Electrical Properties of Materials: Flow of current Electric Current = Flow of electric charges I = ( # of charges per second) = Coulombs/sec 1 ampere = 1 Coulomb/sec What charges flow in materials? Electrons Electrons are Fundamental Particles with me=9 x10−31 kg qe=−1.6 x 10−19 C=−e

Types of materials in nature: Insulators: (glass, air, plastics, wood) --No current flows when electric voltage applied. Electrons are not free to move.

Conductors: ( metals) --- Can carry current. Electrons are its carriers. Photo Conductors – Can sustain current if light shines on them. Semi- Conductors( Silicon) – Intermediate between insulators and conductors.

Before discussing solids, we discuss Atoms: 1. Basic unit of chemical elements 2. Unexcited atoms are charge neutral

atom net

Q

=0

3. Contain equal numbers of positvely charged protons and negatively charged electrons. 4. Mass number of an atom : A = Z +N, where Z represents the number of protons and N the number of neutrons. 5. Charge of an atom = + Ze - Ze = 0 6.

Mass of an atom is M A =Z m p N m n Z me− BE of electrons

7. Sizes: Atomic Size ~ 10^-10 m, Nucleus Size = 10^-14 m

−10

10 m 10 nanometers

10−14 m

Atoms are: 1. Exquisitely balanced electrically. 2. Bound system : Requires input of energy to pull an electron out of the atom. Atomic Properties which cannot be explained with “ classical “ physics and require Quantum physics are: 1. Stability 2. Identity 3. Regeneration

Basic Principles of Quantum Mechanics or Physics 1. Wave Paricle Duality. Physical objects which we call particles are described by mass, momentum and energy Physical objects which we call waves are described by wavelength, frequency, speed and energy. In Quantum description, physical objects are described by an abstract quantity – the wave function. Depending on the experimental observation, the object can be said to behave as a wave or a particle. So for any object to be described correctly we must define For particles their wave length For waves their particle quantum of energy

Quantum physics associates with a particle with momentum p, a wavelength given by =

h p

This is the de Broglie relation. The quantum of energy for a wave of frequency f is given by

E=h f This is the Einstein relation. The constant h is a fundamental constant of nature, called Planck's constant, whose value is:

h =6.64 x 10−34

Joule x seconds

This is very small number !

A particle represented by a wave packet:

Oscillations of wave function define the wavelength of the particle.

What is the wavelength of a fast moving ball ? Take its mass to be 2 kg and speed 10 m/s, then its momentum is 20 kg m/s and its wavelength will be −32

=1.328 x 10

m

A very small number ! Wave nature hard to observe. However consider an electron moving at a speed of

v =0.01 speed of light =3 x 106 m / s Its wavelength 6.64 x10−34 is = =24.6 nano meters −31 6 9 x10 3 x10 

This is comparable to atomic distances in matter, but still tiny. Electron microscopes and tunelling show the wave nature of electrons.

Electron microscope image of a fly !

Max energy of electrons = hf – W Showed light consists of photons or packets of energy with each packet having an energy, hf . Einstein.

Confined Waves and Characteristic Vibrations: We saw that confined waves on a string tied at both ends and under tension can only vibrate in certain characteristic modes : 1 Lowest mode : f 1= v 

2L



Next mode

: f 2=2 f 1=2 v 

Next mode

: f 3=3 f 1 etc

1  2L

With a lowest non-trivial frequency. So when an electron is confined in an atom ( due to electrical attraction between the positively charged nucleus and negatively charged electrons ), it can vibrate only in certain characteristic modes – called quantum energy levels or quantum orbits or quantum paths. This leads to a natural explanation of Identity of atoms of the same kind. All atoms of the same type will have the same orbits.

Confined waves in a hydrogen like atom

Lowest energy orbit

wavelength 0=2 R 0

Emission and absorption of photons between atomic energy levels or orbits.

If one takes apart an atom and puts it together again we wioll get the same orbits and energies – this is the property of regeneration. To get atoms to be stable, requires a new rule of quantum mechanics, which is: Atom in its ground state ( lowest energy state) cannot radiate its energy

Now we extend this to electrons in solids:

Electrons confined in a solid: We use the basic tenet of quantum physics which is: Particle wave duality. Confining particles in solids

Only certain energy levels allowed. Also called orbits.

using =

h p

E= h f

Confining waves in a solid

Only certain characteristic oscillations allowed. Like overtones of a string tied at both ends

1 2 ... n .. f 1, f 2 , ,, f n . . Electrons in solids can only have particular energies. This leads to type of figures : 8.2.2, 8.2.3, 8.2.4 and 8.2.5

Quantum picture of electrons in solids: Concepts needed 1. Only certain paths or levels are available for the electrons. 2. In each level electron has a certain energy. 3. Pauli Exclusion Principle: Each distinguishable electron must have its own level. 4. Electron has an intrinsic property called spin – it can spin clockwise or counterclockwise and no other way : spin up and spin down. Fig 8.2.2 Electron with spin up is distinguishable from one with spin down. Levels

Energy

Each level can accommodate two electrons.

If there are N electrons , start filling the levels starting from the lowest energy level upwards until there are no more electrons left. One reaches a highest level, which is called the FERMI level.

Electrical properties of a material is determined by its level structure. In a real material, levels come in bands, separated by energy gaps where no electrons can reside. Allowed levels Energy gap Allowed levels Energy gap Allowed levels

In photo-conductors gap is small and light photons can raise electrons into levels where they can move or conduct.

Insulator: electrons fill the conduction band completely. No freedom for electrons to move.

Metals: electrons fill conduction band partially. Electrons with slight excess energy can move.

Insulator-- electrons dont move when voltage is applied across an insulator.

Metal – application of voltage causes electrons to transfer across the conductor.

Photo Conductor

Energy of photons of light lift electrons into unfilled levels where they can respond to applied voltage across a photo conductor.

Xerox machine

● ● ●

All things travel as waves All things interact as particles Example 1: Light – –

●

Travels as waves – electromagnetic waves Emitted and absorbed as particles – photons

Example 2: Electrons – –

Detected as particles Travel as waves

Summary for the two kinds of matter found in nature

In nature there are two kinds of basic subatomic objects: They have fundamentally different properties which are important in everyday life. ●

Bosons: Photons – – –

●

Many indistinguishable bosons can Occupy the same energy level. Such sharing leads to lasers & superconductors

Fermions: Electrons, Protons, Neutrons – – –

One indistinguishable fermion allowed per wave or in a given energy level. “Pauli Exclusion Principle” Explain electric conduction properties.

Electrons in Solids Electrons in a solid are confined to move inside the solid. Their movement is influenced by the way the nuclei of atoms are arranged in the solid. Quantum physics requires that electron motion (until they are detected) is described by wave motion. So electrons confined in a solid are electron waves confined to propagate inside the solid satisfying the boundary condition that they cannot be outside the metal. (1) So just as for waves on a string tied at two ends, these electron waves can only oscillate at specific frequencies or we can say that only certain waves fit in the solid.

(2) To each allowed wave there is a certain allowed energy for the electron and no other energies are allowed. This is because the wavelength of an electron moving with a momentum p = m v , is given by the relation =

h p

First formulated by Louis de Broglie. So to each allowed wavelength there is an allowed momentum and therefore an allowed energy. (3) Occupancy: How many electrons can occupy a given energy level ? This is a deep question, the answer to which was furnished by quantum physics – with no classical analogue !

Electrons have, as we have seen, mass and electric charge. They also spin – also an intrinsic property. They can spin in only two ways , either clockwise or anti-clockwise – spin up and spin down. Experimentally these two spin orientations can be distinguished. Only two electrons can be placed in each allowed energy state – one with spin up and another with spin down ! And no more can be put into the level. So these levels can be filled two at a time. Electrons present in the substance can fill these level two at a time starting at the lowest energy. So when we run out of available electrons which reach a high energy level – which is called the Fermi level.

In a metal: ●

●

The Fermi level has empty levels just above it Like patrons in a partly fill theatre who can move around to different seats, electrons can move in response to electric fields

Insulators ●

●

The Fermi level has no empty levels nearby Like patrons in a full theatre, electrons can’t move in response to forces

Semiconductors ● ●

●

Semiconductors are “poor insulators” Valence & conduction bands have narrow gap Like patrons in a theatre with a low balcony, electrons can hop into the balcony and move