Dept. of Mechanical Engineering

F.N.E.

S.C. JUN

1/18

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.

Dept. of Mechanical Engineering

F.N.E.

S.C. JUN

3/18

FROM ATOMIC LEVELS TO ENERGY BANDS

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.)

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.

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

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

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.

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

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

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.

Dept. of Mechanical Engineering

F.N.E.

S.C. JUN

12/18

APPLICATION TO MOLECULES : COVALENT BONDING

(b) (c)

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.

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.

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.

Dept. of Mechanical Engineering

F.N.E.

S.C. JUN

16/18

COVALENT BONDING IN CRYSTALLINE SOLIDS

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

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