DEPARTMENT OF PHYSICS AND NANOTECHNOLOGY SRM INSTITUTE OF SCIENCE AND TECHNOLOGY
21PYB102J –Semiconductor Physics and Computational Methods
MODULE 5
Density of states in 2D, 1D and 0D
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Density of States
• The density of states function describes the number of energy states that are available in a system and is essential for determine the carrier concentrations and energy distributions of carriers within a semiconductor.
• In semiconductors, the free motion of carriers is limited to two, one and zero spatial dimensions. When applying semiconductor statistics to systems of these dimensions, the density of states in quantum well (2D), quantum wires (1D) and quantum dots (0D) must be known.
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Density of states in 3D (Bulk Material)
The density of states is defined as the number of allowed electronic energy states per unit energy range per unit volume of the material.
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Density of states in
lower-dimensional systems
• Three-dimensional electron or hole obtained by doping semiconductors are not ideal for studying quantum effects for two reasons: (i) they are strongly disordered owing to the background of ionized impurities and (ii) the most quantum effects are more pronounced in lower-dimensional systems than those of bulk constituents.
• Therefore, reduction in the dimensionality of a physical system has profound consequences on its profile and new types of electronic and photonic devices can be designed. These devices make use of electron motion through potentials that change rapidly on a length scale comparable to the wavelength associated with the electron and they operate on the rules of quantum mechanics.
• The low dimensional semiconductor systems play a critical role in determining the properties of materials due to the different ways that electrons interact in two-dimensional, one-dimensional and zero dimensional structures.
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Density of states in
lower-dimensional systems
• A low-dimensional system is one where the motion of microscopic degrees-of-freedom, such as electrons, phonons or photons, is restricted from exploring the full three dimensions of the present world.
• In the low dimensional quantum systems such as Quantum well, Quantum wire and Quantum dot, the charge carriers are free to move in two, one and zero dimensions respectively.
• This high confinement brings out new effects of great technological potential applications. Quantum mechanics plays a major role as the semiconductor size approaches the nanoscale.
• The main advantages of these low dimensional semiconductor systems are in the realizations of important devices, like the double heterostructure lasers with low threshold at room temperature, high effective LEDs, bipolar transistors, p-n-p-n switching devices, high electron mobility transistors (HEMT) and many other optoelectronic devices.
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Density of states in 1D
• Quantum effects in systems which confine electrons to regions comparable to their de Broglie wavelength. When such confinement occurs in two dimensions only (say, by two restrictions on the motion of the electron in the z- and y-directions), with free motion in the x direction, a one-dimensional system is created. 6
Density of states in 1D
• Consider a systems where electrons are free to move in one direction and confined in the other two directions (Quantum wire).
• In one dimension two of the k-components are fixed, therefore the area of k-space becomes a length and the area of the annulus becomes a line.
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Density of States in 1D 8
Density of States in 1D 9
Density of States in 1D 10
Density of states in 1D
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Density of states in 2D
• Quantum effects arise in systems which confine electrons to regions comparable to their de Broglie wavelength. When such confinement occurs in one dimension only (say, by a restriction on the motion of the electron in the z-direction), with free motion in the x- and y-directions, a two-dimensional system is created.
• Consider a slab of material that has macroscopic dimensions in the x- and y directions while the thickness is small (in the nanometer range-Quantum Well).
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Density of states in 2D
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Density of states in 2D 14
Density of states in 2D 15
Density of states in 2D
• It is important to notice that the 2D density of states is independent of the energy. However, DOS depends on the number of levels and is thus a sum of the contributions from the discrete levels appearing as a result of the quantization.
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Density of states in 0D
• Electrons can be confined in all three dimensions in a dot. The situation is analogous to that of a hydrogen atom: only discrete energy levels are possible for electrons trapped by such a zero-dimensional potential. The spacing of these levels depends on the precise shape of the potential.
• When considering the density of states for a 0D structure (Quantum dot), no k – space is available as all dimensions are reduced.
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Density of states in 0D
• Therefore DOS of 0D can be expressed as a delta function
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Conclusion
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DEPARTMENT OF PHYSICS AND NANOTECHNOLOGY SRM INSTITUTE OF SCIENCE AND TECHNOLOGY
21PYB103J –Semiconductor Physics and Computational methods
MODULE 5
Lecture – 50
Introduction to Low dimensional systems
Quantum Well, Wire, Dot
21PYB103J Module-V Lecture-2
Nanoscience & Nanotechnology
What is happening at a very, very small length scale?
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What is Nano ? •
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Actual physical dimensions relevant to Nanosystem
Nanoscience
0.1nm 1nm 10nm 100nm 1μm 10 μm
Size and shape dependent
properties
Nanometer scale : The length scale where corresponding property is size & shape dependent.
21PYB103J Module-V Lecture-25
Surface to Volume Ratio
Increases
As surface to volume ratio increases
• A greater amount of a substance
comes in contact with surrounding
material.
• This results in better catalysts,
since a greater proportion of the
material is exposed for potential
reaction.
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What’s interesting about the nanoscale?
• Nano sized particles exhibit different properties than larger particles of the same substance.
• Nano sized particle exhibit size & shape dependent properties.
How do properties
change at the
Nanoscale ?
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Optical Properties: Colour of Gold
• Bulk gold
appears yellow in colour.
• Nano sized gold
appears red in colour.
The particles are so small that electrons are not free to
move about as in bulk gold Because this movement is
restricted, the particles react differently with light.
12 nanometer gold clusters of
particles look red.
Sources:
http://www.sharps-jewellers.co.uk/rings/images/bien-hccncsq5.jpg http://www.foresight.org/Conferences/MNT7/Abstracts/Levi/
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Nanoscience: Nanometer scale science
• A part of science that studies small stuff
So, what is Nano science ?
• It is not only Biology.
• It is not only Physics .
• It is not only Chemistry.
• It is all sciences that work with the very small.
⮚Nanoscience is not physics, chemistry, engineering or biology. It is all of them. S.M. Lindsay, Introduction to Nanoscience,
Oxford University Press (2009).
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Interdisciplinary
• Physicists: physical forces between the individual atoms composing them – quantum effects
Chemists : The interaction of different molecules is governed by chemical forces.
Biologists : creation of small devices (encoding informations in DNA to perform multitasks
Computer Scientists : Steady miniaturization : - Moore’s Law and its corollaries, the phenomena whereby the price performance, speed, and capacity of almost every component of the computer. Electrical Engineers : a steady supply of power. A control of electric signals is also vital to transistor switches and memory storage. Mechanical Engineers: nanolevel issues such as load bearing, wear, material fatigue, and lubrication
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What makes the nanoscale special?
1) High density of structures is possible with small size.
2) Physical and chemical properties can be different at the nano-scale (e.g. electronic, optical, mechanical, thermal, chemical).
3) The physical behavior of material can be different in the nano-regime because of the different ways physical properties scale with dimension (e.g. area vs. volume).
Prof. Richard Feynman
“There’s plenty of room at the bottom”
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21PYB103J Module-V Lecture-2
Dr. Richard P. Feynman
• “Why cannot we write the entire 24
volumes of the Encyclopedia
Britannica on the head of a pin?”
Dr. Richard Feynman, one of
America’s most notable physicists,
1918-1988.
Figure 1.11: Richard Feynman.
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Physical/chemical properties can change as we approach the nano-scale
Melting point of gold particles
Fluorescence of semiconductor nanocrystals
Decreasing crystal size
M. Bawendi, MIT: web.mit.edu/chemistry/nanocluster
Evident, Inc.: www.evidenttech.com K. J. Klabunde, 2001
By controlling nano-scale (1) composition, (2) size, and (3) shape, we can create new materials with new properties �� New technologies
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The Lycurgus Cup
A Roman
Nanotechnology
Reflected, transmitted
The Lycurgus Cup represents one of the outstanding achievements of the ancient glass industry. This late Roman cut glass vessel is extraordinary in several respects, firstly in
the method of fabrication and the exceptional workmanship involved and secondly in terms of the unusual optical effects displayed by the glass. Chemical analysis showed the glass to be of the soda-lime-silica type, similar to most other Roman glass (and to modern window and bottle glass) containing in addition about 0.5% of manganese. In addition, a number of trace elements including silver and gold make up the final 1%. It was further suggested that the unique optical characteristics of the glass might be connected with the presence in the glass of colloidal gold
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• The semiconductors like PbS, GaAs, CdS etc., can be synthesized in the nanometer level and they are called as semiconductor quantum dots. Their properties like band gap, luminescence etc., always differs from their bulk counterpart.
The quantum structures are useful in the fabrication of high efficiency solar cells, infrared detectors, quantum dot lasers etc.
• Properties of Nanomaterials
Unique properties
They have very high magneto resistance
They have lower melting point, high solid state phase transition pressure, lower Debye temperature and high self diffusion coefficient
They have high catalytic activity and lower ferroelectric phase transition temperature
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Variation of physical properties with size
It is well established that mechanical, electrical, optical, chemical, semi conducting and magnetic properties of a material depend strongly upon the size and the arrangement of the constituent clusters or grains.
(i) Electron affinities and chemical properties
Variation in electronic properties with size occurs only when there is a variation in inter particle spacing and geometry. As the size is reduced from the bulk, the electronic bands in metals become narrower and the delocalized electronic states are transformed to more localized molecular bonds.
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• Variation in electronic properties with size occurs only when there is a variation in inter particle spacing and geometry.
As the size is reduced from the bulk, the electronic bands in metals become narrower and the delocalized electronic states are transformed to more localized molecular bonds.
• Fig shows the ionization potential and
reactivity of Fen clusters as a function of
size.
• The ionization potential are higher at
smaller sizes than at the bulk work
function .
• The large surface – to – volume ratio and
the variation in geometry and electronic
structure have a strong effect on catalysis
properties.
Ionization potential and reactivity of Fen clusters as a function of size
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(ii) Magnetic properties
• Nano particles of magnetic and even non magnetic solids exhibit a totally new class of magnetic properties.
• Table gives an account of magnetic behavior of very small particles of various metals.
• Ferro magnetic and anti ferromagnetic multilayers have been found to exhibit Giant Magneto Resistance (GMR).
• Small particles differ from the bulk in that these atoms will have lower co-ordination number.
• From the Fig, it is inferred that the small particles are more magnetic than the bulk material
Metal
Bulk
Cluster
Na, K
Paramagnetic
Ferromagnetic
Fe, Co, Ni
Ferro magnetic
Super paramagnetic
Gd, Tb
Ferromagnetic
Super paramagnetic
Rh
Paramagnetic
Ferromagnetic
Table Magnetism in bulk and in nano particles
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Change in bulk magnetic moment versus co- ordination number
(iii) Mechanical behaviour
Melting point of gold as a function of grain size
From the Fig. it is evident that the melting point reduction is not really significant until the particle size is less than about 10nm.
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• Nanophase metals with their exceptionally small grain size are found to be exceptionally strong.
• It is because clusters and grains in nanophase material are mostly free from dislocations.
• The variation of hardness with diameter of copper nano crystals is shown in Fig.
• From the Fig. it is revealed that when the grains size was 50nm in diameter, the copper was twice as hard as usual.
• Thus the material in nano phase has very high strength and super
hardness.
Fig. Strength of nanophase copper as a function of grain size
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• The basic principles of nanotechnology is positional control. • At the macroscopic scale, it is easy to hold parts in our hands and assemble them by properly positioning them with respect to each other.
• At the molecular scale, the idea of holding and positioning molecules is new and almost shocking.
• It is possible to continue the revolution in computer hardware right down to molecular gates and wires -- something that today's lithographic methods (used to make computer chips) could never hope to do.
• One can inexpensively make very strong and very light materials: shatterproof diamond in precisely the shapes we want, by the ton, and over fifty times lighter than steel of the same strength.
• It is possible to make surgical instruments with high precision and deftness that one could operate on the cells and even molecules from which we are made - something well beyond today's medical technology
• Nanotechnology makes almost every manufactured product faster, lighter, stronger, smarter, safer and cleaner.
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• The general synthetic path
ways to synthesize
nanomaterials are top down and bottom-up
approach
• In the later method,
chemistry plays a unique
role in assembling and
building up nanometric
units from smaller ones.
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Electrical Measurements:
⮚ Measurements of electrical quantities, such as voltage, impedance, current, AC frequency and phase, power, electric energy, electric charge, inductance, and capacitance
⮚ Electrical measurements are among the most widely performed types of measurement.
⮚ The resistivity measurements can be studied by different techniques
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Classification
• Classification is based on the number of dimensions, which are
• not confined to the nanoscale range (<100 nm). • (1) zero-dimensional (0-D), (quantum dot & spherical) • (2) one-dimensional (1-D), (nanorods, nanowires, nanofibers, nanotubes)
• (3) two-dimensional (2-D), and (flat membranes, nanosheets, nanodisc)
• (4) three-dimensional (3-D). (bulk materials) 21PYB103J Module-V Lecture-2 26
Progressive generation of nanostructures
Well : - e-s move only in 2D
Wire : - only in 1 D
Dots: - confined in all directions, 3
movement
curvilin
rectang
ular
Nanomaterials:
Nanomaterials or nanophase
materials are the materials which are made of grains that are about 100nm in diameter and contain less than few ten thousands of atoms
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Quantum well, Quantum wire and Quantum dots
•When the size or dimension of a material is continuously reduced from a large or macroscopic size, such a metre or centimetre, to a very small size, the properties remain the same at first, then small changes begin to occur, until finally when the size drops below100 nm, dramatic changes in properties can occur.
•If one dimension is reduced to the nanorange while the other dimensions remain large, them we obtain a structure known as quantum well.
•If two dimensions are so reduced and one remains large, the resulting structure is referred to as a quantum wire. • The extreme case of this process of size reduction in which all three dimensions reach the low nanometer range is called a quantum dot.
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• The word quantum is associated with the above three types of nanostructures because the changes in properties arise from the quantum mechanical nature of physics in the domain of the ultra small.
The above fig. represents the processes of diminishing the size for the case of rectilinear geometry and the corresponding reductions in curvilinear geometry.
The conduction electrons are confined in a narrow dimension and such a configuration is referred as quantum well.
A quantum wire is a structure such as a copper wire that is long in one dimension, but has a nanometer size as its diameter. In this case, the electrons move freely along the wire but are confined in the transverse directions.
The quantum dot may have the shape of a tiny cube, a short cylinder or a sphere with low nanometre dimensions.
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TECHNOLOGICAL ADVANTAGES OF NANOTECHNOLOGY AND NANOMATERIALS 1. IMPROVED TRANSPORTATION
• Today, most airplanes are made from metal despite the fact that diamond has a strength-to-weight ratio over 50 times that of aerospace aluminum.
• Diamond is expensive, it is not possible to make it in the required shapes, and it shatters. Nanotechnology will let us inexpensively make shatterproof diamond in exactly the shapes we want.
• Nanotechnology will dramatically reduce the costs and increase the capabilities of space ships and space flight.
• The strength-to-weight ratio and the cost of components are absolutely critical to the performance and economy of space ships: with nanotechnology, both of these parameters will be improved.
• Nanotechnology will also provide extremely powerful computers with which to guide both those ships and a wide range of other activities in space.
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2. ATOM COMPUTERS
• Today, computer chips are made using lithography -- literally, "stone writing."
• If the computer hardware revolution is to continue at its current pace, in a decade or so we'll have to move beyond lithography to some new post lithographic manufacturing technology. Ultimately, each logic element will be made from just a few atoms.
• Designs for computer gates with less than 1,000 atoms have already been proposed -- but each atom in such a small device has to be in exactly the right place.
• To economically build and interconnect trillions upon trillions of such small and precise devices in a complex three dimensional pattern we'll need a manufacturing technology well beyond today's lithography: we'll need nanotechnology.
• With it, we should be able to build mass storage devices that can store more than a hundred billion billion bytes in a volume the size of a sugar cube;
• RAM that can store a mere billion billion bytes in such a volume; and massively parallel computers of the same size that can deliver a billion billion instructions per second.
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3. MILITARY APPLICATIONS:
• Today, "smart" weapons are fairly big -- we have the "smart bomb" but not the "smart bullet.“
• In the future, even weapons as small as a single bullet could pack more computer power than the largest supercomputer in existence today, allowing them to perform real time image analysis of their surroundings and communicate with weapons tracking systems to acquire and navigate to targets with greater precision and control.
• We'll also be able to build weapons both inexpensively and much more rapidly, at the same time taking full advantage of the remarkable materials properties of diamond.
• Rapid and inexpensive manufacture of great quantities of stronger more precise weapons guided by massively increased computational power will alter the way we fight wars.
• Changes of this magnitude could destabilize existing power structures in unpredictable ways.
• Military applications of nanotechnology raise a number of concerns that prudence suggests we begin to investigate before, rather than after, we develop this new technology.
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4. SOLAR ENERGY
Nanotechnology will cut costs both of the solar cells and the equipment needed to deploy them, making solar power economical.
In this application we need not make new or technically superior solar cells: making inexpensively what we already know how to make expensively would move solar power into the mainstream.
5. MEDICAL USES
• It is not modern medicine that does the healing, but the cells themselves: we are but onlookers.
• If we had surgical tools that were molecular both in their size and precision, we could develop a medical technology that for the first time would let us directly heal the injuries at the molecular and cellular level that are the root causes of disease and ill health.
• With the precision of drugs combined with the intelligent guidance of the surgeon's scalpel, we can expect a quantum leap in our medical capabilities.
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6. Other Advantages
Less Pollution
The problem with past technologies is that they pollute the environment in cases where we humans would die in years.
A good example of a bad polluting invention would be the automobile. The automobile ran on gas and the gas fumes destroyed the ozone layer.
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DEPARTMENT OF PHYSICS AND NANOTECHNOLOGY SRM INSTITUTE OF SCIENCE AND TECHNOLOGY
21PYB103J –Semiconductor Physics and Computational methods
MODULE 5
Lecture- 51
Introduction to novel low dimensional systems
21PYB103J Module-V Lecture-1
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Dis covered by
Buckyminister fuller
in 1960
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Most abundant form of fullerene is C60 having 32 facets ( 12 pentagons and 20 hexagons
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a substance which has a molecular structure built up chiefly or completely from a large number of similar units bonded together, e.g. many synthetic organic materials used as plastics and resins
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light wire netting with a hexagonal mesh. 21PYB103J Module-V Lecture-1
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