First Brillouin Zone/Wigner-Seitz: Diamond/Zinc-blende Latticeell
The vectors a1, a2, a3 are primitive translation
vectors or primitive basis for the real space
lattice, while b1, b2, b3 are primitive translation
vectors or primitive basis for the reciprocal
lattice. G is called a reciprocal lattice vector.
All reciprocal lattice vectors can be expressed
as a linear combination of b1, b2, b3 using
integer coefficients
volume
How to get the first BZ/W-S cell
a) Select a lattice point and
draw construction lines to the
nearest neighbouring points.
b) Draw lines that
perpendicularly bisect the
construction lines
c) The smallest enclosed area
represents the Wigner-Seitz
cell. Here shown in orange.
Band structure: Diamond
Nanoscience
Dividing Line #1:
ELECTRON WAVES Separate NanoSCIENCE from MicroSCIENCE
The discovery that electrons = waves led to QUANTUM MECHANICS
A weird, new, counter intuitive, non-Newtonian way of looking at the nano world
With a particular impact upon our understanding of electrons: Electrons => Waves
How do you figure out an electron’s wavelength?
electron = h / p
“De Broglie’s Relationship”
( = electron wavelength, h = Planck’s Constant, p = electron’s momentum)
This relationship was based on series of experiments late 1800’s / early 1900’s
To put the size of an electron’s wavelength in perspective:
Size of Things
Millimeters
Ball of a ball point pen
Thickness of paper
Human hair
Talcum Powder
Fiberglass fibers
Carbon fiber
Human red blood cell
E-coli bacterium
Size of a modern transistor
Size of Smallpox virus
0.5
0.1
0.02 - 0.2
(orange = man-made things)
Microns
Nanometers
100
20 – 200
40
10
5
4–6
1
0.25
0.2 – 0.3
250
200 – 300
Electron wavelength: ~10 nm or less
Diameter of Carbon Nanotube
Diameter of DNA spiral
Diameter of C60 Buckyball
Diameter of Benzene ring
Size of one Atom
3
2
0.7
0.28
~0.1
Below that line = Nanoscience!
It’s NOT just about the metric units we prefer to use when measuring things
Things above that line are still often measured using nanometers
It IS about the SCIENCE (QM) => Electrons are mushy clouds of size ~ De Broglie
Above that line, clouds seem small: Electrons ~ hard B-B like dots
Below that line, mushy cloudiness of electrons becomes very important
Controls electrical, optical, mechanical and other properties
Controls bonding and nanostructure
The Science Changes! Microscience ≠ Nanoscience
Dividing Line #2:
LIGHT WAVES Separate NanoTECHNOLOGY form MicroTECHNOLOGY
Technology = The things we make and how we make them
As opposed to the underlying science dictating how they act
Where does light’s wavelength enter into technology?
Micro technology is based on the use of light
How? Light is used for PHOTOENGRAVING:
Use of light images to pattern metal parts =>
Micro projection of light images = Way we make the billions of transistors in the
integrated circuits of our PCs, iPods . . .
(a.k.a. “Microfabrication”)
Size of Things
Millimeters
(orange = man-made things)
Microns
Nanometers
100
20 – 200
40
10
5
4–6
1
1000
Visible Light Wavelength:
0.40 – 0.75 microns
400 – 750 nm
Size of a modern transistor
Size of Smallpox virus
0.25
0.2 – 0.3
250
200 – 300
Ball of a ball point pen
Thickness of paper
Human hair
Talcum Powder
Fiberglass fibers
Carbon fiber
Human red blood cell
E-coli bacterium
0.5
0.1
0.02 - 0.2
Electron wavelength: Upper upper limit ~ 10 nm
Diameter of Carbon Nanotube
Diameter of DNA spiral
Diameter of C60 Buckyball
Diameter of Benzene ring
Size of one Atom
3
2
0.7
0.28
~0.1
Above the new (upper) line:
We can still use light-based “Microfabrication” techniques
And even though they were developed for electronics,
they are now also applied to making all sorts of micro things!
Below that new line:
NO longer able to use Microfabrication
Replacement would be called “Nanofabrication” or “Nanotechnology”
But we don’t yet really know WHAT that replacement will be!
Why Nanoscience research is now such a mix of different techniques
Recurring theme: Hope we can get nano things to ASSEMBLE THEMSELVES (!!!)
Nanoscience and reduced dimensionality
11
Effects of reducing dimension
“Quantum” – electronic properties
become quantized as the size of
a dimension(s) of the structure
diminishes, i.e. change from
being continuous to discrete
Most identifiable aspect of quantum
confinement in semiconductors is
the nanostructure size-dependence
of the band gap – the band gap
increases as the size decreases
The As are the gradients – obtained
by simple model (effective mass, and
particle-in-a-box)
Gallium Nitride-based
Nanostructures
III-Nitrides
InN
III-Nitrides (GaN, AlN, InN) important
semiconductor family for optoelectronic
devices[1] –blue LEDs and laser diodes
GaN
Nanowires hold exciting potential for nano-scale
devices, but little is known about their properties
and of the behaviour of defects and dopants –
InGaN-based nanowires - Potential for achieving
direct white-light emitting diodes
J. Angus, Case Western
Reserve University
Least understood -low thermal
stability. Controversy over
band-gap - was regarded
as 1.9 e; more recent studies
indicate 0.65-1.0 eV
New technological interest - e.g.
high speed electronics and
terahertz devices
GaN Nanowires:
Shape and size dependence
Experimental wires produced with diameters from
4-50nm wurtzite structure [e.g. W.Han et al. Science (1997);
T. Kuykendall, Nat. Mat. (2004)]
Methodology
DFT-PBE GGA functional; full atomic relaxation
- 3D periodic boundary conditions; ~20Å vacuum region
 DMol3
- Valence electron configuration: Ga=3d10,4s2,4p1 N=1s2,2s2,2p3
- DSPP pseudopotentials; Cutoff radius of 9 Bohr
- Double-numerical basis set with polarisation functions
 SIESTA
- Valence electron configuration: Ga=3d10,4s2,4p1 N=2s2,2p3
- Troullier-Martins pseudopotentials; Mesh cutoff of 300 Ry
- Double-zeta basis set with polarisation functions
- Energy shift (due to orbital confinement) of 0.01 Ry
[1] B. Delley, J. Chem. Phys. 92, 508 (1990); ibid 113, 7756 (2000). [2] J.M. Soler, et al., J. Phys.:
Condens. Matter. 14, 2745 (2002).
GaN Nanowire sizes and shapes:
[0001] growth direction
Numbers represent number of atoms in supercell
Hexagonal cross-section wires
Triangular cross-section wires
Band Structure + HOMO, LUMO Ψ
Edge-induced states
H-saturated
unsaturated
9.5 Å (48 atoms)
28.6 Å (300 atoms)
12.7 Å (66 atoms)
25.5 Å (194 atoms)
hex
tri