about band structure

1. Dept. of Mechanical Engineering
F.N.E.
S.C. JUN
1/18

2. Dept. of Mechanical Engineering
F.N.E.
S.C. JUN
2/18
FROM ATOMIC LEVELS TO ENERGY BANDS
 Inside a crystal electrons respond to outer forces as if they have an effective
momentum hk. Near the band edges they respond as if they have an effective mass
m*.
 As atoms are brought closer and closer to each other to form a crystal, the discrete
atomic levels start to broaden to form bands of allowed energies separated by gaps.
The electronic states in the allowed bands are Bloch states,i.e., they are plane wave
states (~eik•r).
 Low lying core levels are relatively unaffected.
 Higher levels are broadened significantly to form bands.

3. Dept. of Mechanical Engineering
F.N.E.
S.C. JUN
3/18
FROM ATOMIC LEVELS TO ENERGY BANDS

4. Dept. of Mechanical Engineering
F.N.E.
S.C. JUN
4/18
The One-Electron Atom
• Another result derived from the analysis of this potential problem (one-electron
atom) is that two additional quantum numbers emerge as a result of the
multidimensional of this problem.
• The solution of Schrodinger’s wave equation for the one-electron potential function
can be designated by ψnlm n,l, and m are the quantum numbers.
• For the lowest energy state, n=1, l=0, and m=0, the wave function is given by equal
to the Bohr radius (from the classical Bohr theory of the atom.)

5. Dept. of Mechanical Engineering
F.N.E.
S.C. JUN
5/18
The One-Electron Atom
• The radial probability density functions, or the probability of finding the eletron at
a particular distance from the nucleus, is proportional to the product ψ100• ψ*100
and also to the differential volume of the shell around the nucleus.
• The probability density function for the lowest energy state is plotted in Figure a.
• The most probable distance from the nucleus is at r=a0, which is the same as the
Bohr theory.
• Considering this spherically symmetric probability functions, we can now begin to
conceive the concept of an electron cloud, or energy shell, surrounding the nucleus
than a discrete particle orbiting around the nucleus.
• The radial probability density function for the next higher spherically symmetric
wave function, corresponding to n=2, l=0, and m=0, is shown in Figure b.

6. Dept. of Mechanical Engineering
F.N.E.
S.C. JUN
6/18
• The one-electron atom plus two additional concepts.
• The fist concept needed is that of electron spin.
• The electron has an intrinsic angular momentum, or spin, that is quantized and
may take on one of two possible values.
• The spin is designated by a quantum numbers: n, l, m, and s.
• The chemical activity of an element is determined primarily by the balance, or
outermost, electrons.
• Since the balance energy shell of helium is full, helium does not react with other
elements and is an inert element.
Periodic Table

7. Dept. of Mechanical Engineering
F.N.E.
S.C. JUN
7/18
• Figure (a)shows the radial probability density function for the lowest electron
energy state of a single, noninteracting hydrogen atom, and Figure (b) shows the
same probability curves for two atoms that are in close proximity to each other.
• The wave functions of the two atoms overlap, which means that the two electrons
will interact.
• This interaction or perturbation results in the discrete quantized energy level
splitting into two discrete energy levels, schematically shown in Figure (c).
• The splitting of the discrete states into two states is consistent with the Pauli
exclusion principle.
Formation of Energy Bands
• The energy of the bound electron is quantized : only discrete values of electron
energy are allowed
ENERGY-BAND THEORY

8. Dept. of Mechanical Engineering
F.N.E.
S.C. JUN
8/18
• So there is a “splitting” of power(energy) of the two
interacting race cars.
• Figure, where the parameter r0 represents the
equilibrium interatomic distance in the crystal.
• At the equilibrium interatomic distance, there is a band
of allowed energies, but within the allowed band, the
energies are at discrete levels.
Formation of Energy Bands
• At any energy level, the number of allowed quantum states is relatively small.
• To accommodate all of the electrons in a crystal, then, we must have many energy
levels within the allowed band.
• A system with 1019 one-electron atoms and also suppose that at the equilibrium
interatomic distance, the width of the allowed energy band is 1 eV.
• For simplicity, we assume that each electron in the system occupies a different
energy level, and if the discrete energy states are equidistant, them the energy levels
are separated by 10-19 eV.
• This energy difference is extremely small, so that for all practical purposes, we have
a quasi-continuous energy distribution through the allowed energy band.
• The fact that 10-19 eV is a very small difference between two energy states.

9. Dept. of Mechanical Engineering
F.N.E.
S.C. JUN
9/18
• If the atoms are initially very far apart, the electrons in adjacent atoms will not
interact and will occupy the discrete energy levels.
• If these atoms are brought close together, the outermost electrons in the n=3
energy shell will begin to interact initially, so that this discrete energy level sill split
into a band of allowed energies.
• If the atoms continue to move closer together, the electrons in the n=2 shell may
begin to interact and will also split into a band of allowed energies.
• Finally, if the atoms become sufficiently close together, the innermost electrons in
the n=1 level may interact, so that this energy level may also split into a band of
allowed energies.
Formation of Energy Bands

10. Dept. of Mechanical Engineering
F.N.E.
S.C. JUN
10/18
• If the equilibrium interatomic distance is r0, the bands of allowed energies that the
electrons may occupy separated by bands of forbidden bands is the energy-band theory
of single-crystal materials.
• A schematic representation of an isolated silicon atom is shown in Figure a (next page).
• Ten of the fourteen silicon atom electrons occupy deep-lying energy levels close to the
nucleus.
• The four remaining valence electrons are relatively weakly bound and are the electrons
involved in chemical reactions.
• the n=3 level for the valence electrons, since the first two energy shells are completely full
and are tightly bound to the nucleus.
• The 3s state corresponds to n=3 and l=0, and contains two quantum states per atom.
• The 3p state corresponds to n=3 and l=1, and contains six quantum states per atom.
• This state will contain the remaining two electrons in the individual silicon atoms.
• As the interatomic distance decreases, the 3s and 3p states interact and overlap.
Formation of Energy Bands

11. Dept. of Mechanical Engineering
F.N.E.
S.C. JUN
11/18
The Energy Band and the Bond Model
• Figure (b) represented the splitting of the discrete silicon energy states into bands of
allowed energies as the crystal is formed.
• At T=0K, the 4N states in the lower band, the valence, are filled with the valence electrons.
• At the equilibrium interatomic distance, the bands have again split, but four quantum states
per atom are in the lower band and four quantum states per atom are in the upper band.
• So that all states in the lower band (the valence band) will be full, and all states in the upper
band(the conduction band) will be empty.
• The band gap energy Eg between the top of the balance band and the bottom of the
conduction band is the width of the forbidden energy band.

12. Dept. of Mechanical Engineering
F.N.E.
S.C. JUN
12/18
APPLICATION TO MOLECULES : COVALENT BONDING
(b)
(c)

13. Dept. of Mechanical Engineering
F.N.E.
S.C. JUN
13/18
APPLICATION TO MOLECULES : COVALENT BONDING
 Where r1 and r2 are the distances between the electron and each of the two nuclei.
Figure (b) shows the allowed electron energy states, which are still quantized.
 Notice that an electron in the ground state (lowest energy state) would be bound to
one of the nuclei, but an electron in an excited stated could travel back and forth
between the nuclei, in effect shared by the two atoms.
 Since electrons tend to seek their lowest allowed energy, this condition of the
electron being in one of the upper levels would not last long – the electron would
quickly revert to the ground state.
 Figure (c) shows the energy band diagram for the case where the separation is small
enough that the potential energy maximum between the nuclei is below
• The ground state energy (E1). In this situation, an electron in the ground state
would be shared by the two nuclei, oscillating between the two positions at which
E = Ep.
• Since each nucleus has a ground state associated with it, it turns out that two
electrons can occupy these ground states for a neutral H2 molecule.
• In the region between the nuclei, the kinetic energy (Ek = E1 - Ep), and thus the
velocity is small. The electrons travel more slowly in this region, or on the average,
the electrons spend most of their time between the two nuclei.

14. Dept. of Mechanical Engineering
F.N.E.
S.C. JUN
14/18
APPLICATION TO MOLECULES : COVALENT BONDING
 The electrons therefore create a negatively charged “electron cloud” in this region
that tends to attract the two nuclei together. If the internuclear spacing is too small.
 However, the potential energy Ep decreases, which increases the kinetic energy Ek
since total energy E is conserved.
 As the kinetic energy and therefore the electron speed increases, the electron cloud
effect is reduced, lessening the attractive force.
 At a particular spacing, the electron-cloud-induced nuclear bonding is stable, and a
stable H2 molecule results. This mechanism is referred to as covalent bonding.

15. Dept. of Mechanical Engineering
F.N.E.
S.C. JUN
15/18
QUANTUM NUMBERS AND THE PAULI EXCLUSION PRINCIPLE
• Quantum mechanics is that the energies in an atom are quantized, or exist only at
certain discrete values.
• The quantum number n, called the principal quantum number. It describes the
energy of an electron in an allowed state.
• The physical meanings of these quantum numbers are not essential to the
understanding of transistors, but the Pauli exclusion principle is essential.
• The lowest energy orbit of an atom, n = 1. This state can bold two electrons; those
two electrons must have different spin quantum numbers, either +1/2 or -1/2.
• In the n = 2 state, there are two possible orbital shapes. One orbit is spherically
symmetric and holds two electrons of opposite spin (the “s” state).
• There are three elliptical orbits with the same shape but different orientations. Each
of these can hold two electrons of opposite spin, bringing the maximum number of
electrons in the second “shell” to eight. The periodic table is built on these quantum
numbers.

16. Dept. of Mechanical Engineering
F.N.E.
S.C. JUN
16/18
COVALENT BONDING IN CRYSTALLINE SOLIDS

17. Dept. of Mechanical Engineering
F.N.E.
S.C. JUN
17/18
• At room temperature, because of thermal agitation, a few electrons are excited
into the conduction band
• Each one eventually falls back down to a vacant state in the valence band, re-
emitting the excess energy as heat or light.
• The average time an electrons spends in the conduction band is called the
“electron lifetime” or just “lifetime” and is on the order of 10-10 to 10-3 seconds,
depending on the material.
• Electrons in the conduction band are free to move around within the crystal.
 They travel at constant energy (between collisions). But now there are many empty
states at the same energy into which an electron can move. This band is called the
conduction band because the moving electrons carry current.
 If an electron were excited to an energy high up in the conduction band, it would
very quickly find a lower energy state.
 Therefore, all of the interesting activity is occurring near the top of the valence
band and near the bottom of the conduction band.
COVALENT BONDING IN CRYSTALLINE SOLIDS

18. Dept. of Mechanical Engineering
F.N.E.
S.C. JUN
18/18
• At nonzero temperatures, there are a few empty states in the valence band. We call
these empty states holes.
• one can see that if an electron moves to a vacant state to the left, that has the same
net effect as one hole moving one step to the right.
• Polycrystalline and Amorphous Materials
• Polycrystalline materials have small regions (grains) of single-crystal material with
different crystalline orientations.
• These grains have dimensions on the order of a few nanometers to a few
millimeters.
• Because of the different crystalline orientations of the grains, the crystal
periodicity at the grain boundaries is interrupted.
• This in turn affects the band structure near the grain boundaries.
COVALENT BONDING IN CRYSTALLINE SOLIDS