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EE 315/ECE 451
N ANOELECTRONICS I
Derived from lecture notes by R. Munden 2010
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8.1 D ENSITY
OF
S TATES
Imagine the hard walled box, and the energy states available:
# states in sphere of radius En:
Octant of sphere
J. DENENBERG- FAIRFIELD UNIV. - EE315
10/13/2015
D O S 3D S YSTEM
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Accounting for Spin. Potential and L3 = 1 for density per unit volume:
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10/19/2015
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8.1.1 D ENSITY OF S TATES
IN L OW D IMENSIONS
For One
Dimensional
Quantum Wire
With spin and potential:
J. DENENBERG- FAIRFIELD UNIV. - EE315
10/19/2015
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8.1.1 D ENSITY OF S TATES
IN L OW D IMENSIONS
For 2D Quantum Well
For 0D Quantum Dots
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Although real materials are 3D,
quantum confinement in small materials
can approximate low dimensional
structures, like quantum dots.
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8.1.2 D ENSITY OF S TATES
IN A S EMICONDUCTOR
In 3D materials:
We can use in semiconductors by substituting effective mass
for band structure and Ec for potential
In Silicon (with transverse and longitudinal effective mass)
and 6 fold symmetry of the conduction band:
J. DENENBERG- FAIRFIELD UNIV. - EE315
10/19/2015
8.2 C LASSICAL AND
Q UANTUM S TATISTICS
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
Classical or Boltzman Distribution
(distinguishable particles – e.g.molecules):

Fermi-Dirac Distribution (indistinguishable,
exclusive particles – e.g. electrons):

Bose-Einstein Distribution (indistinguishable,
non-exclusive particles – e.g. photons and
phonons):
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10/19/2015
F ERMI D ISTRIBUTION
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“Ripples on the Fermi sea”
Occupied
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Unoccupied
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8.2.1 C ARRIER
C ONCENTRATION IN M ATERIALS
3D confined box
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10/19/2015
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8.2.2 T HE I MPORTANCE OF
F ERMI E LECTRONS
R.MUNDEN - FAIRFIELD UNIV. - EE315
11/1/2010
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E LECTRON S TATES
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10/19/2015
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8.2.3 E QUILIBRIUM C ARRIER
C ONCENTRATION
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10/19/2015
8.3 M AIN P OINTS
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
the concept of density of states in various spatial dimensions,
and the significance of the density of states;

how the density of states can be measured;

quantum and classical statistics for collections of large numbers
of particles, including the Boltzmann, Fermi-Dirac, and BoseEinstein distributions;

the role of density of states and quantum statistics in
determining the Fermi level;

applications of density of states and quantum statistics to
determine carrier concentration in materials, including in doped
semiconductors.
R.MUNDEN - FAIRFIELD UNIV. - EE315
11/1/2010
8.4 P ROBLEMS
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R.MUNDEN - FAIRFIELD UNIV. - EE315
11/1/2010