Advanced Functional Properties 6PC20

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Advanced Functional Properties 6PC20

• Lecture 4 Metals

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Last week dielectric response (water)
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Summary
Contributions to dielectric response function
rotation Debye model rotations of polar
molecules vibration Lorentz oscillator model
active nuclear vibrations Various
vibrations may be active each with its own ?0 ,
tvib , ?p Electronic excitation Lorentz
oscillator model active electron
oscillations electrons can oscillate with
different ?0 , tvib , ?p
Parameters Debye trot rotational correlation
time
Parameters Lorentz ?0 natural frequency tvib
damping or friction ?p amplitude or intensity
Increasing frequency
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The electron in an atom as a Lorentz oscillator
Bound electron Harmonic potential Natural
frequency w0
Fspring -kx kmw02
Electric field oscillating at w drives the
electron
e, M (Mgtgtme)
-e, me
Fdrive -e E0 eiwt
Ionized atom e.g. Na
Ffriction - (me/t )(dx/dt)
Friction damps the motion of the electron
Causal relation Newton F m a
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The electron in an atom as a Lorentz oscillator
Oscillatory motion of the electron around an atom
or molecule
Polarization Ne number of oscillators per unit
volume
wp is called the plasma frequency
Plasma gas which is so hot that the atoms
/molecules are ionized
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Putting all kinds of motions together (example)
rotation
vibration 2
excitation 1
vibration 1
excitation 2
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Metals No bonded charges
Suppose that we put w0 0 Electron is then not
bound to the atom core It is a free
electron The material should be a metal !
Lorentz oscillator
Drude model
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Dielectric response of metals Aluminum
Ordal Appl Optics 27 1203 1988
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Dielectric response of metals Gold
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Dielectric response of metals silver
1/w2
1 eV energy needed to accerlate one electron
over 1 volt potential difference
1eV is equivalent to a cycle frequency n of
qevolt /h 2.418 1014 sec-1
1eV corresponds to a wavelength of light in
vacuum of 1240 nm
Handbook of Chemistry and Physics
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Tabulated Drude parameters
t (fs)
wp (1014 s-1)
s0DC (106 ohm-1m-1)
s0 (106 ohm-1m-1)
s0Drude / s0DC
s0DC from static electrical measurements
s0Drude from Drude parameters
Ordal et al. Appl Optics 24, 4493 (1985)
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Family tree of oscillator models

Lorentz oscillator
Overdamped motion
free electrons
Drude model
All these oscillator models Are Kramers-Kronig
consistent !
Debye model
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Conclusion on e(w) for metals
- Low frequency part of e(w) for metals can be
accounted for by the Drude model
Drude model
- In the high frequency part there are many
features which can not be accounted for.

• Questions to be investigated in this lecture
• What is the relation with conductivity ?
• What does t mean in this case ?
• 3. How can we understand w0 0 ? Shouldnt there
  always be some attraction between the electron
  and the atom core ?

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Conductivity
Current I flows though a wire with diameter
A Current density J I / A Conductivity s
defined by
Compare to Ohms Law s ? 1/R s has units 1/(ohm
m)
Electron theory
Where qe charge of the electron Ne density
of electrons me mobility average speed with
which an electron moves at unit electric field
strength
ve velocity of electrons
Combined
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Newtons law for mobile electrons in an
oscillating Field
Oscillating field E to drive the electron.
Linear response
?
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Newtons law for mobile electrons in an
oscillating Field
Thus s0 is the conductivity at static electric
field (dc field)
determined by - Density of free electrons Ne -
Correlation time t
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How to determine t ?

• Au at 0.1 eV
• (e2 / -e1) 0.2
• w 2pn
• w2p (0.2 ? 1014 s-1)1.2 ?1014 s-1
• Thus t (1/w)(-e1/e2)
• 0.8 ?1014 s-1 ? (1/0.2)
• t 4 ? 10-14 s

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How to interpret t ?
Electron will accelerate in applied field until
it collides with a defect and losses all its
kinetic energy Average time between
collisions 2t Velocity after collision
v(t0)0 Average velocity of the electron
Conductivity
acceleration
defects
collision
electron
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Interpretation of t ?
Time constant describing friction t can be
obtained from dielectric measurements Metals t
? 10 femto seconds (10-15 s) t can be
interpreted as the average time between fully
inelastic collisions (speed reduced to
zero) Conductivity of the metal decreases with
increasing temperature The defects controlling
the conductivity are mainly thermal vibrations of
the atoms And/or thermally excited electrons
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Molecules / electrons do not respond
independently but collectively The Lorentz
Local Field approach
The molecule does not only respond to the applied
electric field E but also to the electric field
arising from its neighboring polarized molecules
The applied field E and the contribution of the
neighboring molecules add up to a local field
Eloc Eloc is the actual field to which an
individual molecule responds
Applied Field
E
Dipole field of Neighboring molecule
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How to account for this ? A macroscopic approach
d d d
d- d- d- d- d- d-
d d d d d d
- - - - - -

d- d- d-
Suppose that the molecule is in a solvent cage
What is the local field Eloc experienced by the
molecule in the cage ? Treat the surrounding
solvent macroscopically , i.e. through the
dielectric response function e er e0 The
dielectric medium surrounding the cage polarizes
! See Figure
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Local Lorentz electric field
d d d d d d
d- d- d- d- d- d-
V
Eloc
E0
Ecage
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Polarization
Surface charge density s due to polarization
proportional to component of Eo perpendicular to
the surface E?,0
Et,0
E0
?
E?,0
?
x
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Contribution of an infinitesimal amount of
surface charge at the centre of the cavity
- Infinitesimal amount of charge d s
dS

dS
infinitesimal surface area - Contribution of d
to the electric field in the centre of the sphere

R
?
Er
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Components in the y,z direction average out to
zero
R
y
?
x
Er
z
Only x components add up to a non zero number
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Integration over the Surface S of the cavity
Infinitesimal Volume and surface elements in
various coordinate systems
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Local Lorentz electric field

- - - - - -
V
Lorentz local Field correction
Expresses the local field in terms of the average
field E0
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Bulk property er from properties of individual
molecules
a polarizability of a single molecule in vacuum N
density of molecules
Clausius Mosotti equation Relates the molecular
polarizability a of an isolated molecule
(microscopic quantity ) to the relative
dielectric constant er of the molecular material
(macroscopic quantity ) accounting for the
interaction between the molecules
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Rewrite Clausius Mosotti equation to obtain an
expression for er in terms parameters from the
Lorentz model
?
Lorentz oscillator in vacuum
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Expression for er in terms parameters from the
Lorentz model Implications of the
Clausius-Mosotti equation
Can be rewritten into
Lorentz oscillator with local field correction
Lorentz oscillator without local field
correction
Renormalization (lowering) of the oscillator
frequency from w0 to
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Interpretation Screening of the electrostatic
interactions
- Motion of this electron further away from its
atom core. This increases the electric field. -
The neighboring electrons can respond to this
electric field by moving closer to the middle
atom core - Movement of the neighbors reduces
the electrostatic interaction between the middle
electron and it core
Electron bound to atom by electrostatic force
The presence many polarizable atoms/molecules
close together lowers the electrostatic
interactions in the solid screening
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Snake biting its own tail
Weakly bonded electrons become even more weakly
bonded when grouped close together
In a metal the electrostatic binding of electrons
to the atom core is not absent but reduced by
the surrounding oscillating electrons
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Pressure as a tool to tune the plasma frequency
Lorentz oscillator with local field correction
Frequency shift
Plasma frequency depends on density of
electrons. This can be changed by varying the
pressure
Very high presuures can be reached using a
diamond anvil cell
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High pressure experiments in a diamond anvil cell
By gradually increasing the pressure the plasma
frequency can be raised If
approaches zero the electrons are no longer
bound.
The material becomes a conductor
Prediction essentially any substance should
become metallic at high pressure
Transition to metallic state
Xe
CsI
O2
Si
BaTe
I2
Pressure
0
95
10
380 GPa
132
200
115
20
Pressure in earth core
(20 GPa ? 200kbar)
Angew. Chem. 2007, 119, 3694 3717
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Other examples of metal insulator transitions
Sodium metal dissolved in liquid NH3
non-metal
metal
e1
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Metal nanoparticles
E0
E0
E0


Eloc
Polarization of sphere of dielectric material
must be opposite to that of a hole in the same
material
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Induced dipole moment in a metal sphere
Induced dipole moment
Oscillates at same frequency as field
Intensity of transmitted light Reduced due to
absorption scattering
er
Light
Scattered light emitted by oscillating dipole
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Dielectric response of Gold
Around 2.5 eV e for gold reaches a value of -2
!! For this photon energy there will be very
efficient scattering and absorption of light by
the nanoparticle because the induced
polarization becomes very large
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Synthesis of surfactant stabilized gold
nanoparticles
HAuCl43H2O (octyl)4NBr- (Toluene/Water)
NaBH4
J. van Herrikhuyzen Ph.D. thesis Tu/e 2007
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Transmission Electron Microscopy (TEM)
5 x 103 Au atoms/particle
Matthijn Vos
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Experiment au particles with 4 nm diameter
surface plasmon band
Measure extinction of light due to absorption
and scattering As a function of wavelength of
light
I0
I
Extinction -10Log (I/I0)
Solution of gold nanoparticles
2.5 eV Surface plasmon band
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Surface plasmon band sensitive to e of the
surroundings
Surface plasmon band changes upon exchange of
ligands
(octyl)4NBr- TOAB
2.5 eV Surface plasmon band
OPVS
Closer analysis
es dielectric constant of medium surrounding the
metal nanoparticle
Determining the gold atom concentration from
atomic spectroscopy, the extinction coefficient
for bare gold particles stabilized by
tetra-n-octylammonium bromide (TOAB-Au) before
ligand exchange with OPV can be calculated. Here
we find an extinction coefficient at 2.35 eV of
1.0 ? 107 M-1cm-1, diamter d 4.1 nm
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Surface plasmons at metal surfaces
Surface plasmons on a metal film
Surface plasmons of a metal nanoparticle
Dielectric

es
-
metal

-
er

-
er
1
es
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Surface plasmon waves for detection of binding of
analytes
Without analyte
Angle ?
Reflected intensity
With analyte bound
Gold film
90o
50o
0o
Angle ?
Binding of analytes changes relative dielectric
constant es Resonance conditions change for
surface plasmon change ! Sensitive way of
detecting analyte binding