QM theory of Solids:
Part I
Band structures, bonding,
Fermions
Fig 4.1
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Introduction
•
•
•
•
•
•
We can now develop a QM theory of solids.
We deal with crystals (otherwise too difficult)
Band structure with electron “filling”.
The band structure is determined by the lattice and atomic structure.
Primarily interested in two top most bands (valance and conduction)
Wish to understand the effect of T, applied voltage and incident
light.
•  Three types of materials:
–  Metals – overlapping bands
–  Semiconductors – small band gap ~ 1ev
–  Insulators – large band gap > 2ev
•  Start with quantum bonding of atoms (in simplistic manner).
Fig 4.1
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Quantum Mechanics and Bonding
Bring together two H atoms
and we have an interaction
which produces two possible
solutions with different
energies.
The two electrons pair
their spins and occupy
the bonding orbital.
Using a linear combination of
the single atom we can get an
approximate solution.
A exact solution would
require solving Sch. Eq.
Fig 4.1
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Linear Combination of Atomic Orbitals
Two identical atomic orbitals ψ1s on
atoms A and B can be combined
linearly in two different ways to
generate two separate molecular
orbitals ψσ and ψσ*
ψσ and ψσ* generated from a
linear combination of atomic
orbitals (LCAO)
ψ σ = ψ 1s (rA ) +ψ 1s (rB )
ψ σ * = ψ 1s (rA ) −ψ 1s (rB )
This is a very approximate
model. We will see a variety
of approaches when we look
at material modeling.
Math becomes tough.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Lower Energy level – due to
electron density between
nuclei.
Higher Energy Level
(a)  Electron probability distributions for bonding and antibonding orbitals, ψσ and
(b) ψσ*.
(b) Lines representing contours of constant probability (darker lines represent greater
relative probability).
Fig 4.2
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Splitting of levels as be
form H2
Minimum energy when
both electron (spin up/
down) occupy the lower
state.
This is the covalent
bond from a QM
perspective.
(a)  Energy of !! and !!* vs. the interatomic separation R.
(b)  Schematic diagram showing the changes in the electron energy as two isolated H atoms,
far left and far right, come together to form a hydrogen molecule.
Fig 4.3
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Situation is similar for He, but we
have 4 electron so occupy both
levels
Two He atoms have four electrons. When He atoms come together, two of the electrons enter
the Eσ level and two the Eσ* level, so the overall energy is greater than two isolated He atoms.
Fig 4.5
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Add another atom – get another
energy level
(a) Three molecular orbitals from three Ψ1s atomic orbitals overlapping in three different ways.
(b) The energies of the three molecular orbitals, labeled a, b, and c, in a system with three H
atoms.
Fig 4.7
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Many atoms – many levels
The formation of 2s energy band from the 2s orbitals when N Li atoms come together to form the
Li solid.
There are N 2s electrons, but 2N states in the band. The 2s band is therefore only half full.
The atomic 1s orbital is close to the Li nucleus and remains undisturbed in the solid. Thus, each
Li atom has a closed K shell (full 1s orbital).
Fig 4.8
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Enough atoms and we
have energy bands.
Just as we did for
solving Ψ for a crystal
using Bloch solutions.
Note: the number of
states is the number of
atoms!
As Li atoms are brought together from infinity, the atomic orbitals overlap and give rise to bands.
Outer orbitals overlap first. The 3s orbitals give rise to the 3s band, 2p orbitals to the 2p band,
and so on. The various bands overlap to produce a single band in which the energy is nearly
continuous.
Fig 4.9
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Bands can overlap or
can be separated by
“band-gaps”
In a metal, the various energy bands overlap to give a single energy band that is only partially
full of electrons. There are states with energies up to the vacuum level, where the electron is
free.
Fig 4.10
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Bands and electron transport
• We saw the presence of a periodic electron structure
produces bands when analyzed using Bloch functions and
Sch.Eq.
• Intuitively we can also look at it a scattering problem
with the electrons undergoing Bragg diffraction.
Fig 4.10
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
If the wavelength of the electron is
2a = nλ then we have destructive
interference and no transmission.
Bragg Diffraction
For this case k = nπ/a exactly the
1D edges for the Brillioun zone.
Bragg Diffraction condition
An electron wave propagation through a linear lattice. For certain k values, the reflected
waves at successive atomic planes reinforce each other, giving rise to a reflected wave
traveling in the backward direction. The electron cannot then propagate through the crystal.
Fig 4.50
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
We can also imagine two
configurations for a
electron with k = π/a at
the edge of the zone.
Forward and backward waves in the crystal with k = +/- π/
a give rise to two possible standing
Waves Ψc and Ψs. Their probability density distributions |
Ψc |2 and | Ψs |2 have maxima either
at the ions or between the ions, respectively.
These have two different
energies (as with the
symmetric and antisymmetric solutions for
H2) producing an energy
gap.
Fig 4.51
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
At the edge of the
zones we have
diffraction and an
energy gap.
Away from the zone
edges we have no
diffraction and
basically free
electrons.
The energy of the electron as a function of its wavevectore k inside a one-dimensional crystal.
There are discontinuities in the energy at k = +/-nπ/a, where the waves suffer Bragg reflections
in the crystal. For example, there can be no energy value for the electron between Ec and Es.
therefore, Es-Ec is an energy gap at k = +/- π/a. Away from the critical k values, the E-k vector
is like that of a free electron, with E increasing with k as E = !k 2 / 2me . In a solid, these energies
fall within an energy band.
Fig 4.52
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
In a 2D or 3D crystal the
Bragg Diffraction
condition is function of
the angle of incidence.
For a given direction in
the electron will scatter
off the atom planes.
nπ
k sin θ =
d
This gives rise to 2D/3D
Brillioun zones in kspace.
Diffraction of the electron in a two dimensional cubic crystal. Diffraction
occurs whenever k has a component satisfying k1 = ±nπ/a, k2 = ±nπ/a or k3 =
±21/2nπ/a. In general terms, when ksinθ = nπ/d.
Fig 4.53
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Silicon band structure and bonding.
Single Silicon Atom electronic structure
Si has an unfulfilled outer shell of
electrons. With 4 valence
electrons.
The electronic structure of Si
Fig 4.15
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Bond Hybridization
When Si is solidified
bonding is complicated
but 4 strong hybridized
bonds form with
neighboring atoms in a
tetrahedral lattice.
(a) Si is in Group IV in the Periodic Table. An isolated Si atom has two electrons in the 3s and
two electrons in the 3p orbitals.
(b) When Si is about to bond, the one 3s orbital and the three 3p orbitals become perturbed and
mixed to form four hybridized orbitals, ψhyb, called sp3 orbitals, which are directed toward the
corners of a tetrahedron. The ψhyb orbital has a large major lobe and a small back lobe. Each
ψhyb orbital takes one of the four valence electrons.
Fig 4.16
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
As with H and He
we get symmetric
and anti-symmetric
solutions to Sch. Eq.
Creates a tetrahedral
crystal
(a) Formation of energy bands in the Si crystal first involves
hybridization of 3s and 3p orbitals to four identical ψhyb orbitals which
o
make 109.5 with each other as shown in (b). (c) ψhyb orbitals on two
neighboring Si atoms can overlap to form ψB or ψA. The first is a
bonding orbital (full) and the second is an anti-bonding orbital (empty).
In the crystal yB overlap to give the valence band (full) and ψA overlap
to give the conduction band (empty).
Fig 4.17
And produces two
bands (Conduction
and Valance) each
with N electronic
states.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Simplified Band model
At T=0 the valance band
is full (Si has an even
number of electrons and
each band accepts 2N
electrons.
For T > 0 a few electron
have enough KE to get
excited over the band
gap and become
conduction (free)
electrons.
Energy band diagram of a semiconductor. CB is the conduction band and VB is the valence
band. AT 0 K, the VB is full with all the valence electrons.
Fig 4.18
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Metal, Semiconductor and Insulator Band Structures
(a) For the electron in a metal, there is no apparent energy gap because the second BZ (Brillouin
zone) along [10] overlaps the first BZ along [11]. Bands overlap the energy gaps. Thus, the
electron can always find any energy by changing its direction.
(b) For the electron in a semicondcuctor, there is an energy gap arising from the overlap of the
energy gaps along the [10] and [11] directions. The electron can never have an energy within this
energy gap Eg.
Fig 4.55
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Filling the Bands with Electrons
In a metal we have a half
filled band(s).
Define the Fermi level as
the energy level at which
for T= 0 (no kinetic
energy) all the lower states
are filled.
Define work function as
energy needed to remove
an electron from the Fermi
energy.
Typical electron energy band diagram for a metal.
All the valence electrons are in an energy band, which they only partially fill. The top of the
band is the vacuum level, where the electron is free from the solid (PE = 0).
Fig 4.11
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Work function Φ
The energy required to excite an electron from the Fermi level
to the vacuum level, that is, to liberate the electron from the
metal, is called the work function Φ of the metal.
Electron gas in a metal
The electrons in the energy band of a metal are loosely bound valence
electrons, which become free in the crystal and thereby form a kind of
electron gas within the crystal. It is this electron gas that holds the
metal ions together in the crystal structure and constitutes the metallic
bond.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Effect of a field
•  An applied electric field produces a force on the electron (a term in
SE), but assume a small field and perturbation of the zero field case.
•  This changes the momentum of the electron and it’s energy.
Will only be able to do this if a “space” or state is available in the
band
•  As the field represents a force (a potential energy) the field changes
the zero point energy of the electrons.
•  This means that the band structure moves up or down.
•  When a simplified band structure is plotted versus x this is shown as
“band bending”.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Applied E field increase
the momentum (p) for all
electrons. Sliding them to
the right.
(a) Energy band diagram of a metal. (b) In the absence
of a field, there are as many electrons moving right as
there are moving left. The motions of two electrons at
each energy cancel each other as for a and b. (c) In the
presence of a field in the -x direction, the electron a
accelerates and gains energy to a’ where it is scattered to
an empty state near EFO but moving in the -x direction.
The average of all momenta values is along the +x
direction and results in a net electrical current.
Scattering stops them
sliding all the way to
infinity.
Classical model is useful.
Fig 4.12
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Band structure as function of
position with an applied
potential.
Potential raises the band
structure (potential energy).
Electron move to the right
gaining KE losing PE then
undergoes a scattering and drops
energy by passing heat to the
lattice.
Conduction in a metal is due to the drift of electrons around the Fermi level. When a voltage is
applied, the energy band is bent to be lower at the positive terminal so that the electron’s
potential energy decreases as it moves toward the positive terminal.
Fig 4.13
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Effective Mass
As we saw in QM lecture the
effective mass of the electron
is determined by the E(k)
relationship
Fig 4.19
GaAs Bandstructure
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Semiconductor/Metal Carrier Densities
• Metals have effectively 1⁄2 filled bands
• Semiconductors have a small band gap < 2ev
• This leads to excitation of conduction electrons and production of holes.
• We are interested in calculating the number of electrons (only this for metals)
and holes in the bands.
• To do this we need two things:
• The density of states in a band (the number of states/dE) -- g(E)
• The probability of filling a state at a given temperature -- f(E)
• The electron density in E is then n(E) = g(E)f(E)
• To obtain n integrate over the band! That’s simple!
Fig 4.13
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Density of states.
• We saw that each atom contributes one electronic state to each band.
• However, these states are not distributed uniformly over the band.
• We need a function that prescribes the distribution over the band
• This function provides the number of states/dE – and is usually denoted
by g(E). The density of states function.
• This function is determined by the geometric order of the structure.
• 0D – quantum dot (Quantum well)
• 1D – quantum line (1D electron gas)
• 2D – quantum plane (2D electron gas)
• 3D – quantum solid (3D bulk crystal).
Fig 4.19
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
One ψ state associated
with each atom.
Use a superposition of
single atom states.
Many closely related
states (similar energies)
in the band middle.
(a) In the solid there are N atoms and N extended electron wavefunctions
from ψ1 all the way
To ψN. There are many wavefunctions, states, that have energies that fall in
the central regions
Of the energy band.
(b) The distribution of states in the energy band; darker regions have a
higher number of states.
(c) Schematic representation of the density of states g(E) versus energy E.
g(E) has to go to zero at
band edges
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
To calculate a 2D density of states.
Assume parabolic E(k)
Crystal of size L2 and a 2D
Quantum well which means two
quantum numbers n1 and n2
Each state has an energy of
h2
E=
(n21 + n22 )
2
8me L
And we have constant energy
surfaces given by circles.
Each state, or electron wavefunctions in the
crystal, can be represented by a box at n1,
n2.
We can derive a formula for the
number of states in ring of
thickness dE and that is the density
of states.
Fig 4.21
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Likewise to calculate a 3D density
of states.
Assume parabolic E(k)
Crystal of size L3 and a 3D
Quantum well which means two
quantum numbers n1, n2 and n3
Each state has an energy of
h2
2
2
2
E=
(n
+
n
+
n
1
2
3)
8me L2
And we have constant energy
surfaces given by spheres.
In three dimensions, the volume defined by a
sphere of radius n' and the positive axes n1, n2,
and n3, contains all the possible combinations
of positive n1, n2, and n3 values that satisfy
n12 + n22 + n32 ≤ nʹ′2
Fig 4.22
We can derive a formula for the
number of states in shell of
thickness dE and that is the density
of states.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Density of States – 3D solid
g(E) = Density of states
g(E) dE is the number of states (i.e., wavefunctions) in
the energy interval E to (E + dE) per unit volume of
the sample.
g ( E ) = (8π 2
1/ 2
⎛ me ⎞
⎜ 2 ⎟
⎝ h ⎠
)
3/ 2
E
1/ 2
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Other systems.
•
•
•
•
Blue 3D (solid)
Red 2D (plane)
Green 1D (wire)
What’s 0D (dot)?
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
The Energy Distribution
Classically we know an electron gas should follow a Boltzmann
• distribution
But this distribution is for non-interacting particles and electrons
• strongly
interact as they are fermions.
• The Pauli exclusion principle says only two electrons can occupy a
state (spin up/down)
• The Boltzmann distribution only works when there are many more
states then particles or the particle is not a Fermion.
• This is not true for electrons at the bottom of a band (where they all
are!)
• We have to use Fermi-Dirac statistics which work for fermions.
• The classical expression is still useful at the top of the band.
Fig 4.24
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Boltzmann Classical Statistics
Boltzmann probability function
⎛ E ⎞
P ( E ) = A exp⎜ −
⎟
⎝ kT ⎠
Boltzmann Statistics for two energy levels
N2
⎛ E2 − E1 ⎞
= exp⎜ −
⎟
N1
kT ⎠
⎝
Boltzmann statistics describes
systems of non-interacting
(classical) particles under
thermal equilibrium.
Does not apply to electrons as
they are Fermions subject to the
Pauli exclusion principle.
Exception is when there are
many states and few electrons.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
The Boltzmann energy distribution
describes the statistics of particles,
such as electrons, when there are
many more available states than the
number of particles.
At T = 0 all particles to to E = 0
Fig 4.25
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fermions - electrons
Electrons can only move to
unoccupied states.
So E4 and E3 must be empty.
This changes the
distribution function.
Two electrons with initial wavefunctions ψ1 and ψ2 at E1 and E2 interact and end up
different energies E3 and E4. Their corresponding wavefunctions are ψ3 and ψ4.
Fig 4.24
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fermi-Dirac Function
The distribution is radically different
At T = 0 the distribution is 1.0 from
0 to the Fermi energy EF.
At T > 0 we get smearing of the
function.
The fermi-Dirac f(E) describes the statistics
of electrons in a solid. The electrons
interact with each other and the environment,
obeying the Pauli exclusion principle.
Fig 4.26
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fermi-Dirac Statistics
The Fermi-Dirac function
1
f (E) =
⎛ E − EF ⎞
1 + exp⎜
⎟
⎝ kT ⎠
where EF is a constant called the Fermi energy.
f(E) = the probability of finding an electron in a state with energy
E is given
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fermi-Dirac Implications
•  We can now calculate the number of electrons that contribute to
conduction.
•  This is the main “correction” that QM makes to the classical theory of
conduction in a metal.
•  EF is the chemical potential of the electrons in the material.
•  In general two materials will have different EF
•  When placed in contact electrons will flow from one material to the
other until the Fermi levels align and thermal equilibrium is
established.
•  The Fermi energy EF characterizes the material.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Calculating n
(a) Above 0K, due to thermal excitation, some of the
electrons are at energies above EF.
(b) The density of states, g(E) versus E in the band.
(c) The probability of occupancy of a state at an energy
E is f (E).
(d) The product of g(E) f (E) is the number of electrons
per unit energy per unit volume,
or the electron concentration per unit energy. The area
under the curve on the energy
axis is the concentration of electrons in the band.
n=
n=
Z
1
f (e)g(e)dE
0
3/2
me
h3
1/2
8⇡2
Z
1
0
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
E 1/2 dE
1+e
E
EF
kT
Fermi Energy
Fermi energy at T = 0 K
2
⎛ h ⎞⎛ 3n ⎞
⎟⎟⎜ ⎟
EFO = ⎜⎜
⎝ 8me ⎠⎝ π ⎠
2/3
We can express EF|T=0
in terms of n
n is the concentration of conduction electrons (free carrier concentration)
Fermi energy at T (K)
⎡ π ⎛ kT ⎞
⎟⎟
EF (T ) = EFO ⎢1 − ⎜⎜
⎢⎣ 12 ⎝ EFO ⎠
2
2
⎤
⎥
⎥⎦
And also obtain
an expression
for EF(T)
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Metal Conductivity
QM
classical
1 2
= e vF ⌧ g(EF )
3
⌧
2
=e n
me
g ( E ) = (8π 2
1/ 2
⎛ me ⎞
⎜ 2 ⎟
⎝ h ⎠
)
3/ 2
E 1/ 2
•  QM gives an Ohm’s Law
like conductivity.
Assuming a scattering
process.
•  But only the electrons at
EF (ie. vF) contribute to
the drift current.
•  The σ is proportional to
g(EF) which is the
density of states at the
Fermi Energy.
•  Very important in
determining the
conductivity of a metal.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Average energy per electron
EF is related to the
average energy of the
electrons.
Average energy per electron at 0 K
3
Eav (0) = EFO
5
At thermal
equilibrium EF will
be a constant value.
Average energy per electron at T (K)
⎡ 5π
3
Eav (T ) = EFO ⎢1 +
5
⎢⎣ 12
2
⎛ kT ⎞
⎜⎜
⎟⎟
⎝ EFO ⎠
2
⎤
⎥
⎥⎦
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
At an interface of two metals a
exchange of electrons will
occur to bring the Fermi
Energies into alignment at
thermal equilibrium.
Surface charge and E field is
present at the interface. This
produces a contact potential
drop.
(a)  Electrons are more energetic in Mo, so they tunnel to the surface of Pt.
(b)  Equilibrium is reached when the Fermi levels are lined up.
When two metals are brought together, there is a contact potential !V.
Fig 4.28
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
In equilibrium a closed system
must have I = 0.
So the potential drops must
cancel out.
A simple ring of two metals
show how the potentials will
cancel.
There is no current when a closed circuit is formed by two different metals, even though
there is a contact potential at each contact.
The contact potentials oppose each other.
Fig 4.29
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fermi Energy Significance
For a given metal the Fermi energy represents the
free energy per electron called the electrochemical
potential. The Fermi energy is a measure of the
potential of an electron to do electrical work (e×V) or
non-mechanical work, through chemical or physical
processes.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Summary
•  This lecture primarily dealt with metals we will
deal with semiconductors in detail later.
•  The QM model of the metal provides insight into:
–  Band structure – Fermi energy
–  Free electrons
–  Effective mass
–  Density of states
–  Number of electrons contributing to
conduction
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)