Transcript Quantum Confinement

1
Physics of Low dimensional Materials-1
Prof.P. Ravindran,
Department of Physics, Central University of Tamil
Nadu, India
&
Center for Materials Science and Nanotechnology,
University of Oslo, Norway
http://folk.uio.no/ravi/cutn/NMNT2016
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Nanostructured materials derive their special properties
from having one or more dimensions made small
compared to a length scale critical to the physics of the
process.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Development of electronic properties as a
function of cluster size
Each band has a width that reflects the interaction between atoms, with a
bandgap between the conduction and the valence bands that reflects the
original separation of the bonding ad antibonding states.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Electronic DOS and dimensionality
Size effects are most
evident at band edges
(semiconductor NPs).
DOS (dn/dE) as a function
of dimensionality.
3D case is for free particles.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Quantum Dots: Introduction
Definition:
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Quantum dots (QD) are nanoparticles/structures that exhibit 3
dimensional quantum confinement, which leads to many unique optical
and transport properties.
GaAs Quantum dot containing just 465 atoms.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Quantum Dot: Introduction
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Quantum dots are usually regarded as semiconductors by
definition.
Similar behavior is observed in some metals. Therefore, in some
cases it may be acceptable to speak about metal quantum dots.
Typically, quantum dots are composed of groups II-VI, III-V,
and IV-VI materials.
QDs are bandgap tunable by size which means their optical
and electrical properties can be engineered to meet specific
applications.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Quantum Confinement
Definition:
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Quantum Confinement is the spatial confinement of electron-hole
pairs (excitons) in one or more dimensions within a material.
– 1D confinement: Quantum Wells
– 2D confinement: Quantum Wire
– 3D confinement: Quantum Dot
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Quantum confinement is more prominent in semiconductors
because they have an energy gap in their electronic band
structure.
Metals do not have a bandgap, so quantum size effects are less
prevalent. Quantum confinement is only observed at dimensions
below 2 nm.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Quantum Confinement
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Recall that when atoms are brought together in a bulk material the
number of energy states increases substantially to form nearly continuous
bands of states.
Energy
Energy
N
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Quantum Confinement
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The reduction in the number of atoms in a material results in the
confinement of normally delocalized energy states.
Electron-hole pairs become spatially confined when the diameter of a
particle approaches the de Broglie wavelength of electrons in the
conduction band.
As a result the energy difference between energy bands is increased with
decreasing particle size.
Energy
Eg
Eg
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Quantum Confinement
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This is very similar to the famous particle-in-a-box scenario and can
be understood by examining the Heisenberg Uncertainty Principle.
The Uncertainty Principle states that the more precisely one knows
the position of a particle, the more uncertainty in its momentum (and
vice versa).
Therefore, the more spatially confined and localized a particle
becomes, the broader the range of its momentum/energy.
This is manifested as an increase in the average energy of electrons in
the conduction band = increased energy level spacing = larger
bandgap
The bandgap of a spherical quantum dot is increased from its bulk
value by a factor of 1/R2, where R is the particle radius.*
Based upon single particle solutions of the schrodinger wave equation
valid for R< the exciton bohr radius.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Quantum Confinement
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What does this mean?
– Quantum dots are bandgap tunable by size. We can engineer their optical and
electrical properties.
– Smaller QDs have a large bandgap.
– Absorbance and luminescence spectrums are blue shifted with decreasing
particle size.
Energy
555 nm
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
650 nm
Quantum Dots (QD)
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Nanocrystals (2-10 nm) of
semiconductor compounds
Small size leads to confinement of
excitons (electron-hole pairs)
Quantized energy levels and altered
relaxation dynamics
Examples: CdSe, PbSe, PbTe, InP
Eg
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Quantum Dots: Optics
Absorption and emission occur at specific wavelengths, which
are related to QD size
Eg
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Applications of QDs: Light Emitters
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The discovery of quantum dots has led to the development of an
entirely new gamut of materials for the active regions in LEDs and
laser diodes.
Indirect gap semiconductors that don’t luminesce in their bulk form
such as Si become efficient light emitters at the nanoscale due quantum
confinement effects.
The study of QDs has advanced our understanding of the emission
mechanisms in conventional LED materials such as InGaN, the active
region of blue LEDs.
The high radiative-recombination efficiency of epitaxial InGaN is due
to self-assembled, localized, In rich clusters that behave like QDs.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Quantum Dot Solar Cells
Possible benefits of using quantum dots (QD):
 “Hot carrier” collection: increased voltage due to reduced
thermalization
 Multiple exciton generation: more than one electron-hole
pair per photon absorbed
 Intermediate bands: QDs allow for absorption of light
below the band gap, without sacrificing voltage
MRS Bulletin 2007, 32(3), 236.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
QDs: Collect Hot Carriers
Band structure of bulk semiconductors absorbs light
having energy > Eg. However,
photo-generated carriers
thermalize to band edges.
1.
2.
Tune QD absorption (band gap) to
match incident light.
Extract carriers without loss of
voltage due to thermalization.
Conduction
Band
Eg
Valence
Band
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
QDs: Multiple Exciton Generation
In bulk
semiconductors:
1 photon = 1 exciton
Eg
Absorption of one photon of
light creates one electron-hole
pair, which then relaxes to the
band edges.
In QDs:
1 photon = multiple excitons
Impact ionization
The thermalization of the
original electron-hole pair
creates another pair.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
QDs: Multiple Exciton Generation
Quantum efficiency for exciton
generation: The ratio of excitons
produced to photons absorbed
Quantum Eff (%)
300
250
>100% means multiple exciton
generation
200
Occurs at photon energies (Ehv)
much greater than the band gap
(Eg)
150
100
1
2
3
4
5
Photon Energy (Ehv/Eg)
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
QDs: Intermediate Bands
Conventional band structure
does not absorb light with
energy < Eg
Eg
Intermediate bands in the band gap
allow for absorption of low energy
light
Intermediate band
formed by an
array of QDs
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Expected effects for electrons in
nanostructures
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Quantum confinement effect
Charge discreteness and strong electron-electron Coulomb
interaction effects
Tunneling effects
Strong electric field effects
Ballistic transport effects
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Overview of Quantum Confinement
History: In 1970 Esaki & Tsu proposed fabrication of
an artificial structure, which would consist of
alternating layers of 2 different semiconductors with
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Layer Thickness
 1 nm = 10 Å = 10-9 m  SUPERLATTICE
PHYSICS: The main idea was that introduction of
an artificial periodicity will “fold” the Brillouin
Zones into smaller BZ’s  “mini-zones”.
 The idea was that this would raise the conduction
band minima, which was needed for some device
applications.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
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Modern growth techniques (starting in the 1980’s), especially MBE &
MOCVD, make fabrication of such structures possible!
For the same reason, it is also possible to fabricate many other kinds of
artificial structures on the scale of nm
(nanometers)  “Nanostructures”
Superlattices
= “2 dimensional” structures
Quantum Wells = “2 dimensional” structures
Quantum Wires
= “1 dimensional” structures
Quantum Dots
= “0 dimensional” structures!!
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Clearly, it is not only the electronic properties of materials which can be
drastically altered in this way. Also, vibrational properties (phonons). Here,
only electronic properties & only an overview!
For many years, quantum confinement has been a fast growing field in
both theory & experiment! It is at the forefront of current research!
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Quantum Confinement in Nanostructures: Overview
Electrons Confined in 1 Direction:
Quantum Wells (thin films):
 Electrons can easily move in
2 Dimensions!
ky
kx
nz
Electrons Confined in 2 Directions:
Quantum Wires:
 Electrons can easily move in
1 Dimension!
Electrons Confined in 3 Directions:
ny
kx
2 Dimensional nz
Quantization!
Quantum Dots:
 Electrons can easily move in
0 Dimensions!
1 Dimensional
Quantization!
nz
nx
3 Dimensional
Quantization!
ny
Each further confinement direction changes a continuous k component
to a discrete component characterized by a quantum number n.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Quantum confinement
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Trap particles and restrict their motion
Quantum confinement produces new material behavior/phenomena
“Engineer confinement”- control for specific applications
Structures
 Quantum dots (0-D) only
confined
states, and no freely moving ones
 Nanowires (1-D) particles travel only
along the wire
 Quantum wells (2-D) confines
particles within a thin layer
(Scientific American)
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
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PHYSICS: Back to the bandstructure:
– Consider the 1st Brillouin Zone for the infinite crystal. The
maximum wavevectors are of the order
km  (/a)
a = lattice constant. The potential V is periodic with period a. In the
almost free e- approximation, the bands are free e- like except near the
Brillouin Zone edge. That is, they are of the form:
E  (k)2/(2mo)
So, the energy at the Brillouin Zone edge has the form:
Em  (km)2/(2mo)
or
Em  ()2/(2moa2)
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
PHYSICS
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SUPERLATTICES  Alternating layers of material.
Periodic, with periodicity L (layer thickness). Let kz = wavevector
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perpendicular to the layers.
In a superlattice, the potential V has a new periodicity in the z
direction with periodicity L >> a
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In the z direction, the Brillouin Zone is much smaller than that for an
infinite crystal. The maximum wavevectors are of the order:
ks  (/L)
 At the BZ edge in the z direction, the energy has the form:
Es  ()2/(2moL2) + E2(k)
E2(k) = the 2 dimensional energy for k in the x,y plane.
Note that:
()2/(2moL2) << ()2/(2moa2)
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Primary Qualitative Effects of Quantum Confinement
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Consider electrons confined along 1 direction (say, z) to a layer of width
L:
Energies
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The energy bands are quantized (instead of continuous) in kz &
shifted upward. So kz is quantized:
kz = kn = [(n)/L], n = 1, 2, 3
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So, in the effective mass approximation (m*), the bottom of the
conduction band is quantized (like a particle in a 1 d box) & shifted:
En = (n)2/(2m*L2)
Energies are quantized! Also, the wavefunctions are 2
dimensional Bloch functions (traveling waves) for k in the
x,y plane & standing waves in the z direction.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Quantum Confinement Terminology
Quantum Well  QW
= A single layer of material A (layer thickness L), sandwiched between 2
macroscopically large layers of material B. Usually, the bandgaps satisfy:
EgA < EgB
Multiple Quantum Well  MQW
= Alternating layers of materials A (thickness L) & B (thickness L). In this case:
L >> L
So, the e- & e+ in one A layer are independent of those in other A layers.
Superlattice  SL
= Alternating layers of materials A & B with similar layer thicknesses.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Brief Elementary Quantum Mechanics &
Solid State Physics Review
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Quantum Mechanics of a Free Electron:
– The energies are continuous: E = (k)2/(2mo) (1d, 2d, or 3d)
– The wavefunctions are traveling waves:
ψk(x) = A eikx
(1d)
ψk(r) = A eikr (2d or 3d)
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Solid State Physics: Quantum Mechanics of an Electron in a
Periodic Potential in an infinite crystal :
– The energy bands are (approximately) continuous: E= Enk
– At the bottom of the conduction band or the top of the valence band, in the
effective mass approximation, the bands can be written:
Enk  (k)2/(2m*)
– The wavefunctions are Bloch Functions = traveling waves:
Ψnk(r) = eikr unk(r); unk(r) = unk(r+R)
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Some Basic Physics
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Density of states (DoS)
DoS 
dN dN dk

dE dk dE
N (k ) 
in 3D:
k space vol
vol per state
4 3 k 3

(2 ) 3 V
Structure
Degree of
Confinement
Bulk Material
0D
Quantum Well
1D
Quantum Wire
2D
Quantum Dot
3D
dN
dE
E
1
1/ E
d(E)
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
QM Review: The 1d (infinite) Potential Well
(“particle in a box”) In all QM texts!!
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We want to solve the Schrödinger Equation for:
x < 0, V  ; 0 < x < L, V = 0; x > L, V 
 -[2/(2mo)](d2 ψ/dx2) = Eψ
Boundary Conditions:
ψ = 0 at x = 0 & x = L (V  there)
Energies:
En = (n)2/(2moL2), n = 1,2,3
Wavefunctions:
ψn(x) = (2/L)½sin(nx/L) (a standing wave!)
Qualitative Effects of Quantum Confinement:
Energies are quantized & ψ changes from a traveling
wave to a standing wave.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
In 3Dimensions…
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For the 3D infinite potential well:R
 ( x, y, z ) ~ sin( nLxx ) sin( mLyy ) sin( qLzz ), n, m, q  integer
2 2
Energy levels  8nmLh 2  8mmLh 2  8qmLh 2
2 2
x
2 2
y
z
Real Quantum Structures aren’t this simple!!
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In Superlattices & Quantum Wells, the potential barrier is
obviously not infinite!
In Quantum Dots, there is usually ~ spherical confinement, not
rectangular.
The simple problem only considers a single electron. But, in real
structures, there are many electrons & also holes!
Also, there is often an effective mass mismatch at the boundaries.
That is the boundary conditions we’ve used are too simple!
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Quantization in a Thin Crystal
E
An energy band with continuous k
is quantized into N discrete points kn
in a thin film with N atomic layers.
Electron
Scattering
EVacuum
Inverse
Photoemission
EFermi
Photoemission
0 /d
k
/a
d
= zone
boundary
N atomic layers with the spacing a = d/n
n = 2d / n
N quantized states with kn ≈ n  /d ( n = 1,…,N )
kn = 2 / n = n  /d
Quantization in Thin Graphite Films
E
1 layer =
graphene
2 layers
EVacuum
EFermi
3 layers
Photoemission
0 /d
k
/a
4 layers
N atomic layers with spacing a = d/n :
 N quantized states with kn ≈ N  /d
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
 layers
=
graphite
Quantum Well States
in Thin Films
becoming
continuous for
N
discrete for
small N
Paggel et al.
Science 283, 1709 (1999)
Counting Quantum Well States
n
hAg/Fe(100)
 (eV)
4
13
3
N
15
0.3
14
14
13
13
12
2
14
16
11
10
9
13
8
16
16
16
7
1
14
0.2
6
5
4
16
16
10
3
1
0
Binding Energy (eV)
1
2
3
4
5
6
1
7
8
2
0.1
(b)
Work Function (eV)
2
0
2
1
Periodic Fermi level crossing
of quantum well states with
increasing thickness
(a) Quantum Well States for Ag/Fe(100)
Binding Energy (eV)
14
Line Width (eV)
Photoemission Intensity (arb. units)
11.5
(N, n')
(2, 1)
(3, 1)
(7, 2)
(12, 3)
(13, 3)
100 4.4 200
Number of monolayers N
300
Temperature (K)
4.3
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
n
The Important Electrons in a Metal
Energy  EFermi
Energy Spread  3.5 kBT
Transport (conductivity, magnetoresistance, screening length, ...)
Width of the Fermi function:
FWHM  3.5 kBT
Phase transitions (superconductivity, magnetism, ...)
Superconducting gap:
Eg  3.5 kBTc
(Tc= critical temperature)
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Energy Bands of Ferromagnets
Calculation
Ni
Energy
Relative to EF
[eV]
4
Photoemission data
2
0
-2
0.7
0.9
|| along [011] [Å-1 ]
k
1.1
-4
-6
States near the Fermi level cause the energy
-8
splitting between majority and minority spin
-10

K
X
bands in a ferromagnet (red and green).
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Some Unusual Physical Properties of Nanomaterials
1.
Reduced Melting Point -- Nanomaterials may have a significantly lower melting point or
phase transition temperature and appreciably reduced lattice constants (spacing between
atoms is reduced), due to a huge fraction of surface atoms in the total amount of atoms.
2.
Ultra Hard -- Mechanical properties of nanomaterials may reach the theoretical strength,
which are one or two orders of magnitude higher than that of single crystals in the bulk form.
The enhancement in mechanical strength is simply due to the reduced probability of defects.
3.
Optical properties of nanomaterials can be significantly different from bulk crystals.
--- Semiconductor Blue Shift in adsorption and emission due to an increased band gap.
Quantum Size Effects,
Particle in a box.
--- Metallic Nanoparticles Color Changes in spectra due to Surface Plasmons
Resonances
Lorentz Oscillator Model.
4.
Electrical conductivity decreases with a reduced dimension due to increased surface
scattering.
Electrical conductivity increases due to the better ordering and ballistic transport.
5.
Magnetic properties of nanostructured materials are distinctly different from that of bulk
materials. Ferromagnetism disappears and transfers to superparamagnetism in the
nanometer scale due to the huge surface energy.
6.
Self-purification is an intrinsic thermodynamic property of nanostructures and nanomaterials
due to enhanced diffusion of impurities/defects/dislocations to the nearby surface.
7.
Increased perfection enhances chemical stability.
Most are tunable with size!
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
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2D Nanostructures: Superhydrophobic surfaces
The angle formed by a tangent to a flat surface of a drop of water at the
point of contact (contact angle) is given in terms of the interfacial
energies of the system by the Young equation:
 AB   AC
cos c 
 BC
γAB= air/surface interfacial tension
γAC= water/surface interfacial tension
γBC= air/water interfacial tension
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
cos c  1
Water/surface repulsion (large interfacial tension)
Water drop
Si Nanowires
Coated Si surface
(planar)
Coated
nanostructured
surface (rough)
Roughening on the nanoscale can greatly increase
hydrophobicity.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
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Optical Properties
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The reduction of materials' dimension has
pronounced effects on the optical properties.
The size dependence can be generally classified
into two groups.
One is due to the increased energy level spacing as
the system becomes more confined, and the other
is related to surface plasmon resonance.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
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P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Plasmonics
• The long wavelength of light (≈ m) creates a problem for
extending optoelectronics into the nanometer regime.
• A possible way out is the conversion of light into plasmons.
• They have much shorter wavelengths than light and are
able to propagate electronic signals.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
What is a Plasmon ?
A plasmon is a density wave in an electron gas. It is analogous to a sound
wave, which is a density wave in a gas consisting of molecules.
Plasmons exist mainly in metals, where electrons are weakly bound to the
atoms and free to roam. The free electron gas model provides a good
approximation (also known as jellium model).
The electrons in a metal can wobble like a piece of jelly, pulled back by the
attraction of the positive metal ions that they leave behind.
In contrast to the single electron wave function that we encountered
already, a plasmon is a collective wave where billions of electrons oscil-late
in sync.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
The Plasmon Resonance
Right at the plasmon frequency p the electron gas has a resonance,
it oscillates violently. This resonance frequency increases with the
electron density n , since the electric restoring force is proportional to
the displaced charge (analogous to the force constant of a spring).
Similar to an oscillating spring one obtains the proportionality: p  n
The plasmon resonance can be
observed in electron energy loss
spectroscopy (EELS). Electrons with
and energy of 2 keV are re-flected
from an Al surface and lose energy
by exciting 1, 2, 3,… plasmons. The
larger peaks at multiples of 15.3 eV
are from bulk plasmons, the smaller
peaks at multiples of 10.3 eV from
surface plasmons.
1
2
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
3
4 5
Why are Metals Shiny ?
An electric field cannot exist inside a metal, because metal electrons react to it by creating
an opposing screening field. An example is the image charge, which exactly cancels the
field of any external charge. This is also true for an electromagnetic wave, where
electrons respond to the changing external field and screen it at any given time. As a result,
the electromagnetic wave cannot enter a metal and gets reflected back out.
However, at high frequency (= high photon energy) there comes a point when the external
field oscillates too fast for the electrons to follow. Beyond this frequency a metal loses its
reflectivity.
The corresponding energy is the plasmon energy Ep = ħp (typically 10-30 eV,
deep into the ultraviolet).
The reflectivity of aluminum cuts off at its plasmon energy Data (dashed) are
compared
to the electron gas model (full).
Ep
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Plasmons and Energy-Saving
Window Coatings
The reflectivity cutoff at the plasmon energy can be used for energysaving window coatings which transmit visible sunlight but reflect
thermal radiation back into a heated room.
To get a reflectivity cutoff in the infrared one needs a smaller electron
density than in a metal. A highly-doped semiconductor is just right, such
as indium-tin-oxide (ITO). We encountered this material already as
transparent front electrode for solar cells and LCD screens.
An ITO film transmits visible light and
reflects thermal infrared radiation,
keeping the heat inside a building.
R = Reflectivity
T = Transmission
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Low-Dimensional Plasmons in
Nanostructures
We have showed how single electron waves become quantized by
confinement in a nanostructure. Likewise, collective electron waves
(=
plasmons) are affected by the boundary conditions in a thin film,
a
nano-rod, or a nano-particle.
Plasmons in metal nanoparticles are often called Mie-resonances, after
Gustav Mie who calculated them hundred years ago. Their resonance
energy and color depend strongly on their size, similar
to the color
change induced in semiconductor nanoparticles by confinement of
single electrons. In both cases, smaller particles have higher resonance
energy (blue shift).
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
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Nanotechnology in Roman Times: The Lycurgus Cup
Plasmons of gold nanoparticles in glass reflect green, transmit red.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Quantum Numbers of Plasmons
Like any other particle or wave in a (crystalline) solid, a plasmon has the
energy E and the momentum p as quantum numbers, or the circular
frequency  = E/ħ and the wavevector k = p/ħ . One can use the same E(k)
plots as for single electrons.
Photon
Bulk Plasmon
Surface Plasmon
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Coupling of Light and Plasmons
To combine optoelectronics with plasmonics one has to convert
light (photons)
into plasmons. This is not as simple as it sounds.
Bulk plasmons are longitudinal oscillations (parallel to the propa-gation
direction), while photons are transverse (perpendicular to the propagation). They
don’t match.
Surface plasmons are transverse, but they are mismatched to photons in their
momentum. The two E(k) curves never cross.
It is possible to provide the
necessary momentum by a grating, which transmits the wavevector k = 2/d
(d = line spacing) .
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Attenuated Total Reflection
Another method to couple photons and surface plasmons uses attenuated
total reflection at a metal-coated glass surface. The exponentially
damped (evanescent) light wave escaping from the glass can be matched
to a surface plasmon (or thin film plasmon) in the metal coating. This
technique is surface sensitive and can be used for bio-sensors.
Gold
film
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Plasmons and the Dielectric
Constant 
The dielectric constant is a complex number:
 = 1 + i 2
The real part 1 describes refraction of light,
The imaginary part 2 describes absorption .
The bulk plasmon occurs at an energy Ep where 1 = 0,
the surface plasmon occurs at an energy Es where 1 = -1 .
(More precisely: Im[1/] and Im[1/(+1)] have maxima.)
1
2
Typical behavior of the dielectric
constant versus energy E for a solid
with an optical transition at E=E0 . A
metal has E0=0 .
1
0
-1
0
E
Ep
E0 s
E
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
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P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
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Mesoscopic oscillation of
charge around a positive
lattice
Bandgap (s & p band)
+ Envelope (curvature of wave
function)
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
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Theoretical Tools


Metals:
– Lorentz Oscillator Theory of Materials
– Plasmons and Plasmonics
Semiconductors:
– Band Theory for Crystals
– Band Transport at Nanoscales:
Molecular Metals and Semiconductors
– Microscopic Description of Optical
Properties
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
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P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Optical Properties: Surface Plasmon Resonance
•Surface plasmon resonance is the coherent excitation of all
the "free" electrons within the conduction band, leading to an
in-phase oscillation.
•When the size of a metal nanocrystal is smaller than the
wavelength of incident radiation, a surface plasmon resonance
is generated.
• The figure shows schematically how a surface plasmon
oscillation is created. The electric field of an incoming light
induces a polarization of the free electrons relative to the
cationic lattice.
•The net charge difference occurs at the nanoparticle
boundaries (the surface), which in turn acts as a restoring
force. In this manner a dipolar oscillation of electrons is
created with a certain frequency.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
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60
Surface plasmon resonance




In this manner a dipolar oscillation of electrons is created with a
certain frequency. The energy of the surface plasmon resonance
depends on both the free electron density and the dielectric medium
surrounding the nanoparticle.
The resonance frequency increases with decreasing particle size if the
size of the particles is smaller than the wavelength of the particles.
The width of the resonance varies with the characteristic time before
electron scattering. For larger nanoparticle, the resonance sharpens as
the scattering length increases. Noble metals have the resonance
frequency in the visible light range.
Mie was the first to explain the red color of gold nanoparticle colloidal
in 1908 by solving Maxwell's equation for an electromagnetic light
wave interacting with small metallic spheres.
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1
Bulk plasmon: quantization of collection valence
electron density oscillation at frequency
ne 2
p 
 0m
ħp is typically ~ 10-20 eV
Surface plasmon
sp 
p
2
localized at the surface
and its amplitude
decays with the depth
P.Ravindran, Nanomaterials and Nanotechnology, Spring 2016: Physicls of Low dimensional Materials -1