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Transcript
Nanostructured materials
• 0D: quantum dots
• 1D: Nanowires
• 2D: superlattices and heterostructures
• Nano-Photonics
• Magnetic nanostructures
• Nanofluidic devices and surfaces
Copyright Stuart Lindsay 2009
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.
Copyright Stuart Lindsay 2009
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.
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.
Copyright Stuart Lindsay 2009
k-space is filled with an
uniform grid of points each
separated in units of 2π/L
along any axis.
The volume of k-space
occupied by each point is:
 2 


 L 
r-space:
4 r 2 dr
V
3
k-space:
4 k 2 dk 4 L3 k 2 dk

Vk
8 3
3D DOS
Density of states in a volume V
per unit wave vector:
For a free electron gas:
dn Vk 2

dk 2 2
2k 2
E
2m
dn dn dk Vk 2 m
Vm

 2 2  2 2
dE dk dE 2  k  2
dE  2 k

dk
m
1
2mE
2

E
2
Copyright Stuart Lindsay 2009
2D DOS
dn A2k

dk 2 2
dE  2 k

dk
m
dn dn dk
Am


dE dk dE 2 2
Constant for each
electronic band
Copyright Stuart Lindsay 2009
1D DOS
dn
L

dk 2
dE  2 k

dk
m
dn
Lm

E
2
dE 2 k

1
2
At each atomic level, the DOS
in the 1D solid decreases as
the reciprocal of the square
root of energy.
Copyright Stuart Lindsay 2009
8
0 D DOS
In zero dimensions the energy states are sharp levels corresponding
to the eigenstates of the system.
Copyright Stuart Lindsay 2009
0D Electronic Structures:
Quantum Dots
Light incident on a semiconductor at an energy greater than the
bandgap forms an exciton, i.e. an electron-hole quasiparticle,
representing a bound state.
Excitons can be treated as “Bohr atoms”
2
2
mV
1 e

r
40 r 2
40 
r 2 *
e m
2
When the size of the nanoparticle approaches that of an exciton,
size quantization occurs.
Electronic
energy gap
2 2
2




1
1
e
 *  *   1.8
E  E 0 
2 
2 R  me mh 
R
Intrinsic
band gap
NP radius
electrostatic
correction
1-D Electronic Structures: Carbon Nanotubes
Wrapping vector:
Diameter:



n  n1a1  n2a2
d  n12  n22  n1n2 0.0783nm
The folding of the sheet controls the electronic properties of the
nanotubes.
Conduction in CNTs
pz electrons hybridize to form π e π* valence and conduction bands
that are separated by an energy gap of about 1V (semiconductor).
For certain high simmetry directions (the K points in the reciprocal
lattice) the material behaves like a metal.
k  wave vector perpendicular to the
CNT long axis
Apex: at this point CB
meets VB for graphene
sheets (metal-like
behavior)
Allowed K  states
The component of the wave vector perpendicular to the CNT long
axis is quantized
2n
k 
D
D = diameter of the nanotube
Metallic behavior: the allowed values of k  intersect the k points
at which the conduction and valence bands meet.
CNTs can be either metals or semiconductors depending on their
chirality.
Field effect transistor made from
a single semiconducting CNT
connecting source and drain
connectors.
Semiconductor Nanowires
• Ga-P/Ga-As p/n nanojunctions
TEM images
Line profiles of
(IOP)
the composition
through the
junction region
Copyright Stuart Lindsay 2009
2D Electronic Structures:
superlattices and heterostructures
Superlattice:
alternating layers of small
bandgap semiconductors
(GaAs) interdispersed with
layers of wide bandgap
semiconductors (GaAlAs).
Variation of electron energy in an
MBE grown superlattice
The thickness of each layer
is considerably smaller
than the electron mean free
path.
17
Modulation of the structure on
the length scale d (thickness of
the layer in the superlattice)
gives rise to the formation of
new bands inside the original
Brillouin zone .
Electrons can pass freely from
one small bandgap region to
another without scattering.
Band splitting into sub-bands
Low scattering in 2-D means reaching zone boundaries at
reasonable fields, accelerating electrons at the band edges.
Resonant tunneling through different sub-bands
Negative differential resistance: electrons slow down with increasing
bias when approaching the first sub-band boundary (≈20mV).
Quantum Hall resistance of 2D electron gas
Electrons in a layer are accelerated by an applied magnetic
field at a frequency:
eB
C 
m
Magnetic quantization in
2D electron gas
A series of steps in the Hall
resistance corresponding
exactly at twice the
Landauer frequency were
observed.
Copyright Stuart Lindsay 2009
h
RH  2
ne
von Klitzing resistance
Nobel in Physics 1985
Confinement on optical length scales
Plasmonics
Small (d<<λ) metal particles exhibit a phenomenon called plasma
resonance, i.e. plasma-polariton resonance of the free electrons in
the metal surface.
A resonant metal particle can capture light over a region of many
wavelengths in dimension even if the particle itself is only a fraction
of a wavelength in diameter (resonant antennas).
Free electrons in metals polarize excluding electric fields from the
interior of the metal showing a negative dielectric constant.
The polarizability of a sphere of volume V and dielectric constant εr
is:
 r 1
   0 3V
r  2
When εr →-2
→∞
For d<<λ the resonant frequency is independent on the particle
size, but depends on particle shape.
For a prolate spheroid of eccentricity e:
b
e  1  
a
2
2
 0V
1 r

L (1 L  1)   r
where:
1  e2 
1 1 e   a 
L  2   1  ln
  1  
e 
2e 1  e   b 
1.6
For Ag =-2 at 400 nm, but resonance moves to 700 nm for a/b=6.
The resonance is tunable throughout the visible by engineering the
particle shape.
Plasmon enhanced optical absorption
Placing a chromophore near a resonant metal nanoparticle:
dye layer
Electric field surrounding
a resonant nanoparticle
(E=Ez)
Enhanced fluorescence
Reduced decay times
The plasmon resonance results in local enhancement of the electric
field: doubling the electric field quadruples the light absorption.
Enhanced fluorescence
A single dye
molecule
is
only visible in
fluorescence
when the gold
NP passes over
it.
The increased absorption cross section is accompanied by a
decrease in fluorescence lifetime.
Photonic engineering
3D
2D
Modern semiconductor lasers are
made
from
semiconductors
heterostructures designed to trap
excited electrons and holes in the
optically active part of the laser.
Performance of a solid-state laser
material in various geometries.
1D
Lasing effect: the gain of the laser
medium must exceed the cavity
losses.
Dashed lines: density of states
0D
In quantum dots all of available
gain is squeezed into a narrow
bandwidth.
25
Optical cavity
Quantum dot laser
Quantum dots of the right size can
place all of the exciton energies at
the right value for lasing.
The QDs are chosen to have a
bandgap that is smaller than that of
the medium.
Excitons are stabilized in the
optical cavity, because the
electrons are confined to the lowenergy part of the conduction band
and the holes are confined to the
top of the valence band.
Photonic crystals:
concentrating photon energy into bands
Opalescent materials from colloidal crystal (polystyrene latex beads)
The concentration of modes into bands results in an increase in the
density of states in the allowed bands, particularly near bands edges.
Optical wavelengths require that materials should be structured on
the half-micron scale.
Opalescence comes from sharp (Bragg) reflection in only certain
directions.
Copyright Stuart Lindsay 2009
For a given spacing in some direction in the colloidal crystal lattice ,
the wavelength of the reflected beam is given by the Bragg law.
Bragg’s law
  2nd hkl cos  int 
 2d hkl n 2  sin 2  ext
Colloidal particles spaced
with polymer spacers
sin ext  n sin int
hkl = Miller indices of the colloidal crystal lattice
n = refraction index
dhkl = spacing between Bragg planes in the hkl direction
3D Photonic crystals
Require “non spherical” atoms to give zero “structure factor” in
directions where propagation must be suppressed
Optical dispersion in a crystal made by non spherical atoms. A 3D
bandgap appear when structures are designed to have zero intensity
in directions of allowed Bragg reflections.
Repeat period is on the order of 30μm (3D optical bandgap in the far
IR, limit of today technology).
Photonic crystals convert heat into light! See Optics Letters 28
1909, 2003.
Magnetic properties
• Diamagnetism:
Zero-spin systems give rise to circulating currents that oppose the
applied field (negative magnetic susceptibility, Larmor
diamagnetism).
• Paramagnetism:
Free-electrons are magnetically polarized by an external magnetic
field (positive magnetic susceptibility, Pauli paramagnetism).
• Ferromagnetism:
Spontaneous magnetic ordering due to electron-electron interactions.
Antiferromagnetism: polarization alternates from atom to atom. No
net macroscopic magnetic moment arises.
Magnetic Interactions
• Exchange (electron-electron) interaction (many-particle
wavefunction antisymmetry)
- atomic scales
• Dipole-dipole interactions between locally ordered magnetic
regions
Dipole interaction energy grows with the volume of the
ordered region. The size of the individual domains is set by a
competition between volume and surface energy effects.
- hundreds of atoms to micron scales
• Magnetic Anisotropy energy
Magnetization interacts with angular momentum of the atoms
in the crystal.
– many microns
Super-paramagnetic particles
• Ferromagnetic domains, created by d-electrons exchange
interactions, develop only when a cluster of iron atoms reaches a
critical size (ca. microns).
The magnetic moment per
atom decreases toward the
bulk value as cluster size is
increased.
Stable domains cannot be
established in crystals that
are smaller than the intrinsic
domain size.
• Small particles can have very high magnetic susceptibility with
permanent magnetic dipole.
Small clusters consisting of a single ferromagnetic domain
follow the applied field freely (super-paramagnetism).
The magnetic susceptibility of superparamagnetic particles is
orders of magnitude larger than bulk paramagnetic materials.
Ferromagnetic limit
Magnetic response for
particles of increasing
size (Gd clusters)
Superparamagnetic separations
Induced magnetic moment:
M  H
Magnetic force:
B
Fz  M z
z
B   0 ( H  M)
Magnetic sorting of cells labeled
with superparamagnetic beads
MFS: microfabricated
ferromagnetic strips
Particle were pulled to point of highest field gradient
Copyright Stuart Lindsay 2009
Giant Magnetoresistance
Magnetic hard drives are based on a nanostructured device, called
giant magnetoresistance sensor.
Albert Fert, Peter Grünbers Nobel Prize in Physics 2007
Hitachi hard drive reading head
The magnetization on the surface
of the disk can be read out as
fluctuations in the resistance of
the conducting layer.
Co, magnetic layer
Layers have a width that
is smaller than electron
scattering length.
Cu, electrically conducting layer
NiFe alloy, magnetic layer
An easily re-alignable magnetization
Giant magnetoresistance occurs when the magnetic layers above
and below the conductor are magnetized in opposite direction.
Electron scattering in magnetic
media is strongly dependent on
spin polarization.
I
II
III
When magnetic layers are
parallely magnetized, only one
spin polarization is scattered
(I,III).
For antiparallel magnetic layers both spin polarizations are
scattered, giving rise to super-resistance (II).
Nanofluidics
Fluid flow in small structures is entirely laminar and dominated
by the chemical boundaries of the channel.
Reynolds number (Re), a dimensionless number quantifying the
ratio of inertial to viscous forces that act on the volume of a
liquid:
density
uL
Re 

Re >> 1: turbulence regime
Re << 1: viscous regime
channel narrowest
dimension
viscosity
Kinematic viscosity


u
For water: ν =1·10-6 m2s-1 (25°C)
Re in a L=100nm channel with <u> = 1mm/s not exceeds 10-4
Flow in nanoscale channels is dominated by viscosity!
Fluids do not mix in a nanofluidic device.
The chemistry of the interface becomes critical and aqueous
fluids will not generally enter a channel with hydrophobic
surfaces.
1-D Nanochannel devices
A significant stretching of large molecules can occur in a large ion
gradient or electric field in a channel that is comparable to the
radius of giration of the molecule.
Ex. A 17μm DNA (fully stretched length) is equivalent to 340
freely jointed segments each of 5 nm length.
The relative giration radius is:
r  340  5  92nm
It can be significantly extended in a channel of 100 nm diameter,
owing to the strong interaction between the fluid and the wall.
DNA is introduced through the microchannel and then transported
through the nanochannels by an applied voltage.
DNA cutting starts
Mg ions introduced
Time
Fluorescently labeled
DNA at various times
Microchannels
DNA + cutting
enzymes introduced
distance
Continous time course
of the cutting process
Nanopores: 0D fluidic nanostructure
2-nm diameter holes: only a single DNA molecule can pass
through it at a given time.
electrophoresis
Polystyrene
bead
Optical tweezer
The passage of DNA molecules can be measured by the drops in
currents that occur when a single DNA molecule occludes the hole.
Flow in Carbon Nanotubes
A. Fabrication of a layer of
CNTs penetrating a silicon
nitride membrane;
B. and C. SEM images before
and after filling with silicon
nitride;
D. Individual finished device:
E. An array of devices
The measured rate of water flow through 2nm-diameter CNTs was found
to be 1000 times higher than predicted by classical hydrodynamics. 42
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
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.