Oklahoma

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Oklahoma State University
Lecture 2 Particle in a box in 1, 2, 3 dimensions
Reminder Course web site http//physics.okstat
e.edu/jpw519/phys5110/index.htm
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All Dimesionalities Based on Carbon
Oklahoma State University

• 0D quantum dots C60
• Monodisperse, reproducible, self-assembled,
  doping
• Significant optical, electronic, magnetic
  properties
• High Tc, Micro- nanoelectronics
• 1D quantum wires carbon nanotubes
• Nanotransistors, diode, junction
• Nanoprobe for SPM and NFOM
• Molecular transport nanopipe

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All Dimesionalities Based on Carbon
Oklahoma State University

• 2D layered materials - graphite
• 3D wide gap semicond. - diamond

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Structures of Carbon
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0D
1D
3D
2D
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Fullerene Crystal
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• Fullerene (C60)
• Cage diameter - 0.71 nm
• Optical gap 1.9 eV

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Fullerene Crystal
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• Crystalline Fullerene
• fcc structure with lattice
• constant of 1.417 nm
• C60-C60 distance 1.002 nm
• Melting temperature 118 0 C
• Band gap 1.7 eV

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Lattice Structure of Carbon
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• Graphite Diamond
• SP2 Hybrid - SP3 Hybrid
• 120 for bond angle 109.5 for bond angle

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Non-interacting many electron systems

• Ground state-many electron system having no
• interactions but still obeying Pauli Exclusion
  Principle
• Start filling each single-particle state
  (orbital) from
• the lowest energy, each state with one spin-up
  and
• one spin-down electron

n5
1 dim.
n4
n3
n2
n1
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Fermi energy
Fermi momentum
, electron density in 2-dim.

• Fermi energy and momentum increase with density
• of electrons, n2d
• All single-particle states with EltEF are
  occupied at
• T0. These states are called the Fermi Sea.
  The set
• of points in k-space that divides empty and
  full states
• is the Fermi surface.
• Exactly how EF depends on density depends
  strongly
• on dimensionality.

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Processes we want to determine (e.g., electrical
conductivity) involve transferring electrons from
occupied to unoccupied orbitals (at T0, this is
at EF)
Density of states, g(E)dEdS number of allowed
orbitals with energies between E and EdE in a
length/area/volume of size dS From this
definition, we can find the spacing of orbitals
in a piece of material. Higher g(E) means
orbitals are closer together.
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1-dim.
g
2-dim.
g
3-dim.
g
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If we have 1 cm3 of Na with one valence electron
per atom
Single particle level spacing
Single particle levels are almost continuous But
if we have 1 nm3 Single particle level spacing
Easily measurable!
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Nano-Fabrication and Characterization
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Nanotechnology
Sub?m Tech
Atom Tech
AFM
SCM
SNFOM
SEM
STEM
STM
EB
Electron
FIB
Ion
Excimer, SR
Photon
X-Ray
1000
100
10
1
0.1
Dimension (nm)