter at the nanoscale is the manifestation of 
the wave–particle duality—a fundamental quantum-mechanical principle that states that 
all matter (electrons, nuclei, photons, etc.) behaves as both waves and particles.. The 
quantum effects of confinement become significant when at least one of the dimensions 
of a structure is comparable in length to the deBroglie wavelength. If at least one 
dimension of a solid is comparable to the de Broglie wavelength of the particle, a 
quantum-mechanical treatment of particle motion becomes necessary. In the 
Schrödinger description of quantum mechanics, an elementary particlee.g., an electron, a 
hole and a photon—or even a physical system such as an atom is described by a wave 
function Ψ (r t), which depends on the variables describing the degrees of freedom of the 
particle (system). Thesquare of the wave function is interpreted as the probability of 
finding a particle at spatial location ŕ ( , , )x y z= and time t. 
The wave function contains all of the information that may be obtained about a 
physical entity and is sufficient to describe a particle or syste m of particles. In other 
words, if the wave function of, for example, an ensemble of electrons in a device, is 
known, it is possible in principle—though limited by computational abilities—to 
calculate all of the macroscopic parameters that define the electr onic performance of that 
device. 
The wave function of an uncharged particle with no spin satisfies the Schrödinger 
equation 
 (2) 
where,
2 2 2
2
2 2 2x y z
∂ ∂ ∂
∇= + +
∂ ∂ ∂
 is the Laplacian operator, i= 1− , and V(r t) is the 
spatiotemporally varying potential influencing the particle’s motion. The particle’s mass 
m in the equation has to be carefully handled. For a particle (electron or hole) in a solid, 
this mass is its effective mass m, which is usually less than the mass of an isolated 
electron. In the above equation the action of Hamiltonian operator 
 on the wave function yields the total energy of the 
particle. The first part of ( , ) ( , )H r t r tψ is the kinetic energy, and the second part is the 
potential energy. For many real systems, the potential does not depend on time, ie 
V(r,t)=V(r). Then, the dependences on time and spatial coordinates of ( )zψ (r t) are 
separated as 
 (3) 
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Where ( )rψ is a complex-valued function of space only, and E is the energy of the 
system. Using this representation of the wave function in the Schrödinger equation, the 
time harmonic Shcrodinger equation is obtained and can be written as, 
 (4) 
Analytical solutions of the time-harmonic Schrödinger equation can be obtained for a 
variety of relatively simple conditions. These solutions provide an insight into the 
nature of quantum phenomena and sometimes provide a reasonable approximation 
of the behaviour of more complex systems—e.g., in statistical mechanics, 
molecular vibrations are often approximated as harmonic oscillators. Several of the more 
common analytical solutions are for a free (isolated) particle: a particle in a box, a finite 
potential well, 1D lattice, ring, or spherically symmetric potential; the hydrogen atom or 
hydrogen-like atom; the quantum harmonic oscillator; the linear rigid rotor; and the 
symmetric top. 
 
For many systems, however, there is no analytic solution to the 
Schrödinger equation, and the use of approximate solutions becomes necessary. 
Some commonly used numerical techniques are: perturbation theory, density functional 
theory, variational methods (such as the popular Hartree–Fock method which is the basis 
of many post-Hartree–Fock methods), quantum Monte Carlo methods, the Wentzel–
Kramers–Brillouin (WKB) approximation, and the discrete delta- potential method. The 
interested reader is encouraged to consult specialized books on these methods. 
 
 
2 INTRODUCTION ABOUT QUANTUM WELLS, 
QUANTUM WIRES AND QUANTUM DOTS 
 
The most significant nanostructures required to design nanoelectronic devices are 
Quantum Wells, Quantum Wires and Quantum Dots. They are the basic building blocks 
of nanoelctronic devices. In nanoelectronics also we are going to control the transfer of 
electrons. But how to confine them?, how to activate them?, how to fix the threshold 
level for conductance?. All these questions will be answered when we understand the 
physics of these three quantum structures. 
 
Before discussing about the three types of fundamental nanostructures, let us 
discuss the analogy of these structures for a basic understanding: 
 
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Consider the examination results of say 10th standard in the Tamil Nadu scenario 
in which around 6, 00,000 number of students appear every year. In this case consider the 
result of the whole state as the first case, results of a district, a school and a class room as 
the second, third and fourth cases respectively. Let us draw the bar diagram similar to the 
energy level diagram for number of students having scored marks in different bands like 
99-100, 98-99, 97-98 and so on. 
 
Just visualize this bar diagram for the whole state, district, school and a class, 
how it will look like? 
 
When you look at the state level mark band diagram, each band will be crowded 
and placed very close to each other. Here the spatial dimension of the state is larger and 
the number of students considered is more. This case is very similar to bulk materials 
where electrons are free to move in all the directions and no limitations or no 
confinement. In this case electronic energy level bands will be crowded as well as will be 
almost continuous. 
 
When you look at the district level mark band diagram, each band will be less 
crowded and placed close to each other but not like state level. This is because of the 
reduced dimension and in-turn lesser number of students. This case is very similar to 
Quantum Well where electrons are free to move only in two directions and confined to 
one direction. In this case electronic energy level bands will not be crowded like bulk and 
with an increased gap between bands. 
 
When you look at the school level mark band diagram, each band will be having 
still smaller number of students and bands are placed wider to each other. Here, further 
reduction in the dimension of the sample resulting in band widening with minimum 
number of students in each band. This case is very similar to Quantum Wire where 
electrons are free to move in only one direction and confined to two directions. In this 
case electronic energy level bands will be still widened and with much increased gap 
between bands. 
 
Finally, when you look at the Class level mark band diagram, each band will be 
having very few numbers of students and bands are placed much wider to each other. 
Here a class representing the smallest dimension in terms of sample considered. This 
case is an analogy for the Quantum Dot. Since Quantum Dot is a nano-sized particle 
where electrons are totally confined and cannot move anywhere. In this case electronic 
energy level bands are widened to the maximum. 
 
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The number students in a band and number electrons in a band are the analogy of 
this example. As we reduce the size of the materials more and more we can confine 
electrons more and more and so controlling, activating can be done at very low voltage 
and current lelvels. 
 
We can make use of these dots, wires and wells to fabricate a device say a 
transistor or a gate or a memory device and they can work with voltage levels less than 
0.2 V and current in the nano to pico Amps. 
 
Most of the nanoelectronic devices are based on the semiconductor nanostructures 
fabricated by tailoring the band gaps of desired level. The major focus of the band gap 
engineering is to design non- traditional devices with unusual electron transport and 
optical effects. 
 
When we think about how to engineer the band gap?, Quantum well is the first answer. 
 
Quantum Well? 
 
The term “well” refers to a semiconductor region that is grown to possess a lower 
energy, so that it acts as a trap for electrons and holes (electrons and holes gravitate 
towards their lowest possible energy positions). They are referred to as “quantum” wells 
because these semiconductor regions are only a few atomic layers thick; in turn, this 
means that their properties are governed by quantum mechanics, allowing only specific 
energies and band gaps. Because QW structures are very thin, they can be modified very 
easily. 
 
In other words 
 
Quantum wells are real-world implementation of the “particle in the box” problem; they 
act as potential wells for charge carriers and are typically experimentally realized by 
epitaxial growth of a sequence of ultrathin layers consisting of semiconducting materials 
of varying composition. 
 
 
 
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Two dissimilar semiconductors with different band gaps can be joined to form a 
heterojunction. The discontinuity in either the conduction or the valence band can be used 
to form a potential well. If a thin layer of a narrower-band gap material 'A' say, is 
sandwiched between two layers of a wider-band gap material 'B', then they form a double 
heterojunction. If layer 'A' is sufficiently thin for quantum properties to be exhibited, then 
such a band alignment is called a single quantum well. 
 
Additional semiconductor layers can be included in the heterostructure, for 
example a stepped or asymmetric quantum well can be formed by the inclusion of an 
alloy between materials A and B. 
 
Still more complex structures can be formed, such as symmetric or asymmetric 
double quantum well and multiple quantum wells or superlattices. The difference 
between the latter is the extent of the interaction between the quantum wells; in 
particular, a multiple quantum well exhibits the properties of a collection of isolated 
single quantum wells, whereas in a superlattice the quantum wells do interact. The 
motivation behind introducing increasingly complicated structures is an attempt to tailor 
the electronic and optical properties of these materials for exploitation in devices. 
 
All of the structures illustrated so far have been examples of Type-I systems. In 
this type, the band gap of one material is nestled entirely within that of the wider-band 
gap material. The consequence of this is that any electrons or holes fall into quantum 
wells which are within the same layer of material. Thus both types of charge carrier are 
localised in the same region of space, which makes for efficient (fast) recombination. 
 
In Type-II systems the band gaps of the materials, say 'A' and 'C', are aligned such 
that the quantum wells formed in the conduction and valence bands are in different 
materials. This leads to the electrons and holes being confined in different layers of the 
semiconductor. The consequence of this is that the recombination times of electrons and 
holes are long. 
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How it will look like? 
 
AGaAs/Ga1 -x Alx As layered structure and the in-plane motion of a charge 
carrier 
If the one-dimensional potential V(z) is constructed from alternating thin layers of 
dissimilar semiconductors, then the particle, whether it be an electron or a hole, can move 
in the plane of the layers. 
 
Further reducing the dimensionality of the electron's environment from a two-
dimensional quantum well to a one-dimensional quantum wire and eventually to a zero-
dimensional quantum dot. The dimensionality refers to the number of degrees of freedom 
in the electron momentum; in fact, within a quantum wire, the electron is confined across 
two directions, rather than just the one in a quantum well, and, so, therefore, reducing the 
degrees of freedom to one. In a quantum dot, the electron is confined in all three-
dimensions, thus reducing the degrees of freedom to zero. 
 
The number of degrees of freedom Df in the electron motion, together with the extent of 
the confinement Dc
Sl. 
No. 
, for the four basic dimensionality systems. 
 
 
System D Df c 
1 Bulk 0 3 0 3 
2 Quantum well 1 2 1 2 
3 Quantum wire 2 1 2 1 
4 Quantum dot 3 0 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
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2.1 Quantum Wire 
 
2.1.1 How to prepare Quantum Wire? 
 
A standard quantum well layer can be patterned with photolithography or perhaps 
electron-beam lithography, and etched to leave a free standing strip of quantum well 
material; the latter may or may not be filled in with an overgrowth of the barrier material 
(in this case, Ga(1-x)Alx
 
 
 
The following Fig. shows an expanded view of a single quantum wire, where clearly the 
electron (or hole) is free to move in only one direction, in this case along the y-axis. 
 
As ). Any charge carriers are still confined along the 
heterostructure growth (z-) axis, as they were in the quantum well, but in addition 
(provided the strip is narrow enough) they are now confined along an additional 
direction, either the x- or the y-axis, depending on the lithography. 
 
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2.2 Quantum Dot 
 
Quantum dots can again be formed by further lithography and etching, e.g. if a 
quantum well sample is etched to leave pillars rather than wires, then a charge carrier can 
become confined in all three dimensions, as shown in Fig. 
 
 
 
 
2.3 Two-Dimensional Structures: Quantum Wells 
 
In a 2D structure, particles are confined to a thin sheet of thickness L z
 
 along the z 
axis by infinite potential barriers that create a quantum well, as illustrated in the below 
figure. 
A particle cannot escape from the quantum well 0 z L≤ ≤ z and loses no energy 
on colliding with its walls z=0 and z=Lz. In real systems, this confinement is due to 
electrostatic potentials (generated by external electrodes, doping, strain, impurities, 
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etc.), the presence of interfaces between different materials (e.g., in core-shell 
nanocrystals), the presence of surfaces (e.g., semiconductor nanocrystals), or a 
combination of these agents. Motion of the particle in the other two directions (i.e., in 
the xy plane) inside the quantum well is free. It is generally accepted that quantum 
confinement of electrons by the potential wells of nano meter-sized structures provides 
one of the most powerful and versatile means to control the electrical, optical, 
magnetic, and thermoelectric properties of solid state functional materials. A 1D 
potential profile for electrons can be physically implemented by using two 
heterojunctions. The below figure shows a quantum well structure. 
 
 
 
 
 
 
 
 
 
 
 
A GaAs Quantum well inserted between two AluGa1-u
 (5) 
 
 
As as barrier layers. The 
layer of GaAs is a quantum well because the barrier layers are made of a material with a 
larger bandgap than GaAs; the energy difference between the valence band and 
conduction band in a semiconductor is called the bandgap. By adjusting the aluminum 
content of the barrier layers and the thickness of the GaAs layer at the time of growth, 
quantum wells with electronic properties tailored to the user’s specifications can be 
created. This practice is referred to as quantum engineering. 
The infinitely deep 1D potential well is the simplest confinement potential to treat in 
quantum mechanics. In classical mechanics, the solution to the problem is trivial, since 
the particle will move in a straight line and always at the same speed until it reflects 
from a wall at an equal but opposite angle. However, in order to find the quantum -
mechanical solution, many fundamental concepts need to be introduced. 
After restricting analysis to an infinitely deep 1D potential well aligned alon g the z axis 
will be of the form 
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 (6) 
 
Thetime harmonic Schrodinger wave equation can be written as 
 
Outside the chosen potential well, the potential is infinite; hence, the only possible 
solution is ( )zψ =0, 0z ≤ or zz L≥ , which in turn implies that all values of the energy 
E are allowed. Within the infinitely deep potential well, the Schrödinger equation 
simplifies to 
 (7) 
Note that ( )zψ must be continuous inside the well and must be zero on both walls. 
Furthermore, since the particle must exist somewhere on the z axis, and because 
2( ( ( )))mod zψ is the probability of finding the particle at a particular value of z, it follows 
that 2(mod( ( ))) 1zψ
+∞
−∞
=∫ . With these stipulations the solutions of the simplified 
Schrodinger equations are many and these solutions are called Eigen functions and 
maybe written as 
2 / sin( )
z
z
n z
z
n zL
L
π
ψ = , 0 , 1, 2,3......z zz L n< < = (8) 
The index nz ( ) 0zψ ==0 is ruled out since then for all ( , )z∈ −∞ ∞ , corresponding to the 
case where there is no particle in the infinitely deep potential well. Negative values of nz
 (9) 
Where, A
 
are also neglected, since they merely change the sign of the sine function. The 
complete solution is the superposition of all the Eigen functions and is given as 
n are the coefficients of expansion i ndicating the relative importance of the 
eigen functions in the solution. Each eigen function describes a state of electron 
confinement. The Eigen energy associated with nz
 (10) 
Where, n
’th Eigen function is given by 
z is called principle quantum number. 
 
 
 
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2.3.1 Zero-Point Energy 
 
Quantum size effects are apparent for reduced size. Thus, a particle confined to an 
infinitely deep potential well has only specific (discrete) energy levels, and the zero-
energy level is not one of them. The lowest possible energy level of the particle is usually 
called the zero-point energy or confinement energy, which can be understood in terms of 
the Heisenberg uncertainty principle as follows: Because the particle is constrained 
within a finite region, the variability in its position has an upper bound. As the variability 
in the particle’s momentum cannot then be zero due to the uncertainty principle, the 
particle must contain some energy that increases as the width Lz
 
 
 
Thus, the potential V(r) is written as the sum of a 2D confinement potential (yz plane) 
plus a potential along the wire (x axis) as 
 of the infinitely deep 
potential well decreases. 
 
3 ONE-DIMENSIONAL STRUCTURES: QUANTUM WIRES 
AND NANOWIRES 
 
For 1D structures—usually called quantum wires , although other systems such as rods, 
belts, and tubes also fall within this category—it is possible to decouple the motion along 
the length of the wire, which is taken to be along the x axis, as shown in Fig. 
 (11) 
Accordingly, the wave function is written as the following product of two components: 
 (12) 
Substituting the above two equations in time harmonic Schrodinger equation, we get 
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 (13) 
From this equation it is possible to obtain the following two autonomous equations of 
motion: 
(14) 
The above equation is satisfied by a plane wave of the form 
being the particle’s momentum along the x axis, thus 
leading to the dispersion relationship 
 (15) 
This equation gives the energy level along the x axis, in which direction the particle is 
free to move, and resembles that of a 3D structure (wherein confinement is not possible). 
The potential V(2,3)
 (16) 
Outside the rectangular region, the wave function 
(y,z) is given by 
(2,3) ( , )y zψ is identically zero. 
Therefore the second equation of motion needs to solved in the rectangular region and 
can be written as 
 (17) 
The form of the potential in this equation allows the wave function dependencies on y 
and z to be decoupled, ie., . The method of separation of 
variables can be used which allows the energy superposition E(2, 3)=E2 + E3 and leads to 
the following decoupled equations. 
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 (18) 
 
These equations are identical to the Schrodinger equation in the deep potential well and 
are subject to similar boundary conditions. Basically, since the potential energy outside 
the wire is infinite, the standard boundary condition of continuity of the wave function at 
the walls implies that the product of 2 ( )yψ and 3( )zψ must be zero on the walls. Hence, 
the Eigen solutions are 
 
Thus, 
 (19) 
The quantum states in a quantum wire are described by two principal quantum numbers 
(ny and nz
 (20) 
 
Where, n
), while only one principal quantum number is needed for a quantum well. 
Similarly to that for a quantum well, the Eigen energy in a quantum wire increases for 
decreasing size. Also, a lower effective mass results in a larger Eigen energy for a given 
size. 
The corresponding Energy levels are given by 
y and nz are principle quantum numbers. 
 
 
 
 
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4 Zero-Dimensional Structures: Quantum Dots and 
Nanodots 
 
A special 0D structure is now considered: the cuboid quantum dot, more often 
designated as the quantum box shown in Fig. This special case can be used for a 
qualitative description of the response of quantum dots of many shapes. Zero-
dimensional structures of other shapes, such as spherical quantum dots, require numerical 
solution of the Schrödinger equation. The quantum box is a generalization of a quantum 
wire of rectangular cross- section, in that there is additional confinement along the x axis 
to 0 xx L< < . This additional confinement removes the only degree of freedom 
remaining in the particle’s momentum, thus localizing it in all three directions. 
According to the Heisenberg uncertainty principle, increased spatial confinement will 
result in increased energy of the confined states. 
 
 
For simplicity, let the potential be zero inside the quantum box but infinite 
everywhere else; i.e., 
 (21) 
The 3D time-harmonic Schrödinger equation within the quantum box becomes 
 (22) 
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The method of separation of variables applies, leading to Eigen functions described by 
three principal quantum numbers (nxnynz
 (23) 
The Eigen energy for a specific Eigen function is given by 
 ) as follows: 
 (24) 
Of fundamental importance is the fact that Enxnynz is the total particle energy because of 
3D confinement, in contrast with the previous two cases in which the solutions for 
the confined states in a quantum well and a quantum wire gave us only the Eigen 
energies associated with transverse confinement. The discrete energy spectrum in 
a quantum box and the lack of free propagation are the main features 
distinguishing quantum boxes from quantum wells and quantum wires. As these 
features are typical for atoms, quantum dots and quantum boxes are often called 
artificial atoms. 
A remarkable feature of a quantum box is that whentwo or more of the 
dimensions are the same (e.g., Lx= Ly ), more than one wave function 
corresponds to the same total energy. When exactly two wave functions have 
the same energy, that energy level is said to be doubly degenerate. Degeneracy 
results from the symmetry of the structure. 
 
 
5 Quantum conductance 
Conductance is the measure of transport of electrons in the medium. Then, what is 
mean by quantum conductance? It is the limit of the material in low dimension. 
Before discuss the quantum conductance, we know the transport properties in the 
classical and quantum level. 
In classical (i.e., size not in the range of nanometers) obeys the scattering 
transport mechanism. Moreover, scattering transport failed in quantum level called as 
ballistic transport. Table shows the difference in both transports as follows 
 
 
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Ballistic Transport Scattering transport 
In low dimensional material, travelling of 
electron without scattering is known as 
Ballistic transport 
Normally in three dimensional material, 
travelling of electrons deflected by various 
centres like stable ion, defects, electrons, etc. 
causes the restriction for the flow is known as 
scattering transport 
Energy and Momentum does not change Energy and Momentum changes 
Electrons are not necessarily in equilibrium 
with the material 
Electrons are equilibrium with the material 
Electrons are not restricted to the lowest 
unoccupied energy states in the material 
Scattering can makes an electron to transport 
from high energy state to low energy states 
There is no interaction between the electrons, 
hence electrons are treated to move in free 
space 
Electron interaction attributed to the scattering 
effect, classified as elastic and inelastic 
Elastic scattering: This affect only the 
momentum of the electron, not the energy 
Inelastic scattering : This affect both the 
energy and momentum 
 
Consider the quantum wire a two dimensional material, L is the length of the 
wire, a potential flow from one end to other; it is shown in the Figure. 
 
To flow a charge then should be difference in Fermi level (Chemical Potential) at the two 
end of wire. 
 
 L 
V 
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Charge will flow from the contact to quantum wire, depends upon the Fermi level 
difference and it leads to the accumulation and depletion of electrons from the end of the 
contact. It is shown in the Figure. 
 
where is the Chemical potential of Drain region, is the chemical potential of Source 
region, 
 
is the potential difference between the two contacts. 
Let the potential difference 
(25) 
 
Total current in the Quantum wire can be represented as 
 (26) 
 
We know that total energy expressed as 
 (27) 
 
Differentiate the equation with respect to wave vector, we have 
 (28) 
Upon substitution of wave vector value, we get the relation 
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 (29) 
Hence, velocity of the electrons in the quantum wire can be calculated from the equation, 
also we know that by classically distance travelled by unit time. 
Therefore, 
 
 (30) 
 (31) 
Now, number of electrons present in the quantum wire is the number of states between 
the two levels and is represented as 
 (32) 
Substitute Eqs. 8& 7 in 2 
 
 (33) 
 (34) 
 (35) 
 (36) 
Quantum wire of conductance has some constant with applied voltage, it is compared 
classical mechanics, then it follows the ohms law, resistance will be 
 (37) 
By substituting, the constants value h(Planck’s constant) and q (charge) 
 (38) 
where quantum conductance is the inverse of resistance, then 
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 (39) 
If the quantum wire has multiple anode, then the total conductance 
 (40) 
Thus, Ballistic conductance principle also has a resistance, which is independent of 
length of the wire. But, velocity of electrons depends on the state (Dimension) of the 
wire. 
 
6 QUANTUM CAPACITNECE 
 
Quantum capacitance (density) is a physical value first introduced by Serge Luryi 
(1988)[1] to describe the 2D-electronic systems in silicon surfaces and GaAs junctions. 
This capacitance was defined through standard density of states in the solids. Quantum 
capacitance could be used in the quantum Hall Effect (integer and fractional) 
investigations as a new approach which uses quantum LC circuit. 
 The quantum capacitance in grapheneis given by 
 
CQL= ᶓCQO ~ 2,187.10^-3 F/m^2 (41) 
 
6.1 QUANTUM CONDUCTANCE 
 
The conductance is defined as Current per potential difference 
 
G = I/ΔV (42) 
Current is defined as: 
 
I =nq/Δt (43) 
 
Where 
nis the number of electrons and 
q is the charge of an electron 
 
Potential difference is defined as: ΔV =ΔU/q 
 
Wherenis the number of electrons and 
U is the electrostatic charge 
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In the quantum system electrons propagate as waves. 
o The electrons get energy from collisions with ions. 
o The ions have energy of kT and only the electrons that are within kT of 
the Fermi level can be promoted to the conduction band. 
o The exclusion principle dictates that there will only be a few electrons at 
that level. 
o Thermal vibrations cause scattering of the electrons. 
 
6.2 Effect of Quantum Conductance 
Consider a clock speed of 3GHz.This would require as many as 3·109electrons to 
pass though a single molecule transistor. The current required to support this clock speed 
would be focused onto a very small area which could exceed the energy of the molecular 
bonds. 
 
What we need 1A = 1C/1s, V=IR, V=1J/1C, R=1027.3Ω, e‐= 1.602·10 ‐19C, 1 amp = 
6.242·1018e‐/s 
 
 
7 QUANTUM HALL EFFECT 
 
The quantumHall effect (or integer quantum Hall effect) is a quantum-
mechanical version of the Hall effect, observed in two-dimensional electron 
systems subjected to low temperatures and strong magnetic fields, in which the 
Hall conductivity 
 (44) 
σ takes on the quantized values 
where e is the elementary charge and h is Planck's constant. 
 
The prefactor ν is known as the "filling factor", and can take on either integer (ν 
= 1, 2, 3, .. ) or rational (ν = 1/3, 2/5, 3/7, 2/3, 3/5, 1/5, 2/9, 3/13, 5/2, 12/5 ...) values. 
The quantum Hall effect is referred to as the integer or fractional quantum Hall effect 
depending on whether ν is an integer or fraction respectively. The integer quantum Hall 
effect is very well understood, and can be simply explained in terms of single-particle 
orbitals of an electron in a magnetic field (see Landau quantization). The fractional 
quantum Hall effect is more complicated, as its existence relies fundamentally on 
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electron–electron interactions. It is also very well understood as an integer quantum Hall 
effect, not of electrons but of charge-flux composites known as composite fermions 
 
 (45) 
 
Where7.1 Integer Quantum Hall Effect – Landau Levels 
 
In two dimensions, when classical electrons are subjected to a magnetic field 
they follow circular cyclotron orbits. When the system is treated quantum mechanically, 
these orbits are quantized. 
 The energy levels of these quantized orbitals take on discrete values: 
 
 
 
 ωc = eB/m 
These orbital’s are known as
is the cyclotron frequency. 
 
 Landau levels, and at weak magnetic fields, their 
existence gives rise to many interesting "quantum oscillations" such as the Shubnikov–
de Haas oscillations and the de Haas–van Alphen effect (which is often used to map the 
Fermi surface of metals). For strong magnetic fields, each Landau level is highly 
degenerate (i.e. there are many single particle states which have the same energy En). 
Specifically, for a sample of area A, in magnetic field B, the degeneracy of each 
Landau level is 
 
 (46) 
 
Wheregs represent a factor of 2 for spin degeneracy, and φ0 is the magnetic flux 
quantum. 
 For sufficiently strong B-fields, each Landau level may have so many states that 
all of the free electrons in the system sit in only a few Landau levels; it is in this regime 
where one observes the quantum Hall Effect. 
 
 
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• Integral Quantum Hall effect can be accurately modeled as a non-interacting 
electron gas in a magnetic field 
• The Fractional Quantum Hall effect involves composite fermion quasi-particles 
which are electrons with attached flux quanta 
 
Transport properties of semiconductor are a complicated process because of their 
dependence on the actual size of the samples. Based on the size of the semiconductor 
structure the need for quantum treatment can be desired. In order to define the limits of 
various transport regimes one may scale the size of the sample against the de-Broglie 
wavelength. This De- Brogliewavelength for an electron travelling with the thermal 
kinetic energy in a semiconductor as 
 (47) 
whereh=Planck’s constant 
 p=momentum 
 E=energy 
 =de Broglie wavelength of a free electron 
 = electron effective mass in semiconductor 
The room temperature De-Broglie wavelength of a free electron is ~76Å, and that 
of electron in GaAs is ~295Å.For several semiconducting materials, the D-B wavelength 
is comparably with the size of semiconductor structures in the nanostructure limit, hence 
a Quantum mechanical treatment of the transport properties in nanostructures must be 
considered.Quantum Transport in low dimensional is very interesting and offers the 
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investigation remarkable properties such as the Quantum Hall effect, shubnikov-de Haus 
effect, ballistic transport and the fractional quantum hall effect. As we know, the 
shubnikov-de Hauseffectwill help us to precisely measurer the carrier concentration 
formed at the hetrojunctions interfaces. 
 
The investigation of two dimensional system in a perpendicular magnetic field 
provides quantisation hall resistance, which results from the quantisation of energy in 
series of landau levels. Hence understanding these effects will give a clear picture of 
transport properties especially conductance of low dimensional conducting 
semiconductor materials, which are the building blocks of nanoelectronic devises. 
 
7.2 Quantum Hall Effect 
 
As a starting point, it is very beneficial to understand the Drude classical model of 
the magnetoresistance in semiconductors. The classical equation of motion of an electron 
in the presence ofmagnetic (B) and electric ( ) fields can be expressed as 
 (48) 
where is the drift velocity and is the scattering time. The magnetic field is applied 
along the z axis, and and are assumed to vary with time as . This equation 
can be expressed in its three components as 
 
(49) 
 
By multiplying Eq. 3 by the carrier concentration and the electron charge– , and 
comparing the results with the relation 
 (50) 
where is the conductivity tensor, one can obtain the components of the conductivity 
tensor as 
 
(51) 
 
 
where is the conductivity in the absence of the magnetic field. For the 
steady-state case where , the conductivity tensor can be written as 
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 (52) 
Thus, the conductivity in a two dimensional system in the presence of a magnetic field 
applied along the z direction can be expressed as 
 (53) 
and the resistivity tensor is related to the conductivity tensor as 
 (54) 
The resistivity tensor can now be written as 
 (55) 
The condition implies that the carriers are collision less. By applying 
this condition to Eq. 5, one can obtain and . In the presence of 
collisions, where , we have 
 
(56) 
 
where these conductivity components are simply the sum of the collision and 
collisionless parts. When the Fermi energy level is between Landau levels labelled n and 
n +1,no elastic scattering can occur at low temperatures (T ≤ 4.2K), and the energy 
separation between consecutive Landau levels is . This case is thus equivalent to the 
condition , which gives , and is given by its classical collisionless 
value. From the density of states per Landau level, , one can write the carrier 
density ns
 
 as , where n is the nth Landau level. The Hall conductivity can 
be expressed as 
(57) 
 
This equation shows that the Hall resistivity takes quantized values of 25812.87/n 
whenever the Fermi energy level lies between filled-broadened Landau levels. This is 
called the quantum Hall effect. 
The quantum Hall effect is observed for integer filling factors as . 
However, at low temperature (T< 5.2 K), a fractional value of the filling factor v has been 
observed for the lowest Landau level in many hetrojunction systems with high mobility. 
In this case, v can take on values of p/q. where p and q are integers. This is called the 
fractional quantum Hall effect. Laughlin (1983) provided an explanation of the fractional 
quantum Hall effect based on the condensation of electrons or holes into a collective 
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ground state due to electron-electron or hole-hole interactions. This ground state is 
separated from the nearest excited by an energy of , where is the Landau 
magnetic length. The possibility of a repulsive interaction between carriers of the same 
charge, leading to a condensation, is related to the two dimensional character of the 
system. The condensed phase consists of quasi-particles called anyons, of fractional 
charge , where that follow statistics intermediate between Fermi-Dirac 
and Bose-Einstein formalisms. 
 
8 REFERENCES 
 
1. Tsui, D. C., H. L. St¨ormer, J. C. M. Huang, J. S. Brooks, and M. J. 
Naughton.Observation of a Fractional Quantum Number.Phys. Rev. B28 (1983): 
2274. 
 
2. Laughlin, R. B..Anomalous Quantum Hall Effect: An Incompressible Quantum 
Fluidwith Fractionally Charged Excitations. Phys. Rev. Lett. 50 (1983): 1395. 
 
 
 
 
 
 
 
	1. INTRODUCTION
	1.1 What are Low-Dimensional Structures?
	1.2 Classification of Low-dimensional Materials
	1.3 Why we need Quantum Mechanics?
	2 INTRODUCTION ABOUT QUANTUM WELLS, QUANTUM WIRES AND QUANTUM DOTS
	2.1 Quantum Wire
	2.1.1 How to prepare Quantum Wire?
	2.2 Quantum Dot
	2.3Two-Dimensional Structures: Quantum Wells
	2.3.1 Zero-Point Energy
	3 ONE-DIMENSIONAL STRUCTURES: QUANTUM WIRES AND NANOWIRES
	4 Zero-Dimensional Structures: Quantum Dots and Nanodots
	5 Quantum conductance
	6 QUANTUM CAPACITNECE
	6.1 QUANTUM CONDUCTANCE
	6.2 Effect of Quantum Conductance
	7 QUANTUM HALL EFFECT
	7.1 Integer Quantum Hall Effect – Landau Levels
	7.2 Quantum Hall Effect
	8 REFERENCES