Download Size-Dependent Properties Classes of Materials

Nanophysics
&
Lectures Notes
http://www1.na.infn.it/TIMSI/materialicorsi/iavarone/Nanophysics&Nanotechnology/Nano&Nano.htm
Nanotechnology
Size-Dependent Properties
Classes of Materials
Metallic Materials
Good thermal and
electrical conductors,
ductile
Ceramic Materials
Hard, brittle, good
insulators
Polymeric Materials
Good insulators, good
resistance to corrosion
Composite Materials
Matrix-reinforcing
phase. Properties vary
At the nanometer scale property become size dependent.
For example:
1. Mechanical properties-adhesion, capillary forces
2. Chemical Properties-reactivity, catalysis
3. Thermal Properties-melting point
4. Electrical properties-tunneling current
5. Optical properties-absorption, scattering
6. Magnetic properties-superparamagnetic effect
New properties enable new applications!
Electronic Materials
Metal, ceramic,
polymers,
semiconductors
Biomaterials
Bio-compatible
Nanomaterials
Nanoparticles,
nanotubes, nanowires,
nanocoatings…
1
Thermal Behavior
Heat Conduction
Heat affects mechanical, electrical, and optical
properties .
As temperature rises, materials expand, the
elastic modulus decreases, the strength falls, and
the material start to creep, deforming slowly until
the melting point is reached.
Electrical resistivity rises with temperature, the
refractive index falls, color may change.
Heat Conduction
Gases: hotter air molecules (with larger kinetic energy)
randomly pass their excess energy to cooler molecules.
Heat is transported by occasional collisions with each
other. The average velocity of a molecule can be as large as
500 m/s.
Radiation
Solids: Heat is transmitted in three ways, thermal vibrations,
movement of free electrons in metals, and if they are
transparent by radiation.
Thermal waves are really not different from e-m waves that
we use in communication (radio/TV signals).
There is one fundamental difference which is the wavelength:
λ=
Average
velocity 105106m/s
c 3 ×108 m / s 1
=
= m
f 900 × 10 6 H 3
λ = 0.5µm
Much shorter
The major difference is not in the propagation but in the
generation.
generation
Thermal radiation typically refers to e-m waves that are
generated by the oscillation charges in the atoms, while
radio/TV signals are generated by artificial current
oscillations in circuits.
2
Phonons
Thermal Conductivity:
The rate at which heat is conducted in a solid is measured by
the thermal conductivity, which is a material property (for
macroscopic samples)
k[W / m ⋅ K ]
Fourier’
Fourier’s Law:
Thermal conductivity is calculated from Fourier’s law
The phonons travel through the material, and like any elastic
wave, they move with the speed of sound (few thousand per
meter per second).
c0 = E ρ
q = −k
q = [W / m 2 ]
dT
(T − T )
= −k 1 2
dx
x1 − x2
Heat flux
Phonon Conduction
Why does heat not diffuse at the same speed of sound?
Because of scattering they travel a short distance before
bouncing. This distance is called mean free path and it is
tipically 0.01 microns.
Net flux model:
The rate at which heat is conducted in
a solid is measured by the thermal
conductivity, which is a material
property (for macroscopic samples)
1
dT
q = ρC p lm c0
3
dx
Elastic waves contribute little to conductivity of pure
metals such as copper or aluminum because the heat is
carried more rapidly by the electrons. The equation
shown earlier is still valid, but Cp, c0 and lm become the
thermal capacity, the velocity, and the mean free path
of the electrons.
1
k = ρC p lm c0
3
3
Thermal Transport of
Nanomaterials
Classical size effects.
In bulk materials the characteristic length of the box is
much longer than the mean free path. Therefore, the
collisions between the molecules (electrons, phonons)
and the wall are neglected and thermal conductivity is
regarded as a bulk property.
In nanomaterials the mean free path becomes
comparable or larger than the size of the system.
Quantum size effects.
According to quantum mechanics, electrons and phonons are
also material waves. In the case of a bulk material the
wavelength of phonons are much smaller than the length
scale of the sample. However, in nanomaterials the length
scale of the material is similar to the wavelength of the
phonons. Therefore, quantum confinement occurs.
In 0D nanomaterials (nanoparticles) confinement occurs
in 3 dimensions
In 1D nanomaterials (nanowires, and nanotubes)
confinement is restricted to 2 dimensions
In 2D nanomaterials (nanofilms, nanocoatings) quantum
confinement takes place in one dimension
Quantum size effects.
The presence of the surface in nanostructures causes a
change in the distribution of the phonon frequencies, and
creates new modes that are not present in bulk.
The same happens for electron.
One-dimensional materials behave as phonon waveguide,
similar to optical one for light. For example for carbon
nanotubes, several authors have predicted very high thermal
conductivity along the nanotube, close to 3000 Wm-1K-1.
As a comparison we should keep in mind that the thermal
conductivity of copper is approximately 400 Wm-1K-1.
Despite these results, there are still open questions about
phonon transport in 1D nanostructures, particularly
regarding phonon/phonon interactions and the role of
defects.
4
Thermal conductivity of
suspended Platinum nanofilms
Thermal conductivity of
materials with nanoscale grains
Polycrystalline silicon films show a
strong reduction of thermal
conductivity with respect to the bulk.
Pt 28 nm
X. Zhang et al., Applied Physics Letters, 86, 171912 (2005)
Thermal conductivity is less than those of the corresponding
bulk materials. In fact, the thinner the film the lower the
thermal conductivity
For nanoporous materials, the nanosize
effect is determined by the number
and the size of the pores.
S. Uma et al., Int. J.
Thermophys., 22 , 605
(2001)
Low thermal conductivity, in the case
of microelectronic components, leads
to an increase in the operation
temperature and earlier circuit
failure.
A brief review of quantum
mechanics
Quantum mechanics tells us that the fundamental building
blocks of nature can be thought as both particles and waves.
What determines when we must take into account the wave
nature of particles?
λ=
h
p
de Broglie wavelength
If the de Broglie wavelength of the particle is less than its
physical dimensions, then the classical mechanics provides a
good description.
5
Electrical Behavior
Resistivity and Conductivity
I
Electrical Materials
Normal metal
S
l
Conductors
Insulators
Non-piezoelectric
dielectrics
Quartz
(Ultrasonic oscillator
ink-jet print-heads)
Piezoelectrics
Non-pyroelectric
Pyroelectric
(Barium Titanate)
Non-ferroelectric
Ferroelectric
R=
V
i
ρ =R
ΘD
S
l
ρ [Ω ⋅ m ]
For good conductors the resistivity is of the
order of 1 µΩcm while for best insulators is
more than 1024 µΩcm.
The electrical conductivity is simply the
reciprocal of the resistivity. Its units are
Siemens per meter (S/m or (Ωm)-1).
PZT ( Diesel injector)
Electrical Behavior
How come some materials have mobile electrons and
some do not?
To answer this question we need quantum mechanics
Electrical Behavior
When n atoms are brought together to form a solid the inner
electrons remain in within the atom, the outer ones start
interacting. As a consequence the discrete level of an isolated
atom broadens to form bands (of closely spaced energy levels).
The electrons fill the bands from the bottom. The topmost
filled energy level is called the Fermi level.
The electrons of an atom occupy discrete energy levels.
The electrons fills the shells with the lowest energy
according to the Pauli principle (two electrons of
opposite spin in each orbit).
6
Electrical Resistance
If a field E exerts a force Ee on an electron, why
does it not accelerate forever, giving a current that
continuously increases with time?
Electrical Behavior: Quantum Size Effects
This overall behavior of bulk materials changes when the
dimensions are reduced to nanoscale.
For 0-D nanomaterials an electron is confined in 3D space
For 1-D nanomaterials an electron is confined in 2-D, whereas
delocalization takes place along the long axis (nanotube…)
Normal metal
For 2-D nanomaterials the conduction electrons will be
confined along the thickness but delocalized in the plane.
ΘD
Electrical Behavior: Quantum Size Effects
For example let’s consider electrons in a thin film.
They can be approximated as standing waves inside a potential
well. The condition for the formation of such a standing wave
is that the wavelength, λ, satisfies the following relation:
Electrical Behavior: Quantum Size Effects
λ=
nh
2D
The energy of the electron is:
E=
D = λn 2 ( n = 1,2...)
p=
h 2D
=
p
n
p2
h2n2
h2
=
=
n2
2m (2 D )2 2m 8mD 2
Keeping in mind that me=9.1x10-31Kg, h=6.6 x 10-34 :
1) D=1mm
2
E=
where D is the width of the potential well (thickness of the
film). According to the de Broglie relation, the wavelength is:
λ=
h
p
 6.6 × 10 −34 
n2

 ≈ 10 −34 n 2 J << k BT = 4.14 × 10 −21 J
8 × 9.1× 10 −31  10 −3 
room temperature
2) D=10-8 m
2
E=
 6.6 ×10 −34 
n2

 ≈ 10 −21 n 2 J ≥ k BT = 4.14 × 10 −21 J
8 × 9.1× 10−31  10−8 
7
Electrical Behavior: Quantum Size Effects
There are critical questions to answer. When to consider classical size effect or
quantum size effect?
2-D Nanomaterials: Potential Well
The electrons become trapped in a so called
potential well of width equal to the thickness.
Keeping in mind that ћ =h/2pπ
100
En =
10
1
 π 2h 2  2
h2
n
n 2 = 
2
2 
8mD
 2mD 
E/kBT
0.1
0.01
MacroMacroscopic
Quantum
1E-3
The electrons are free to move in the plane x-y.
Therefore the total energy of the lectrons has two
components. The unrestricted motion in plane by two
wave vectors kx and ky, which are related to the
momentum along x and y, px= ћkx and py= ћky.
1E-4
1E-5
Mesoscopic
1E-6
1E-7
1E-8 0
10
1
10
2
3
10
10
4
10
D (nm)
 π 2h 2  2  h 2 k F2
n + 
En = 
2 
 2mD 
 2m



The boundaries between these regions will shift to larger sizes as the
temperature is reduced and if we use semiconductors (lower effective mass)
1-D and 0-D Nanomaterials: Energy Levels
For 1-D nanomaterials an electron is confined in 2-D, whereas delocalization
takes place along the long axis (nanotube, rod, wire…)
 π 2 h 2 n y 2   π 2h 2 n z 2 
+

Eny ,nz = 
 2mL 2   2mL 2 
y
z


 
 π 2 h 2 n y 2   π 2 h 2 nz 2   h 2 k x 2 
+
+

E =
 2mL 2   2mL 2   2m 
 
y
z


 
Density of States
If we consider a single electron traveling through a conductor. It will have a
wave-function of the form ψ(r)=eik r. It is instructive to think of the occupied
states as represented by points within a sphere in k-space. The surface of the
sphere with k=kF is known as the Fermi surface, and represents states at the
Fermi energy.
Energy of electrons at the Fermi surface
EF =
h 2 kF
2m
2
 2π 
V1 = 

 L 
Volume of one state
For 0-D nanomaterials the motion of electrons is totally confined alonfg the
three directions Lx, Ly, Lz. Therefore, the energy is given by:
Enx , ny ,nz
Number of electrons in the conductor
2
 π 2 h 2 nx 2   π 2 h 2 n y   π 2 h 2 nz 2 
+
+

= 
2 
 2mL 2   2mL 2 
2
mL
x
y
z

 

 
Under these conditions, metallic systems can behave like insulators, due to the
formation of an energy band gap!
3
N =2
 3π 2 N 

k F = 
 V 
1
4π k F
3
3 = Vk F 3
3
3π 2
(2π L )
3
EF =
h 2  3π 2 N 


2m  V 
2
3
8
Calculated Conduction Parameters
Density of States in 3-D
The number of states between k and k+dk is:
∆N =
4πk 2 dk Vk 2 ∆k
=
2π 2
V1
The density of states is defined as the number of quantum states per unit
interval of energy and per unit volume:
D( E ) =
E=
h2k 2
2m
1 ∆N
k 2 ∆k
k 2 dk
=
=
V ∆E 2π 2 ∆E 2π 2 dE
k=
2m
E
h2
D(E ) = 2
Density of States in Nanomaterials
The number of conduction electrons that exist at a given energy level, the
number of electrons dN in a narrow energy range dE: D(E)=dN/dE. This
quantity strongly depends upon dimensionality of the structure.
3
k 2 dk
1  2m  2 12
=

 E
2π 2 dE 2π 2  h 2 
Size effect on Electrical Resistance
How does the resistance change as we make a conductor smaller?
What is so interesting about nanoscale that a wire is not simply a passive element!
Various physical properties depend upon the density of states: specific heat,
thermopower effect, superconductivity energy gap. Overall the ability to
control the density of states is crucial for applications such as infrared
detectors, lasers, optical memories……
9
Magnetic Behavior
Magnetic Fields in Materials
Magnetic fields are created by moving electric chargeelectric current in electromagnets, electron spin in atoms
of magnetic materials.
If the space inside the coil is filled with a material the
induction changes, because its atoms respond to the field
by forming magnetic dipoles. The material acquires a
macroscopic dipole moment or magnetization.
H=
n=number of turns in the coil
L= length
i=current
H=magnetic field generated
ni
L
B = µ 0 (H + M )
B = µ0 µ R H
H is a vector and is measured in A/m
M = (µ R − 1)H = χH
Magnetic fields exert forces on a wire carrying an electric
current. A current i flowing in a loop of area S generates a
dipole moment m where:
m = iS
M is a vector measured in Am2
B = µ0 H
Saturation magnetization decreases
with temperature, falling to zero at
the Curie temperature
B is measured in Tesla
Origin of Magnetic Behavior
Orbital Motion of Charged Particles
Flow of charge gives rise to magnetic field
Charges flow in response to magnetic field
There is, therefore, a magnetic dipole associated with
each orbiting electron.
χ is the magnetic susceptibility
Different types of Magnetic Behavior
Paramagnetic
The interaction energy between a
magnetic dipole and magnetic
field is given by:
Emagnetic = µ ⋅ B = µB cos θ
Spin angular momentum
Spin is a pure quantum mechanical property
It can be thought as a particle turning on its own axis
To minimize the energy, the dipole
moment points along the field
10
Different types of Magnetic Behavior
Different types of Magnetic Behavior
Ferromagnetism
Paramagnetic
For a paramagnet the interaction between spin is negligible.
There is no magnetization in absence of external
magnetic field.
Various types of interaction may lead to alignment of
adjacent spins.
If the interaction prefers parallel
alignment, then the interaction is
called ferromagnetic.
If the interaction prefers antiparallel alignment, then the
interaction is called
antiferromagnetic.
Note: there is no external magnetic field yet !
Energy Scales
Energy Scales
Paramagnets
Ferromagnets
Does the fact that the interactions between magnetic
moments in a paramagnet is weak mean that all moments line
up exactly in even a small external field?
Instead of the magnetic energy of a magnetic moment in an
external magnetic field, the relevant energy is the
interaction energy between the moments, J (exchange
energy).
The moments will be randomizedJ < k BT
no net magnetization
No. While the external field would like to align them, finite
temperature likes to randomize their directions.
Alignment of all magnetic moments depends on the balance
between the energy associated with a moment in magnetic
field, and the thermal energy.
Emagnetic < k BT
Emagnetic > k BT
The moments will be randomized
J > k BT
The moments will order spontaneously
Phase transition
The moments will be aligned
11
Other magnetic systems
Paramagnetic
Ferromagnetic
Antiferromagnetic
Even a ferromagnet below the Curie temperature may not
have a net magnetization. This is because the ferromagnet
breaks up into domains, each with its own magnetization
pointing in a different direction. A magnetic field is required
to align the moments of all the domains in the same direction.
Ferrimagnetic
In a system with antiferromagnetic
interactions the crystal structure
might be such that the system
cannot order antiferromagnetically.
Ferromagnet: Domains
Hysteresis curve of a ferromagnet
An external magnetic field
progressively aligns the
direction of the
magnetization of the domains.
The magnetization saturates
when all the domains are
aligned. Moving domains costs
energy. Therefore, the
magnetization is not a single
valued function of the
external field.
12
Ferromagnet: Domain Walls
Between two magnetic domains, the average magnetization
must switch direction:
Energy Considerations
Etotal = Eexc + Eani + Edem + Eapp
The area over which the
magnetization reverses is
called a domain wall. The
domain wall is typically
determined by the material
properties.
Etotal = M ⋅ H
Typical domain wall width
for conventional
ferromagnets are of the
order of 20-100 nm.
Closure domain structure
Open domain structure
Nanoferromagnets
Nanoferromagnets
What happens if the dimensions of a ferromagnet are smaller than a
domain wall?
Exchanges forces are dominant, therefore there is a critical grain size
below which the materials will be single domain. For a spherical grain:
Dcri =
9γ B
µ 0 M S2
γ B = 4( AK1 )1 2
A exchange stiffness
K1 anisotropic constant
MS saturation magnet.
If the particle size becomes too small (typically few nm), the
magnetization can become unstable and loss of magnetization occurs due
to thermal fluctuations. This is called superparamagnetic limit.
This determines the minimum size of the domains that can be used to
store information.
This can be controlled by controlling materials parameters
SD
MD
Critical diameter for Co is 70 nm, for Fe is 15 nm.
Which way will the magnetization point?
Coercitivity field
versus particle size
HC
A number of factors control the direction of the magnetization. Two
are particularly important: crystalline anisotropy, shape anisotropy.
DSP
Dcrit
13
Giant and Colossal Magnetoresistance
Optical Behavior
Many aspects of radiation are
understood by thinking of radiation as a
wave. Others need a different picture.
Radiation is discrete packets of energy,
photons.
Change in resistance 5%, while in CMR
materials (manganese base perovskite
oxides) there are changes of order of
magnitudes
Why aren’t metals transparent?
E ph = hν =
hc
λ
How does light get through dielectrics?
Metals absorb photons
by promoting an electron in
an empty level. When the
electron falls back a photon
is reemitted.
Dielectrics have a full band separated by the conduction band by an energy
gap. The material cannot capture photons with an energy less than the gap,
for those frequencies the material is transparent
14
Color
Excitons
Optical Behavior in Nanostructures
Optical Behavior in Nanostructures
The effect of nanoscale on optical absorption are associated with:
1. The density of states in valence and conduction bands
2. The quantized energy levels of the nanostructures
3. The influence of excitonic effects.
When moving from 3D to 0D quantum confinement is more severe,
band gap shift to higher energies, i.e. shorter wavelengths. A blue
is expected whe the size is decreased and red when the size is
increased.
Cd-Se quantum dots of various size
Also emission of photons can be tuned with the nanoparticle size. Typical trend is
a shift of the emission peak toward shorter wavelength (blue shift).
15
Optical Behavior in Metallic
Nanostructures: Plasmons
Plasmons are quantized waves that propagate in
materials through electrons. Plasmons can exist in
the bulk and on the surface. Among these surface
plasmons are more relevant for nanomaterials.
Surface plasmons have lower frequencies than
bulk plasmons and thus they can interact with
photons.
When photons couple with surface plasmons
(surface plasmons polaritons), alternating regions
of positive and negative charges are produced on
at the surface.
Surface plasmons-polaritons can be used to cause
extraordinary transmission
Light absorption and emission can be enhanced by using
nanoparticles. As a result, metallic nanoparticles acan be
used as structural and chemical label.
16