Electricity at
nanoscale
Lecture 4
MTX9100
Nanomaterjalid
OUTLINE
-What is difference between semiconductor and
insulator?
-How large is the band gap?
-What happens at nanoscale?
-Why GD lasers are amazing device?
The hierarchy of electrical behavior
Energy zones
Valence electrons are delocalized,
interact and interpenetrate each other.
Their sharp energy levels are
broadened into energy bands.
• Example: Sodium has 1 valence electron (3S1). If
there are N sodium atoms, there are N distinct 3S1
energy levels in 3S band.
• Sodium is a good conductor since it has half filled
outer3 orbital
Energy bands
Instead of having discrete
energies as in the case of
free atoms, the available
energy states form bands
of very closely spaced
levels.
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Band Theory of solids
A useful way to visualize the difference between
conductors, insulators and semiconductors is to plot the
available energies for electrons in the materials.
Instead of having discrete energies as in the case of
free atoms, the available energy states form bands.
Crucial to the conduction process is whether or not
there are electrons in the conduction band.
An important parameter in the band theory is
the Fermi level,
the top of the available electron energy levels at low
temperatures. The position of the Fermi level with the
relation to the conduction band is a crucial factor in
determining electrical properties.
Band theory
The position of the Fermi level with the relation to the
conduction band is a crucial factor in determining
electrical properties.
6
Conductor or insulator?
Insulator or conductor?
In insulators the electrons in the valence band
are separated by a large gap from the
conduction band,
in conductors like metals the valence band
overlaps the conduction band, and
in semiconductors there is a small enough gap
between the valence and conduction bands that
thermal or other excitations can bridge the
gap. With such a small gap, the presence of a
small percentage of a doping material can
increase conductivity dramatically.
Band theory
For the total number N of
atoms in a solid (1023 cm–3),
N energy levels split apart
within a width ∆E.
– Leads to a band of
energies for each initial
atomic energy level
(e.g. 1s energy band for 1s
energy level).
Band zones in metals
As solid atoms are brought together from infinity, the atomic orbitals overlap
and give rise to bands. Outer orbitals overlap first. The 3s orbitals give rise
to the 3s band, 2p orbitals to the 2p band and so on. The various bands
overlap to produce a single band in which the energy is nearly continuous.
In metals, several energy levels such as E2s, E2p, and E3s (separated from E1s ) in Li are
overlapped up to the vacuum level (above which electrons are "free").
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Fermi level
"Fermi level" is the term used to describe the top of the collection
of electron energy levels at absolute zero temperature.
This concept comes from Fermi-Dirac statistics.
Electrons are fermions and by the Pauli exclusion principle cannot
exist in identical energy states. So at absolute zero they pack into
the lowest available energy states and build up a "Fermi sea" of
electron energy states. The Fermi level is the surface of that sea
at absolute zero where no electrons will have enough energy to rise
above the surface.
The concept of the Fermi energy is a crucially important concept
for the understanding of the electrical and thermal properties of
solids. Both ordinary electrical and thermal processes involve
energies of a small fraction of an electron volt. But the Fermi
energies of metals are on the order of electron volts. This implies
that the vast majority of the electrons cannot receive energy from
those processes because there are no available energy states for
them to go to within a fraction of an electron volt of their present
energy. Limited to a tiny depth of energy, these interactions are
limited to "ripples on the Fermi sea".
Fermi function
The Fermi function f(E) gives the probability that a given available
electron energy state will be occupied at a given temperature.
The basic nature of this function dictates that at ordinary temperatures, most of the
levels up to the Fermi level EF are filled, and relatively few electrons have energies above
the Fermi level. The Fermi level is on the order of electron volts (e.g., 7 eV for copper),
whereas the thermal energy kT is only about 0.026 eV at 300K. If you put those numbers
into the Fermi function at ordinary temperatures, you find that its value is essentially 1
up to the Fermi level, and rapidly approaches zero above it.
Electronic band structure
How large is the gap?
• At T = 0, all levels in conduction band below the Fermi energy are
filled with electrons.
• Electrons are free to move into “empty” states of conduction band
with only a small electric field E, leading to high electrical conductivity!
• At T > 0, electrons have a probability to be thermally “excited” from
below the Fermi energy to above it.
How difficult to jump over the gap?
Insulator
The large energy gap between the valence
and conduction bands in an insulator says
that at ordinary temperatures no
electrons can reach the conduction band.
Semiconductor
• At T = 0, lower valence band is
filled with electrons and upper
conduction band is empty, leading
to zero conductivity.
• At T > 0, electrons thermally
“excited” from valence to
conduction band, leading to
measurable conductivity.
Conductors, Insulators and
Semiconductors
Semiconductor band structure
In semiconductors and insulators,
electrons are confined to a number
of bands of energy, and forbidden
from other regions. The term "band
gap" refers to the energy difference
between the top of the valence band
and the bottom of the conduction
band; electrons are able to jump
from one band to another. In order
for an electron to jump from a
valence band to a conduction band, it
requires a specific amount of energy
for the transition. The required
energy differs with different
materials.
Density of states
In statistical and condensed matter physics,
the density of states (DOS) of a system describes the number of
states at each energy level that are available to be occupied.
A high DOS at a specific energy level means that there are many states
available for occupation.
A DOS of zero - no states can be occupied at that energy level.
While the DOS in a band could be very large for some materials, it may not be
uniform. It approaches zero at the band boundaries, and is generally higher
near the middle of a band. The density of states for the free electron model in
three dimensions is given by
f(E) is the probability that a
state at energy E is
occupied
How do electrons and holes populate
the bands?
Insulator energy bands
Most solid substances are insulators, and in terms of the band theory of
solids this implies that there is a large forbidden gap between the energies
of the valence electrons and the energy at which the electrons can move
freely through the material (the conduction band).
Large energy
between valence
and conduction
bands
Glass is an insulating material which may be transparent to
visible light for reasons closely correlated with its nature as
an electrical insulator. The visible light photons do not have
enough quantum energy to bridge the band gap and get the
electrons up to an available energy level in the conduction
band. The visible properties of glass can also give some insight
into the effects of "doping" on the properties of solids. A very
small percentage of impurity atoms in the glass can give it
color by providing specific available energy levels which absorb
certain colors of visible light. The ruby mineral (corundum) is
aluminum oxide with a small amount (about 0.05%) of chromium
which gives it its characteristic pink or red color by absorbing
green and blue light.
While the doping of insulators can dramatically change their optical properties,
it is not enough to overcome the large band gap to make them good conductors
of electricity. However, the doping of semiconductors has a much more
dramatic effect on their electrical conductivity and is the basis for solid state
electronics.
Group IV elements: Insulators or
Semiconductors?
Semiconductor energy bands
For intrinsic semiconductors like silicon and germanium, the Fermi level is
essentially halfway between the valence and conduction bands. Although no
conduction occurs at 0 K, at higher temperatures a finite number of
electrons can reach the conduction band and provide some current.
In doped semiconductors, extra energy levels are added.
In n-type material there are electron energy levels near the
top of the band gap so that they can be easily excited into the
conduction band.
In p-type material, extra holes in the band gap allow excitation Extrinsic semiconductor: (a)
of valence band electrons, leaving mobile holes in the valence
n-type, e.g. P doped Si
band.
(b) p-type, e.g. Ga doped Si.
Conductor energy bands
In terms of the band theory of solids, metals are unique as good conductors
of electricity. This can be seen to be a result of their valence electrons
being essentially free. In the band theory, this is depicted as an overlap of
the valence band and the conduction band so that at least a fraction of the
valence electrons can move through the material.
The free electron density in a metal is a factor in
determining its electrical conductivity.
It is involved in the Ohm's law behavior of
metals on a microscopic scale. Because
electrons are fermions and obey the Pauli
exclusion principle, then at 0 K temperature
the electrons fill all available energy levels up
to the Fermi level. Therefore the free
electron density of a metal is related to the
Fermi level and can be calculated from
Typical band gaps
Materials electro-resistance
A zero
resistance
state!
Conductivity of graphite
What’s different at the nanoscale?
As the system length scale is reduced to the
nanoscale,
two effects are of importance:
(1) the quantum effect,
where due to electron confinement the energy
bands are replaced by discreet energy states,
leading to cases where conducting materials can
behave as semiconductors or insulators, and
(2) the classical effect,
where the mean-free path for inelastic scattering
becomes comparable with the size of the system,
leading to a reduction in scattering events.
Semiconductor nanostructures
In an unconfined (bulk) semiconductor, an electron-hole pair is
typically bound within a characteristic length called the Bohr excitation
radius. If the electron and hole are constrained further, then the
semiconductor's properties change.
This effect is a form of quantum confinement, and it is a key feature
in many emerging electronic structures.
Other quantum confined semiconductors include:
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.
Quantum well, or quantum wire confinements give the electron at least
one degree of freedom.
Although this kind of confinement leads to quantization of the electron
spectrum which changes the density of states, and results in one or twodimensional energy sub-bands, it still gives the electron at least one direction
to propagate. On the other hand, today’s technology allows us to create
Quantum Dots structures, in which all existing degrees of freedom of electron
propagation are quantized.
Geometries of the different structures
A plot of Eg against
length/diameter ratio
for the InAs quantum
rods
a. Geometries of the different structures.
b. Plots of Eg (the increase in the band gap over the bulk value) against d (the thickness or diameter) for
rectangular quantum wells, cylindrical quantum wires and spherical QDs obtained from particle-in-a-box
approximations.The grey area between the dot and wire curves is the intermediate zone corresponding to quantum
rods.The vertical dotted line and points qualitatively represent the expected variation in the band gap for InAs
quantum rods of varying length/diameter ratio.
Comparison of the quantization of
density of states
(a) bulk, (b) quantum well, (c) quantum wire, (d) quantum dot.
The conduction and valence bands split into overlapping sub-bands
that get successively narrower as the electron motion is
restricted in more dimensions
Quantum well lasers
If the middle layer is made thin enough, it acts as a quantum well.
This means that the vertical variation of the electron's wavefunction, and
thus a component of its energy, is quantized.
The efficiency of a quantum well laser is greater than that of a bulk
laser because the density of states function of electrons in the quantum
well system has an abrupt edge that concentrates electrons in energy
states that contribute to laser action.
A quantum well is a
potential well that confines
particles, which were
originally free to move in
three dimensions, to two
dimensions, forcing them to
occupy a planar region.
Schematic of a semiconductor laser
Further improvements in the laser efficiency have also been demonstrated by reducing
the quantum well layer to a quantum wire or to a "sea" of quantum dots.
Quantum dot laser
A quantum dot laser is a semiconductor laser that uses quantum dots as
the active laser medium in its light emitting region. Due to the tight
confinement of charge carriers in quantum dots, they exhibit an
electronic structure similar to atoms.
Lasers fabricated from such an active media exhibit device
performance that is closer to gas lasers, and avoid some of the negative
aspects of device performance associated with traditional
semiconductor lasers based on bulk or quantum well active media.
The quantum dot active region may also be engineered to operate at
different wavelengths by varying dot size and composition. This allows
quantum dot lasers to be fabricated to operate at wavelengths
previously not possible using semiconductor laser technology.
Image of the actual laser diode chip (shown on
the eye of a needle for scale) contained within
the package shown in the above image Wikipedia
Application of QD laser
Devices based on quantum dot active media are finding commercial
application in medicine (laser scalpel, optical coherence tomography),
display technologies (projection, laser TV), spectroscopy and
telecommunications.
Colloidal semiconductor nanocrystals
irradiated with ultraviolet light. Quantum
confinement causes the band gap energy to
vary with the nanocrystal's size. Each vial
contains a monodisperse sample of
nanocrystals dispersed in a liquid solvent. Wikipedia
Advantages of QD-lasers
1. Emits light at wavelengths determined by the energy levels of the dots, rather than
the band gap energy. Thus, they offer the possibility of improved device performance
and increased flexibility to adjust the wavelength
2. Has the maximum material gain and differential gain, at least 2-3 orders higher
than QW lasers
3. Advantages of small volume:
a. low power high frequency operation,
b. large modulation bandwidth,
c. small dynamic chirp,
d. small linewidth enhancement factor,
e. and low threshold current.
4. Shows superior temperature stability of the threshold current. The threshold
current is given by the relation:
where T is the active region temperature, (T ref) is the reference temperature, and (T 0) is an empirically-determined
"characteristic temperature", which is itself a function of temperature and device length. In QDLs T0 can be high,
because one can effectively decouple electronphonon interaction by increasing the intersubband separation. This leads
to undiminished room-temperature performance without external thermal stabilization.
The quantum states and dot
photoluminescence
Functional device scales