Transcript photonic crystals - Homepages | The University of Aberdeen

Non-crystalline materials and
other things
By the end of this section you should:
• know the difference between crystalline and
amorphous solids and some applications for the latter
• understand how the different states affect the X-ray
patterns
• be able to show the Ewald sphere construction for an
amorphous solid
• be aware of different types of mesophases
• know the background to photonic crystals
Amorphous Solids
So far we have discussed crystalline solids.
Many solids are not crystalline - i.e. have no long range
order.
They can be thought of as “solid liquids”
Amorphous Solids
The arrangement in an amorphous solid is not completely
random:
1) Coordination of atoms satisfied (?)
2) Bond lengths sensible
3) Each atom excludes others from the space it occupies.
 represented by radial distribution function, g(r) *
g(r) is probability of finding an atom at a distance between
r and r+r from centre of a reference atom
* Sometimes known as pair distribution function
Radial Distribution Function
Take a reference atoms with radius a
g(r) = 0 for r<a
g(r)  1 for large r
At intermediate distances, g(r) oscillates around unity short range order.
From any central atoms, the nearest neighbours tend to
have a certain pattern - though not so rigidly as in a crystal
SiO4 - angles tend to
109.5º but are not exact
Radial Distribution Function
As we move out, the pattern becomes more and more
varied until we reach complete disorder
X-ray diffraction can still give information on the structure.
X-rays scattered from atoms (not planes) and interference
effects will occur.
We use angle , though this does not relate to any lattice
plane as in Bragg’s law.
2
K    sin 
 
Radial Distribution Function
Scattered intensity depends on modulus - not direction of K for an amorphous material.
This means that diffraction patterns have circular
symmetry rather than spots.
Interference Function
The interference function (i.e. “scattering factor” for
amorphous materials) S(K) is given by:

S(K)  1  n 0 4r 2{g(r)  1} sinc Kr dr
where n is the no. of atoms per unit volume and
sinc  = sin / 
S(K) is a Fourier transform of {g(r)-1} and
1 
g(r )  1 
4K 2{S(K) 1} sinc Kr dK
n 0

Measurements
We can measure the intensity, I(K), which (we assume) is
directly related to S(K). Thus g(r) can be calculated from
the interference effects in the (circular) diffraction pattern,
and hence interatomic distances can be estimated.
e.g. taking a radial cut from the centre of the pattern:
Measurements
Assignments made on
expected distances
between atoms
As we get further out,
becomes less “ideal” due
to increased disorder
“Solid Liquids”
Diffraction patterns of an amorphous solid and a liquid
of the same composition are very similar:
The average structures are
more or less the same.
Short range order less well
developed in liquid (peaks
not so well defined)
RDF in crystals
We can also calculate this for a perfect crystal
a2
a3
Polonium, a = 3.359 Å
a
3a
2a
This can allow analysis of “not so perfect” crystals – disorder
“Total diffraction”
Ewald Sphere for amorphous solids
From previously:
2
K    sin 
 
i.e. scattering depends only on modulus of K. So we have
a reciprocal “sphere” of radius |K| intersecting with the
Ewald sphere:
This gives a circle
where they intersect
= diffraction pattern.
(circle perp. to page)
Back to EXAFS
• The Fourier transform of the EXAFS spectrum is also
a radial distribution function
Intensity vs R (radius from central atom)
Free volume
Free volume (VF) defined as:
SV of glass/liquid - SV of corresponding crystal
SV = Volume per unit mass
Volume
liquid
Glass
transition
VF
glass
melting
crystal
Tg
Tm
Temperature
Amorphous silicon
• Amorphous materials often not good conductors –
pathways blocked
• Crystalline silicon – diamond structure, 4-fold
coordination, regular (corner-sharing) tetrahedra
• Amorphous silicon – mostly 4-fold coordination, fairly
regular tetrahedra BUT…
• …not all atoms 4-fold coordinated
• …“dangling bonds”
Can be terminated by H atoms
kypros.physics.uoc.gr/resproj.htm
Uses
• Method of production means it can be deposited over
large areas – thin films, flexible substrates
• Photovoltaics – e.g. solar cells
Energy conversion not so efficient as crystalline Si, but
more energy efficient to produce
Photovoltaics
• Instead of heat, light causes electron/hole pairs
• Cell made of pn junction - photons absorbed in p-layer.
• p-layer is tuned to the type of light - absorbs as many
photons as possible
• move to n-layer and out to circuit.
http://www.nrel.gov/data/pix/Jpegs/07786.jpg
http://solarcellstringer.com/
Mesophases
Normally a solid melts to give a liquid.
In some cases, an intermediate state exists called the
mesophase (middle).
Substances with a mesophase are called liquid
crystals
Liquid Crystals and Mesophases
Crystal 145.5 °C LC
178.5°C
I
Friedrich Reinitzer
(1857-1927)
Cholesteryl
benzoate
Thanks to Toby Donaldson
What types of molecules show liquid
crystalline behaviour?
• Anisometric molecular shape
OR
NC
OCnH2n+1
RO
OR
Calamitic liquid crystals
OR
RO
OR
Discotic liquid crystals
Thanks to Toby Donaldson
Polarised light microscopy
Mostly now used in geology
Gases, liquids, unstressed glasses and
cubic crystals are all isotropic
Otto Lehmann
(1855-1922)
One refractive index – same optical
properties in all directions
Most (90%) solids are anisotropic
and their optical properties vary
depending on direction.
Birefringent Lysozyme crystals viewed by
polarised light microscopy
http://www.ph.ed.ac.uk/~pbeales/research.html
Polarised light microscopy
Crystalline
Liquid crystalline
cholesteric
Isotropic
Thanks to Toby Donaldson
Mesophases
If we increase temperature, we can see how the
disordering occurs:
Mesophases - more detail
(a) smectic phase - from the Greek for soap, smegm
A
C
Layers are preserved, but order between and
within layers is lost
Smectic
Thanks to Toby Donaldson
Mesophases - more detail
(b) nematic phase - from the Greek for thread, nemos
Layers are lost, but the molecules remain aligned
If we looked at this end on, it would look like a liquid
Nematic Phase, N
Thanks to Toby Donaldson
Isotropic Liquid
Mesophases - XRD
Example - mix of powder (circles) and ordering (arcs)
LCDs
• LCs sandwiched between
two cross polarisers
• “twist” in LC allows light to
pass through
• Applied voltage removes
twist and light no longer
passes through
http://www.geocities.com/Omegaman_UK/lcd.html
http://www.edinformatics.com/inventions_inventors/lcd.htm
Photonic Crystals
1887: Lord Rayleigh noted Bragg Diffraction in 1-D
Photonic Crystals
1987: Eli Yablonovitch: “Inhibited spontaneous emission
in solid state physics and electronics”
Physical Review Letters, 58, 2059, 1987
Sajeev John: “Strong localization of photons in
certain disordered dielectric super lattices”
Physical Review Letters, 58, 2486, 1987
Basics of photonics
• Periodic structures with alternating refractive index
1887
1987
1-D
2-D
periodic in
one direction
periodic in
two directions
3-D
periodic in
three directions
Photonic band gap analogous to electronic band gap
Weakly interacting bosons vs strongly interacting fermions
http://ab-initio.mit.edu/photons/tutorial/ - S.G Johnston
Bragg’s Law – wider applications
n = 2d sin 
This is a general truth for any 3-d array.
If we imagine the “atoms” as larger spheres, then:
d becomes larger
 becomes larger – visible light
This is the basis for photonic crystals
Opal (SiO2.nH2O)
A fossilised bone!
Silica spheres 150-300 nm
in diameter – ccp/hcp
http://www.mindat.org/gallery.php?min=3004
Bragg’s Law – wider applications
We replace the d-spacing, from Bragg’s law,
with the “optical thickness”
nrd
where nr is the refractive index (e.g. of the silica in opal)
n = 2nrd sin 
nr is ~1.45 in opal so
n = 2.9 d sin 
This gives max = 2.9 d for normal incidence
Geometry of packed spheres
If we assume the spheres “close pack”, then we can
calculate d:
2r
sin 60 = d/2r
d = 1.73 r
r
max = 2.9 d
So max = 5r (approx.) for
normal incidence
We now need to manipulate d!!
Photonic band gap
From above: max = 2nrd
And from de Broglie
at this , no light propagates
E = hc/ 
So in photonic crystals, we define the photonic band
gap:
hc
E
2n r d
Photonics – in nature
J. Zi et al, Proc. Nat. Acad. Sci. USA, 100, 12576 (2003);
Blau, Physics Today 57, 18 (2004)
http://newton.ex.ac.uk/research/
Artificial Photonics
Massive research area (esp. in Scotland!)
Control areas of differing refractive index, e.g.

d
The first experiment
An array of small holes 1mm apart were drilled into a
piece of material which had refractive index 3.6.
Calculate the wavelength of light “trapped” by this
material
max
= 2nrd
= 2 x 3.6 x 0.001
= 7.2 x 10-3 m
Microwaves
Woodpile crystal
[]
“Logs” of Si 1.2 mm wide
K. Ho et al., Solid State Comm. 89, 413 (1994)
H. S. Sözüer et al., J. Mod. Opt. 41, 231 (1994)
http://www.sandia.gov/media/photonic.htm
“Artificial” photonic crystals
S. G. Johnson et al., Nature. 429, 538 (2004)
From amorphous silicon
– 3D, 1.3 – 1.5 mm
T. Baba et al, Yokohama National University
Artificial Opal
D. Norris, University of Minnesota: http://www.cems.umn.edu/research/norris/index.html
Inverse Opal
Templating to
produce…
Yurii A. Vlasov, Xiang-Zheng Bo, James C. Sturm & David J. Norris., Nature 414, 289-293 (2001)
Inverse Opal
• Silica spheres with a
refractive index of 1.45
• ~ 1.3 mm
Q: Calculate d (and hence
the radius of the spheres)
from this information.
Uses
From: K Inoue & K. Ohtaka: “Photonic crystals” ( Springer, NewYork,2003).
Summary
 Amorphous materials show short range order and have
have various applications e.g. in photovoltaics
 X-ray interference effects still occur, leading to circular
diffraction patterns which relate to g(r), the radial
distribution function and the scattered X-ray intensity
depends on the modulus of the scattering vector, K
 States intermediate between crystalline and liquid exist
- mesophases - such as nematic and smectic
 These have wide applications, an example being LCDs
 Extension of Bragg’s law to a different scale length
leads us to consider photonic crystals