Electrical Properties of Materials: Flow of current

Electric Current = Flow of electric charges

I = ( # of charges per second) = Coulombs/sec1 ampere = 1 Coulomb/sec

What charges flow in materials? Electrons

Electrons are Fundamental Particles with

me=9x10−31kg

qe=−1.6x10−19C=−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 shineson them.

Semi- Conductors( Silicon) – Intermediate between insulators and conductors.

Before discussing solids, we discuss Atoms:

1. Basic unit of chemical elements2. Unexcited atoms are charge neutral Qnet

atom=0

3. Contain equal numbers of positvely charged protonsand negatively charged electrons.

4. Mass number of an atom : A = Z +N, where Z representsthe number of protons and N the number of neutrons.

5. Charge of an atom = + Ze - Ze = 0

6. Mass of an atom is MA=ZmpNmnZme− BE of electrons

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

10−10m 10 nanometers

10−14m

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 describedby an abstract quantity – the wave function. Dependingon the experimental observation, the object can be said to behave as a wave or a particle. So for any objectto be described correctly we must define

For particles their wave lengthFor waves their particle quantum of energy

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

=hp

This is the de Broglie relation.

The quantum of energy for a wave of frequency f isgiven by

E=hfThis is the Einstein relation.

The constant h is a fundamental constant of nature,called Planck's constant, whose value is:

h=6.64 x10−34 Joule x seconds

This is very small number !

A particle represented by a wave packet:

Oscillations of wave function define the wavelengthof the particle.

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

=1.328 x10−32m

A very small number ! Wave nature hard to observe.

However consider an electron moving at a speed of

v=0.01speed of light =3x106m /s

Its wavelength

is = 6.64 x10−34

9x10−313x106=24.6nano meters

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

Electron microscope image of a fly !

Max energy of electrons = hf – WShowed light consists of photons or packets of energywith each packet having an energy, hf . Einstein.

Confined Waves and Characteristic Vibrations:

We saw that confined waves on a string tied at both ends andunder tension can only vibrate in certain characteristicmodes :

Lowest mode : f1=v 12L

Next mode : f2=2f1=2v 12L

Next mode : f 3=3f1 etc

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 vibrateonly in certain characteristic modes – called quantum energylevels 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=2R0

Emission and absorption of photons between atomicenergy levels or orbits.

If one takes apart an atom and puts it together again wewioll get the same orbits and energies – this is the property of regeneration.

To get atoms to be stable, requires a new rule of quantummechanics, 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 insolids

Confining waves in a solid

Only certain energy levels allowed. Also called orbits.

Only certain characteristicoscillations allowed.Like overtones of a stringtied at both ends

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

using =hp

12...n ..f 1, f2 , , , f n ..

E=hf

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 haveits own level. 4. Electron has an intrinsic property called spin – it can spin clockwiseor 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

EnergyEach level can accommodate two electrons.

If there are N electrons , start filling the levels starting from the lowest energylevel 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 itslevel structure.

In a real material, levels come in bands, separated byenergy gaps where no electrons can reside.

Energy gap

Energy gap

Allowed levelsAllowed levels

Allowed levels

Allowed levels

Insulator: electrons fillthe conduction band completely. No freedomfor electrons to move.

Metals: electronsfill conduction bandpartially. Electronswith slight excess energy can move.

In photo-conductors gap is smalland light photons can raise electrons into levels where theycan move or conduct.

Insulator-- electrons dontmove when voltage is appliedacross an insulator.

Metal – applicationof voltage causes electrons to transferacross the conductor.

Photo Conductor

Energy of photons of light lift electrons into unfilled levels where they can respond to appliedvoltage 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

● 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.

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

Summary for the two kinds of matter found in nature

Electrons in SolidsElectrons 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 requiresthat electron motion (until they are detected) is described by wave motion. So electrons confined in a solidare electron waves confined to propagate inside the solidsatisfying the boundary condition that they cannot be outside the metal.

(1) So just as for waves on a string tied at two ends, theseelectron waves can only oscillate at specific frequencies orwe 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 witha momentum p = m v , is given by the relation

=hp

First formulated by Louis de Broglie. So to each allowedwavelength there is an allowed momentum and thereforean allowed energy.

(3) Occupancy: How many electrons can occupy a givenenergy level ? This is a deep question, the answer to whichwas 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 oranti-clockwise – spin up and spin down. Experimentallythese 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 theselevels can be filled two at a time.

Electrons present in the substance can fill these leveltwo at a time starting at the lowest energy. So whenwe run out of available electrons which reach a highenergy level – which is called the Fermi level.

● 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

In a metal:

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