1. The document discusses density of states (DOS), which describes the number of accessible quantum states at each energy level in a system. It explains how electrons populate energy bands based on DOS and the Fermi distribution function. 2. Calculation of DOS for a semiconductor is shown, and applications like quantization in low-dimensional structures and photonic crystals are described. Impurity bands formed by dopants are also discussed. 3. In summary, the document provides an overview of density of states, how it is calculated, and its applications in areas like quantization effects and photonic crystals.

1. Aryabhatta Knowledge University
Aryabhatta Centre of Nanoscience &
Nanotechnology
Manu Shreshtha
19601601008
MTech Nanoscience & Nanotechnology (2019-21)
Aryabhatta Centre of Nanoscience & Nanotechnology
Aryabhatta Knowledge University, Patna
Presentation
on
Density of States & its Application
Guided by:
Dr Rakesh Kumar Singh

2. Density
of
States
Engineering

3. Content
1. Density of States
2. How do Electrons and Holes Populate the Bands?
i. Density of States Concept
ii. Probability of Occupation (Fermi Function) Concept
3. Calculation of the Density of States
4. Application of Density of States
i. Quantization
ii. Photonic crystals
iii. The Impurity bands
5. References

4. Density of States
 In measurable and consolidated matter physics, the density
of states (DOS) of a system portrays the number of states
at every energy level that is accessible to be involved.
 A high DOS at a particular energy level implies that there
are numerous states accessible for occupation. A DOS of
zero implies that no states can be involved at that energy
level.
 The number of electrons at every energy is then
gotten by duplicating the number of states with the
likelihood that a state is involved by an electron.
Since the number of energy levels is enormous and
reliant on the size of the semiconductor, we will
ascertain the number of states per unit of energy and
per unit volume.

5. How do Electrons and Holes Populate
the Bands?
dE
E
gc )
(
The number of conduction band
states/cm3 lying in the energy
range between E and E + dE
(if E  Ec).
The number of valence band
states/cm3 lying in the energy
range between E and E + dE
(if E  Ev).
dE
E
gv )
(
 Density of States Concept
General energy dependence of gc (E)
and gv (E) near the band edges.

6.  Density of States Concept
Quantum Mechanics tells us that the number of available
states in a cm3 per unit of energy, the density of states, is
given by:
Density of States
in Conduction Band
Density of States
in Valence Band

7.  Probability of Occupation (Fermi Function) Concept
 Now that we know the number of available states at each energy,
then how do the electrons occupy these states?
 We need to know how the electrons are “distributed in
energy”.
 Again, Quantum Mechanics tells us that the electrons follow the
“Fermi-distribution function”.
Ef ≡ Fermi energy (average energy in the
crystal)
k ≡ Boltzmann constant (k=8.61710-5eV/K)
T ≡Temperature in Kelvin (K)
 f(E) is the probability that a state at energy E is occupied.
 1-f(E) is the probability that a state at energy E is unoccupied.
kT
E
E f
e
E
f /
)
(
1
1
)
( 


 Fermi function applies only under equilibrium conditions, however, is
universal in the sense that it applies with all materials-insulators,
semiconductors, and metals.

8. Calculation of the Density of States
 The density of states in a semiconductor rises to the density per unit volume and
energy of the number of answers for Schrödinger's equation. We will accept that the
semiconductor can be displayed as a boundless quantum well in which electrons with
powerful mass, m*, are allowed to move. The energy in the well is set to zero. The
semiconductor has accepted a cube with side L. This suspicion doesn't influence the
outcome since the density of states per unit volume ought not to rely upon the
real size or shape of the semiconductor.
 The solutions to the wave equation where V(x) = 0 are sine and cosine functions:
…(a)

9.  Where A and B are to be resolved. The wave function should be zero at the infinite
barriers of the well. At x = 0 the wave function should be zero so just sine
functions can be legitimate arrangements or B should rise to zero. At x = L, the
wavefunction should likewise be zero yielding the accompanying potential qualities for
the wavenumber, kx.
…(b)

10.  This analysis can now be repeated in the y and z-direction. Each possible solution
then corresponds to a cube in k-space with size np/L as indicated in the figure 1:
Figure 1: Calculation of the number of states with wavenumber less than k.

11.  An example of the density of states in 3, 2 and 1 dimensions is shown in the figure
2:
Figure 2: Density of states per unit volume and energy for a 3-D semiconductor (blue curve), a 10 nm quantum well with infinite barriers (red
curve) and a 10 nm by 10 nm quantum wire with infinite barriers (green curve). m*/m0 = 0.8.

12. Application of Density of States
Quantization
Calculating the density of states for small structures shows that the distribution of
electrons changes as dimensionality is reduced. For quantum wires, the DOS for certain
energies actually becomes higher than the DOS for bulk semiconductors, and
for quantum dots, the electrons become quantized to certain energies.

13. Photonic crystals
The photon density of states can be manipulated by using periodic structures with
length scales on the order of the wavelength of light. Some structures can completely
inhibit the propagation of light of certain colours (energies), creating a photonic
bandgap: the DOS is zero for those photon energies. Other structures can inhibit the
propagation of light only in certain directions to create mirrors, waveguides, and
cavities. Such periodic structures are known as photonic crystals. In nanostructured
media, the concept of the local density of states (LDOS) is often more relevant than
that of DOS, as the DOS varies considerably from point to point.

14. The Impurity bands
It is well known that a group V impurity atom can give rise to an energy level in the gap of
a group IV semiconductor (Bonch-Bruevitch 1966), the wavefunction of the donor impurity
being located near the nucleus. When the number of impurities increases, the impurity level
broadens into a band due to the weak overlap of the wavefunction and to the fluctuations of
concentration (Lifshitz 1964). Above some critical concentration Nc, the Mott
concentration, the conduction becomes metallic. Just above this critical concentration, the
overlap between the wavefunction remains weak enough to justify the use of a tight-binding
approximation. The density of states will be obtained from its moments as described above,
so as to analyse its precise shape near its centre. We shall consider various concentrations in
impurities, above the Mott concentration, for which one can justify the use of a simple
tight-binding approximation, where the excited states of the impurities and the band
structure of the matrix are neglected. We will show that, in this range of concentration, the
impurity band has no ‘erosion’ near the impurity level.

15. Reference
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https://eng.libretexts.org/Bookshelves/Materials_Science/Supplemental_Modules_(Mate
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https://www.bbc.co.uk/bitesize/guides/ztb3h39/revision/2.
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https://www.sciencedirect.com/topics/chemistry/density-of-state.
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http://alan.ece.gatech.edu/ECE6451/Lectures/StudentLectures/King_Notes_Density_o
f_States_2D1D0D.pdf.

16. 6. "Photonic Band Gap - an overview | ScienceDirect Topics."
https://www.sciencedirect.com/topics/materials-science/photonic-band-gap.
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17. Thank You.