11.1 Band Theory of Solids 11.2 Semiconductor Theory 11.3 Semiconductor Devices 11.4 Nanotechnology CHAPTER 11 Semiconductor Theory and Devices It is evident that many years of research by a great many people, both before and after the discovery of the transistor effect, has been required to bring our knowledge of semiconductors to its present development. We were fortunate to be involved at a particularly opportune time and to add another small step in the control of Nature for the benefit of mankind. - John Bardeen, 1956 Nobel lecture
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11.1 Band Theory of Solids 11.2 Semiconductor Theory 11.3 Semiconductor Devices 11.4 Nanotechnology
CHAPTER 11Semiconductor Theory and Devices
It is evident that many years of research by a great many people, both before and after the discovery of the transistor effect, has been required to bring our knowledge of semiconductors to its present development. We were fortunate to be involved at a particularly opportune time and to add another small step in the control of Nature for the benefit of mankind.
- John Bardeen, 1956 Nobel lecture
11.1: Band Theory of Solids
Three categories of solids, based on electrical conductivity conductors semiconductors insulators
Energy
Insulator Metal Semimetal Semiconductor Semiconductor
Can be classified by Energy band Theory
Electrical Resistivity and Conductivity at 293 K
The electrical conductivity at room temperature is quite different Metals and alloys have the highest conductivities followed by semiconductors and then by insulators
Semiconductor Conduction The free-electron model from Chapter 9 does not apply to
semiconductors and insulators, since these materials simply lack enough free electrons to conduct in a free-electron mode.
There is a different conduction mechanism for semiconductors than for normal conductors.
Typical conductor Semiconductor
The resistivity increases dramatically as T → 0.
Resistivity vs. Temperature
Band Theory of Solids In order to account for decreasing resistivity
with increasing temperature as well as other properties of semiconductors,a new theory known as the band theory is introduced.
How Energy Band formed?
In solids with a large number of atoms,
the energy levels are split into nearly continuous energy bands.
Each band consists of a number of closely spaced energy levels.
Consider initially the known wave functions of two hydrogen atoms far enough apart so that they do not interact.
Interaction of the wave functions occurs as the atoms get closer
Band Theory of Solids
An atom in the symmetric state has a nonzero probability of being halfway between the two atoms,
while an electron in the antisymmetric state has a zero probability of being at that location.
Symmetric state Antisymmetric state
Band Theory of Solids
In the symmetric case the binding energy is slightly stronger resulting in a lower energy state, than antisymmetric.
In a real solid with a large number of atoms, nearly continuous energy bands, with each band consisting of a number of closely spaced energy levels.
When more atoms are added,further splitting of energy levels.
Symmetric
Antisymmetric
Those Energy splitting occurs at all possible energy levels (1s, 2s, and so on)
Solid: Carbon Diamond (2S2 3P2) and Silicon (3S2 3P2)
Carbon Silicon Internuclear distance
Conduction Band
Valence Band
Forbidden
4N atoms in (2 x 4N) possible levels Insulator
4N
4N
Valance Band, Conduction Band, Forbidden Gap
Silicon (3S2 3P2): Insulator, but Semiconductor
Silicon
The bandgap energy (1 eV) is small, Many Electrons can easily be excited to the conduction band in Room T. Fermi Energy is placed at the gap center. Its conductivity is in between conductor and insulator, “Semiconductor”
1 eV
EF
Kronig-Penney Model for Energy gap An effective way to understand the energy gap in semiconductors is to
model the interaction between the electrons and the lattice of atoms.
In 1931 Kronig and Penney assumed that an electron experiences an infinite one-dimensional array of finite potential wells.
Each potential well models attraction to an atom in the lattice, the well size must correspond roughly to the lattice spacing.
Since the electrons are not free, their energies are less than the height V0 of each of the potentials, but the electron is essentially free in the gap 0 < x < a, where it has a wave function of the form
E In the region between a < x < a + b the electron can tunnel through the wave function loses its oscillatory solution and it becomes exponential:
where
where
Matching solutions at the boundaries; x = 0 and x = a,
K is another wave number.
There exist restricted forbidden zones
-1 < < +1
The gaps occur regularly at ka = n
forbidden
forbidden
Kronig-Penney Model
The allowed ranges k are referred to as Brillouin zones.
Condition for Bragg Diffraction:
Energy Band Theory in Momentum (k) Space- Brillouin Zones defined by Bragg Diffraction -
• p-type (doped by Acceptor) and n-type (doped by Donor)
Semiconductor Theory At T = 0 we expect all of the atoms in a solid to be in the ground
state. The distribution of electrons (fermions) at the various energy levels is governed by the Fermi-Dirac distribution:
β = (kT)−1 and EF is the Fermi energy.
When the temperature is increased from T = 0, more and more atoms are found in excited states. The increased number of electrons in excited states explains the temperature
dependence of the resistivity of semiconductors. Only those electrons that have jumped from the valence band to
the conduction band are available to participate in the conductionprocess in a semiconductor.
More and more electrons are found in the conduction band as the temperature is increased, the resistivity of the semiconductor therefore decreases.
Semiconductor TheoryFind the relative number of electrons with energies 0.10 eV, 1.0 eV, and 10 eV above the valence band at room temperature (293 K).
Conduction
Valence
Forbidden
Note that 1.0 eV = 1.60 x 10-19 J.The Fermi energy is at the top of the valence band, so the energy above the valence band is E - EF. EF
This implies that the number of electrons available for conduction drops off sharply as the band gap increases.
Holes and Intrinsic Semiconductors
When electrons move into the conduction band, they leave behind vacancies in the valence band. These vacancies are called holes.
Because holes represent the absence of negative charges, it is useful to think of holes as positive charges.
Whereas the electrons move opposite to the applied E field, the holes move in the direction of the electric field.
A semiconductor in which there is a balance between the number of electrons in the conduction band and the number of holes in the valence band is called an intrinsic semiconductor.
Examples of intrinsic semiconductors include pure carbon and germanium.
Impurity Semiconductor It is possible to fine-tune a semiconductor’s properties by adding a
small amount of another material, called a dopant, to the semiconductor creating what is a called an impurity semiconductor.
As an example, silicon has four electrons in its outermost shell (this corresponds to the valence band) and arsenic has five.
Thus while four of arsenic’s outer-shell electrons participate in covalent bonding with its nearest neighbors (just as another silicon atom would), the fifth electron is very weakly bound.
It takes only about 0.05 eV to move this extra electron into the conduction band.
Adding (or, Doping) only a small amount of arsenic to silicon greatly increases the electrical conductivity.
In addition to intrinsic and impurity semiconductors, there are many compound semiconductors, which consist of equal numbers of two kinds of atoms
n-type and Donor Impurity levels Addition of arsenic (V) to silicon (IV) creates what is known as
an n-type semiconductor (n for negative), because it is the electrons close to the conduction band that will eventually carry electrical current.
The new arsenic energy levels just below the conduction band are called donor levels because an electron there is easily donated to the conduction band.
When indium (III) is added to silicon (IV). Indium has one less electron one extra hole per indium atom. These holes creates extra energy levels just above the valence
band, because it takes relatively little energy to move another electron into a hole
Those new indium levels are called acceptor levels because they can easily accept an electron from the valence band.
Easier to think in terms of the flow of positive charges (holes) in the direction of the applied field, so we call this a p-type semiconductor (p for positive).
p-type and Acceptor Impurity levels
N-Type (Electrons) and p-type (Holes)
n-Type
p-Type
Hall effectThe magnitude and sign of the Hall voltage allow one to calculate the density and sign of the charge carriers in a conductor or semiconductor and verify that charges in carriers in p-type and n-type materials have different signs.
x
z
A thin strip of metal immersed in a magnetic field is used to test the Hall effect. (a) Here negative charge carriers are forced to the right. (b) In this configuration, the buildup of negative charge on the right side (with a corresponding positive charge on the left) sets up the electric field as shown. This creates an electric force on the charge carriers equal and opposite to the magnetic force. The voltmeter (reading VH) can detect the magnitudeand sign of the potential difference across the strip.
In equilibrium,
The sign of the Hall voltage define the sign of the charge carriers.
Electrons, Holes, and Dispersion curve (or, Dispersion relation)
A dispersion relation is the relationship between wavevector (k-vector) and energy (E) in a band structure.
E = E (k), or = (k)
The dispersion relation determines how electrons respond to forces (via the concept of effective mass).
An electron floating in space has the dispersion relation E= 2k2/(2m), where m is the (real) electron mass.
In the conduction band of a semiconductor, the dispersion relation is instead E= 2k2/(2m*) (m* is the effective mass), so a conduction-band electron responds to forces as if it had the mass m*.
Electrons near the top of the valence band behave as if they have negative mass:(When a force pulls the electrons to the right, these electrons actually move left.)
Electrons near the bottom of the band have positive effective mass.
The concept of holes is useful as a shortcut for calculating the total current of an almost-full band. Hole can be regarded as a positive-charge, positive-mass quasiparticle. A hole with positive charge and positive mass responds to electric and magnetic fields
in the same way as an electron with negative charge and negative mass. An analogy is a bubble underwater: The bubble moves the same direction as the water, not opposite.