nanoelectronics note

7/29/2019 nanoelectronics note

1/7

1. The carrier density in a semiconductor, is obtained by integratingthe product of the density of states and the probability densityfunction over all possible states.

2. Non-degenerate semiconductors are defined as semiconductors

for which the Fermi energy is at least 3kTaway from either bandedge. The reason we restrict ourselves to non-degeneratesemiconductors is that this definition allows the Fermi function tobe replaced with a simple exponential function, i.e. the Maxwell-Boltzmann distribution function.

3.f(E), under equilibrium conditions, the probability that an availablestate at an energy E will be occupied by an electron

4. 1-f(E), under equilibrium conditions, the probability that anavailable state at an energy E will NOT be occupied by an electron

5. At T=0K (above), No occupation of states above EF andcomplete occupation of states below EF6.At T>0K (below), occupation probability is reduced with

increasing energy f(E=EF) = 1/2 regardless of temperature.7. At higher temperatures, higher energy states can be occupied,

leavingmore lower energy states unoccupied (1-f(E)).

Bose-Einstein statistics: The Bose-Einstein probability distribution function is

used to find the probability that a boson occupies a specific quantum state in a

system at thermal equilibrium. Bosons are particles which do not obey the PauliExclusion Principle (e.g. phonons and photons). The distribution function can

be written as

The Bose-Einstein distribution

describes the statistical behaviour of integerspinparticles (bosons). At low

temperatures, bosons can behave very differently than fermions because anunlimited number of them can collect into the same energy state, a phenomenon

called "condensation". Bosons are the particles with zero or integer spins. An

important characteristic of bosons is that there is no limit to the number that can

occupy the same quantum state.Bosons contrast with fermions, which obey

FermiDirac statistics. Two or more fermions cannot occupy the same quantum

state .Since bosons with the same energy can occupy the same place in space,

bosons are often force carrier particles. In contrast, fermions are usually

associated with matter (although in quantum physics the distinction between the

two concepts is not clear cut).


2/7

Fermi-Dirac statistics: The Fermi-Dirac probability distribution function is

used to find the probability that a fermion occupies a specific quantum state in a

Exclusion Principle (e.g. electrons, protons, neutrons). The distribution function

system at thermal equilibrium. Fermions are particles which obey the Pauli can

be written as

In solid-state and condensed matter physics, the density of states (DOS) of asystem describes the number of states per interval of energy at each energy level

that are available to be occupied by electrons. A high DOS at a specific energy

level means that there are many states available for occupation. A DOS of zero

means that no states can be occupied at that energy level. In general a DOS is an

average over the space and time domains occupied by the system.

The theorem of equipartition of energy states that molecules in thermalequilibrium have the same average energy associated with each independent

degree of freedom of their motion and that the energy is

The density of states function describes the number of states that are available

in a system and is essential for determining the carrier concentrations andenergy 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 wells (2D), quantum wires

(1D), and quantum dots (0D) must be known.

A quantum dot is a nano crystal made of semiconductor materialsthat aresmall enough to display quantum mechanical properties, specifically its excitons

are confined in all three spatial dimensions. The electronic properties of these

materials are intermediate between those of bulk semiconductors and of discrete

molecules. Quantum dots are semiconductors whose electronic characteristicsare closely related to the size and shape of the individual crystal. Size and band

gap are inversely related in quantum dots.

Quantum Dots Small devices that contain a tiny droplet of free electrons.

Dimensions between nanometers to a few microns. Contains single electron to a collection of several thousands


3/7

Size, shape, and number ofelectrons can be precisely controlled

Exciton: bound electron hole pair (EHP)

Attractivepotential between electron and hole

Excitons generated inside the dot

Excitons confined to the dotDegree of confinement determined by dot sizeDiscrete energies

Quantum wires, which confine electrons or holes in two spatial dimensions

and allow free propagation in the third.

Quantum wells, which confine electrons or holes in one dimension and allow

free propagation in two dimensions

A quantum well is a potential well with only discrete energy values. Onetechnology to create quantization is to confine particles, which were originally

free to move in three dimensions, to two dimensions, forcing them to occupy a

planar region. The effects of quantum confinement take place when the

quantum well thickness becomes comparable to the de Broglie wavelength ofthe carriers (generally electrons and holes), leading to energy levels called

"energy subbands", i.e., the carriers can only have discrete energy values. thedensity of states for a quantum well is a step function with stepsoccurring at the energy of each quantized levelIn condensed matter physics, a quantum wire is an electrically conductingwire in which quantum effects influence the transport properties. If the diameter

of a wire is sufficiently small, electrons will experience quantum confinement inthe transverse direction. As a result, their transverse energy will be quantized

into a series of discrete values. One consequence of this quantization is that the

classical formula for calculating the electrical resistance of a wire: is not valid

for quantum wires


4/7

Thermionic emission is the heat-induced flow of charge carriers from asurface or over a potential-energy barrier. This occurs because the thermal

energy given to the carrier overcomes the binding potential, also known as work

function of the metal. Thermionic emission must not be confused with thermic

emission (of black body for example) where photons are emitted from heatedbody. In the case of thermionic emission we observe emitted electrons from ametal due to its high temperature. Mobile electrons in metals, also called

valence electrons, are responsible for electric current conduction. If we increase

the temperature of the metal, electrons start to move faster and some may have

enough energy to escape (evaporate) from the metal.The higher the temperature,

the higher will be the current of escaping electrons. This temperature inducedelectron flow is called thermionic emission. If we use the hot metal from which

electrons evaporate as a cathode on a high potential difference to the anode,

then we get electric field between cathode and anode and emission of electrons

can be measured as a anode current.let us consider the electron to move only on xy plane under a magnetic field pointingalong the z direction.The energy of the electrons motion in the plane perpendicular to B is completely quantised. These

quantised levels are known as Landau levels.The k-space areas of the orbits in the plane perpendicular to B are also quantised (this is easyto work out in the present case and is left as an exercise); thus, allowed orbits fall on Landautubes in k-space with quantised cross-sectional area. The energy quantum for the in-plane motionappears to be h(the semiclassical cyclotronfrequency).The motion parallel to B is unaffected

Under a magnetic field B, electrons move in helical paths (orbits around the fielddirection, uniform motion parallel to B), with angular frequency

the cyclotron frequency. Electrons can absorb energy from anelectromagnetic field of the appropriate frequency cyclotron resonance this is how

effective masses can be measured.

When considering the density of states for a

quantum dot, no free motion is possible. Because there is no k-space to be filled

with electrons and all available states exist only at discrete energies, we


5/7

describe the density of states for 0D with the delta function.


6/7

The effective mass is defined as: In a solid, the electron (hole) effective massrepresents how electrons move in an applied field.The effective mass reflects the inverse of the band curvature: the lower m*, thelarger is the curvature.Flatter bands have larger effective mass; electrons are more accelerated.In quantum physics, the spinorbit interaction (also called spinorbit effect orspinorbit coupling) is an interaction of a particle's spin with its motion. The first and bestknown example of this is that spin-orbit interaction causes shifts in an electron's atomic

energy levels due to electromagnetic interaction between the electron's spin and the magnetic

field generated by the electron's orbit around the nucleus. This is detectable as a splitting ofspectral lines. A similar effect, due to the relationship between angular momentum and the


7/7

strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a

shift in their energy levels in the nucleus shell model. In the field of spintronics, spin-orbit

effects for electrons in semiconductors and other materials are explored for technological

applications.

Richardson's law tells us what is the current densityJx [A/m2] of thermically escaped

electrons in the direction perpendicular to heated metal. Let us say we have a flow of thermalelectrons with charge q in thex direction. We consider empty space in front of metal plate.

Then the current density of this electrons is

whereAR is Richardson's constant with a valueAR =

1,2106 A/m2K2. Work function W= UEFis the minimum amount of energy needed for an

electron to leave the metal and depends on a metalA Fermi gas is an ensemble of a large number offermions. Fermions, named afterEnrico Fermi, are

particlesthat obeyFermi

Dirac statistics. These statistics determine the energy distribution offermions in a Fermi gas inthermal equilibrium, and is characterized by theirnumber density,

temperature, and the set of available energy states.

is anyparticlecharacterized byFermiDirac statisticsand following thePauli exclusion

principle; fermions include allquarksandleptons, as well as anycomposite particlemade of

anodd numberof these, such as allbaryonsand manyatomsandnuclei. Fermions contrast

withbosonswhich obeyBoseEinstein statistics.

A fermion can be anelementary particle, such as theelectron; or it can be acomposite

particle, such as theproton. According to thespin-statistics theoremin any reasonablerelativisticquantum field theory, particles withintegerspinarebosons, while particles with

half-integerspin are fermions.

Besides this spin characteristic fermions have another specific property: they possess

conserved baryon or lepton quantum numbers. Therefore what is usually referred as the spin-

statistics relation is in fact a spin-statistics-quantum number relation.[2]

In contrast to bosons, as a consequence of the Pauli principle only one fermion can occupy a

particularquantum stateat any given time. If multiple fermions have the same spatial

probability distribution, then at least one property of each fermion, such as its spin, must be

different. Fermions are usually associated withmatter, whereas bosons are generallyforcecarrierparticles; although in the current state of particle physics the distinction between the

two concepts is unclear.

Composite fermions, such as protons andneutrons, are key building blocks of matter. Weakly

interacting[clarification needed] fermions can also display bosonic behavior under extreme

conditions, such as insuperconductivity.

The Pauli exclusion principle is thequantum mechanicalprinciple that no twoidenticalfermions

(particles with half-integerspin) may occupy the samequantum statesimultaneously.

no two electrons in a single atom can have the same fourquantum numbers; ifn, , and m are the

same, ms must be different such that the electrons have opposite spins, and so on.
http://en.wikipedia.org/wiki/Fermionshttp://en.wikipedia.org/wiki/Fermionshttp://en.wikipedia.org/wiki/Fermionshttp://en.wikipedia.org/wiki/Enrico_Fermihttp://en.wikipedia.org/wiki/Enrico_Fermihttp://en.wikipedia.org/wiki/Enrico_Fermihttp://en.wikipedia.org/wiki/Subatomic_particlehttp://en.wikipedia.org/wiki/Subatomic_particlehttp://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statisticshttp://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statisticshttp://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statisticshttp://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statisticshttp://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statisticshttp://en.wikipedia.org/wiki/Thermal_equilibriumhttp://en.wikipedia.org/wiki/Thermal_equilibriumhttp://en.wikipedia.org/wiki/Thermal_equilibriumhttp://en.wikipedia.org/wiki/Number_densityhttp://en.wikipedia.org/wiki/Number_densityhttp://en.wikipedia.org/wiki/Number_densityhttp://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Subatomic_particlehttp://en.wikipedia.org/wiki/Subatomic_particlehttp://en.wikipedia.org/wiki/Subatomic_particlehttp://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statisticshttp://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statisticshttp://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statisticshttp://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statisticshttp://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statisticshttp://en.wikipedia.org/wiki/Pauli_exclusion_principlehttp://en.wikipedia.org/wiki/Pauli_exclusion_principlehttp://en.wikipedia.org/wiki/Pauli_exclusion_principlehttp://en.wikipedia.org/wiki/Pauli_exclusion_principlehttp://en.wikipedia.org/wiki/Quarkhttp://en.wikipedia.org/wiki/Quarkhttp://en.wikipedia.org/wiki/Quarkhttp://en.wikipedia.org/wiki/Leptonhttp://en.wikipedia.org/wiki/Leptonhttp://en.wikipedia.org/wiki/Leptonhttp://en.wikipedia.org/wiki/Composite_particlehttp://en.wikipedia.org/wiki/Composite_particlehttp://en.wikipedia.org/wiki/Composite_particlehttp://en.wikipedia.org/wiki/Even_and_oddhttp://en.wikipedia.org/wiki/Even_and_oddhttp://en.wikipedia.org/wiki/Even_and_oddhttp://en.wikipedia.org/wiki/Baryonhttp://en.wikipedia.org/wiki/Baryonhttp://en.wikipedia.org/wiki/Baryonhttp://en.wikipedia.org/wiki/Atomhttp://en.wikipedia.org/wiki/Atomhttp://en.wikipedia.org/wiki/Atomhttp://en.wikipedia.org/wiki/Atomic_nucleushttp://en.wikipedia.org/wiki/Atomic_nucleushttp://en.wikipedia.org/wiki/Atomic_nucleushttp://en.wikipedia.org/wiki/Bosonhttp://en.wikipedia.org/wiki/Bosonhttp://en.wikipedia.org/wiki/Bosonhttp://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statisticshttp://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statisticshttp://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statisticshttp://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statisticshttp://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statisticshttp://en.wikipedia.org/wiki/Elementary_particlehttp://en.wikipedia.org/wiki/Elementary_particlehttp://en.wikipedia.org/wiki/Elementary_particlehttp://en.wikipedia.org/wiki/Electronhttp://en.wikipedia.org/wiki/Electronhttp://en.wikipedia.org/wiki/Electronhttp://en.wikipedia.org/wiki/Composite_particlehttp://en.wikipedia.org/wiki/Composite_particlehttp://en.wikipedia.org/wiki/Composite_particlehttp://en.wikipedia.org/wiki/Composite_particlehttp://en.wikipedia.org/wiki/Protonhttp://en.wikipedia.org/wiki/Protonhttp://en.wikipedia.org/wiki/Protonhttp://en.wikipedia.org/wiki/Spin-statistics_theoremhttp://en.wikipedia.org/wiki/Spin-statistics_theoremhttp://en.wikipedia.org/wiki/Spin-statistics_theoremhttp://en.wikipedia.org/wiki/Theory_of_relativityhttp://en.wikipedia.org/wiki/Quantum_field_theoryhttp://en.wikipedia.org/wiki/Quantum_field_theoryhttp://en.wikipedia.org/wiki/Quantum_field_theoryhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Spin_%28physics%29http://en.wikipedia.org/wiki/Spin_%28physics%29http://en.wikipedia.org/wiki/Spin_%28physics%29http://en.wikipedia.org/wiki/Bosonhttp://en.wikipedia.org/wiki/Bosonhttp://en.wikipedia.org/wiki/Bosonhttp://en.wikipedia.org/wiki/Half-integerhttp://en.wikipedia.org/wiki/Half-integerhttp://en.wikipedia.org/wiki/Fermions#cite_note-2http://en.wikipedia.org/wiki/Fermions#cite_note-2http://en.wikipedia.org/wiki/Fermions#cite_note-2http://en.wikipedia.org/wiki/Quantum_statehttp://en.wikipedia.org/wiki/Quantum_statehttp://en.wikipedia.org/wiki/Quantum_statehttp://en.wikipedia.org/wiki/Matterhttp://en.wikipedia.org/wiki/Matterhttp://en.wikipedia.org/wiki/Matterhttp://en.wikipedia.org/wiki/Force_carrierhttp://en.wikipedia.org/wiki/Force_carrierhttp://en.wikipedia.org/wiki/Force_carrierhttp://en.wikipedia.org/wiki/Force_carrierhttp://en.wikipedia.org/wiki/Neutronhttp://en.wikipedia.org/wiki/Neutronhttp://en.wikipedia.org/wiki/Neutronhttp://en.wikipedia.org/wiki/Wikipedia:Please_clarifyhttp://en.wikipedia.org/wiki/Wikipedia:Please_clarifyhttp://en.wikipedia.org/wiki/Wikipedia:Please_clarifyhttp://en.wikipedia.org/wiki/Superconductivityhttp://en.wikipedia.org/wiki/Superconductivityhttp://en.wikipedia.org/wiki/Superconductivityhttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Identical_particleshttp://en.wikipedia.org/wiki/Identical_particleshttp://en.wikipedia.org/wiki/Fermionshttp://en.wikipedia.org/wiki/Fermionshttp://en.wikipedia.org/wiki/Fermionshttp://en.wikipedia.org/wiki/Spin_%28physics%29http://en.wikipedia.org/wiki/Spin_%28physics%29http://en.wikipedia.org/wiki/Spin_%28physics%29http://en.wikipedia.org/wiki/Quantum_statehttp://en.wikipedia.org/wiki/Quantum_statehttp://en.wikipedia.org/wiki/Quantum_statehttp://en.wikipedia.org/wiki/Quantum_numberhttp://en.wikipedia.org/wiki/Quantum_numberhttp://en.wikipedia.org/wiki/Quantum_numberhttp://en.wikipedia.org/wiki/Quantum_numberhttp://en.wikipedia.org/wiki/Quantum_statehttp://en.wikipedia.org/wiki/Spin_%28physics%29http://en.wikipedia.org/wiki/Fermionshttp://en.wikipedia.org/wiki/Identical_particleshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Superconductivityhttp://en.wikipedia.org/wiki/Wikipedia:Please_clarifyhttp://en.wikipedia.org/wiki/Neutronhttp://en.wikipedia.org/wiki/Force_carrierhttp://en.wikipedia.org/wiki/Force_carrierhttp://en.wikipedia.org/wiki/Matterhttp://en.wikipedia.org/wiki/Quantum_statehttp://en.wikipedia.org/wiki/Fermions#cite_note-2http://en.wikipedia.org/wiki/Half-integerhttp://en.wikipedia.org/wiki/Bosonhttp://en.wikipedia.org/wiki/Spin_%28physics%29http://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Quantum_field_theoryhttp://en.wikipedia.org/wiki/Theory_of_relativityhttp://en.wikipedia.org/wiki/Spin-statistics_theoremhttp://en.wikipedia.org/wiki/Protonhttp://en.wikipedia.org/wiki/Composite_particlehttp://en.wikipedia.org/wiki/Composite_particlehttp://en.wikipedia.org/wiki/Electronhttp://en.wikipedia.org/wiki/Elementary_particlehttp://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statisticshttp://en.wikipedia.org/wiki/Bosonhttp://en.wikipedia.org/wiki/Atomic_nucleushttp://en.wikipedia.org/wiki/Atomhttp://en.wikipedia.org/wiki/Baryonhttp://en.wikipedia.org/wiki/Even_and_oddhttp://en.wikipedia.org/wiki/Composite_particlehttp://en.wikipedia.org/wiki/Leptonhttp://en.wikipedia.org/wiki/Quarkhttp://en.wikipedia.org/wiki/Pauli_exclusion_principlehttp://en.wikipedia.org/wiki/Pauli_exclusion_principlehttp://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statisticshttp://en.wikipedia.org/wiki/Subatomic_particlehttp://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Number_densityhttp://en.wikipedia.org/wiki/Thermal_equilibriumhttp://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statisticshttp://en.wikipedia.org/wiki/Subatomic_particlehttp://en.wikipedia.org/wiki/Enrico_Fermihttp://en.wikipedia.org/wiki/Fermions

nanoelectronics note

Documents

Transcript of nanoelectronics note