Semi-Conducting & Magnetic Materials-Week 1-Jan 9-2012

8/2/2019 Semi-Conducting & Magnetic Materials-Week 1-Jan 9-2012

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Semi-conducting & Magnetic Materials

Prof S. B. Sant

Department of Metallurgical & Materials EngineeringIIT Kharagpur

MT41016


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Behaviour of Electrons in SolidsOr

Electron Theory of Solids

Capable of explaining:

Optical

MagneticThermalElectrical

Properties of materials

Chapter 1


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Applications of materials :

Optical Lasers, Lenses, Windows, Solar collectors,Communication.

Magnetic Electric Generators, Motors, Loudspeakers,Transformers, Data Storage.

Thermal Refrigeration & Heating devices, Heat shieldsfor Spacecrafts.

Electrical Conductors, Insulators & Semi-conductors.

Understanding of electronic properties of materials:

3 approaches over the last century.


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Continuum Theory Laws were Empirically Derived.

Structure of matter ignored.

Examples:

Ohms Law, Newtons Law, Maxwells Equations

Classical Electron Theory Matter described by

Atomistic Principles. Free Electrons in metals drift inresponse to external force and interact with lattice atoms.

Quantum Theory Newtonian mechanics - inaccurate atomic dimensions i.e., interactions of electrons with

solids.

Lacks visualization difficult to comprehend.


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Newtons Law: F = ma (1.1)

Kinetic Energy: (1.2)

( is the particle velocity)

Momentum: (1.3)

Combining (1.1) & (1.2) yields (1.4)

Speed of light: c = v(v = frequency of the light wave & its wavelength) (1.5)

Velocity of a wave: (1.6)

Angular frequency: (1.7)

Einsteins law: E = mc2 (1.8)

m

p

Ek 2

2

=

2

2

1mEk =

m=

v=

v 2=


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The Wave-Particle Duality:What is an electron? No one has seen an electron!

We experience the actions of electrons

e.g., on a TV screen or in an electron microscope.

Minimum energy of light, one quantum (a photon) with energy E = v.h = needs to impinge on a metal to free an electron by overcoming its binding energy

to its positively charged nucleus into free space.

h is the Planck constant (6.626 x 10-34 J.s)

Reduced Planck constant is used (=h/2) with the angular frequency, = 2.v

Particle Wave

Similar duality is with lightWave Electromagnetic wave color is f ()Particle Photoelectric effect emission of electrons from a metallic surface

that has been illuminated by light of high energy.


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Particle property of electrons with rest mass mo and charge e:

J.J. Thomson at the Cavendish Lab in Cambridge noticed the deviation

of a cathode ray by electric & magnetic field.

E. Rutherford (one of Thomsons students) suggested that an atom

resembled a solar system electrons orbited around a massive

positively charged center.

Today we know that the electron is the lightest stable elementary particle

of matter & that it carries the basic charge of electricity.

It was also found that electrons in metals can move around freely

under certain conditions.


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In 1924, de Broglie, postulated that electrons should possess

The Wave-Particle Duality and suggested the wave nature of electrons.

He connected the wavelength , of an electron and the momentum, p,of the particle by:

p = h (2.3)

Assignment: Derive the above using Equations & definitions given earlier.

In 1926, Schrdinger gave de Broglies idea a mathematical form.

In 1927, Davisson & Gerner and independently, in 1928, G. P. Thomson

discovered electron diffraction by a crystal, thereby proving the

wave nature of electrons.

Q: What is a wave?

A wave is a disturbance periodic in position and time.

In contrast a vibration is a disturbance periodic in position ortime.


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Waves are characterized by

a velocity, , a frequency, v and a wavelength, that are interrelated by

(2.4)

Often, the wavelength is replaced by the inverse quantity the wave number

(2.5)


v=

2=k

And the frequency v, by the angular frequency

Equation (2.4) becomes: (2.6)

Simplest waveform expressed mathematically Sine (or cosine)

function called Harmonic Wave.

v 2=

k

=


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Properties of an electron can be described by a Harmonic Wave i.e. aWave Function (time & space dependent):

(2.7)

Where k is the wave number and the angular frequency.

N.B. This wave function does not represent any physical wave orquantity just a mathematical description of a particle.

The Wave-Particle Duality can be understood by combining severalwave trains having slightly different frequencies, andand different wave numbers, kand


)sin( tkx =

( ) +( )kk +

Let us consider 2 waves such as:

(2.7)

and (2.8)

)sin(1 tkx =

( )[ ]))(sin2 txkk ++=


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Superimposing 1 and 2 yields a new wave .

Using

We obtain:

( ) ( ) +=+2

1sin.

2

1cos2sinsin

+

+

=+= tx

kkx

kt

22sin.

22cos221

Modulated

amplitudeSine wave

(2.9)

Equation (2.9) describes a sine wave

frequency between and ( + ) amplitude slowly modulated by a cosine function.


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Fig 2.1: Combination of two waves of slightly different frequencies.

Fig 2.2: Monochromatic matter wave ( and k = 0). Wave has constant

amplitude & travels with the phase velocity


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Boundary conditions:

1. No variation in and k. (ie, = 0 and k= 0) - Infinitelylong wave packet

monochromatic wave i,.e., wave picture of an electron.

2. Alternatively, and kcould be assumed to be very large short wave packet.

If a large number of waves are considered, having frequencies ( +n ) (where n= 1, 2, 3, 4) then the string of wave packetsreduces to one packet only.

The electron is then represented as a particle.


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Fig 2.3: Superposition of-waves.


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Different velocities need to be distinguished:

1. The velocity of the matter wave is called the wave velocity orphase velocity.

Matter wave is a monochromatic wave a stream of particles ofequal velocity whose

frequency, ,wavelength, ,momentum,p,

or energy, E,

can be exactly determined (Fig 2.2).

Cannot determine location of the particles.


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Using the sine wave component of equation (2.9) we can deducethat:

We obtain the velocity of matter wave with frequencyand a wave number

The phase velocity varies for different wavelengths called

dispersione.g. rainbow colors from a prism

'

'

2/

2/

kkkt

x =++== (2.6a)

( )2/ +( )2/kk +

+

+

=+= tx

kkx

kt

22

sin.

22

cos221

(2.9)


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2. A particle can be understood to be composed of a group of waves or awave packet.

Each individual wave has a slightly different frequency.

Therefore, Velocity of a particle is called a group velocity

In figure 2.1, the envelope propagates with the group velocity

From the left hand portion of Equation 2.9 (modulation amplitude)We obtain the group velocity of the particles:

Equation (2.10) is the velocity of a pulse wave, i.e., of a moving particle.

The location X of a particle is known precisely, while the frequency is not.

g

g

dk

d

kt

xg

=

== (2.10)


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A Wave Packet can be thought to consist of several Wave Functions 1, 2, n, withslightly different frequencies.

Looked at differently, we can perform a Fourier analysis of a pulse wave (insert Fig 2.4) results in a series of sine & cosine functions (waves) with different wavelengths.

The better the location X, of a particle can be determined, the wider the frequencyrange, , of its wave.

Leads to one form of Heisenbergs Uncertainty Principle:

This means that the product of

the distance over which there is a finite probability of finding an electron, X, and

the range of momenta, p (or wavelength) of the electron wave

is greater than or equal to a constant.

The location and the Frequency of an Electron cannot be accurately determined at thesame time.


,. hXp


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The question then is:

What do we do with Wave Functions?

To interpret them, we shall use the Bohrs postulate:

The square of the wave function (as is a complex function, we

shall use . *) is the probability of finding a particle at a certainlocation. That is,

. * dx dy dz = . * d

is the probability of finding an electron in the volume element d.

N.B.

In wave mechanics, we encounter probability statements;

In classical mechanics, we can exactly find the location of a particle.


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Another question then is:

Is an electron wave the same an electromagnetic wave?

No!

Electromagnetic waves propagate by interaction of electrical &magnetic fields.

And then, are particles & waves completely, unrelated phenomena?

Conceptually, yes; however, looking at Equation (2.9), we find thatboth waves and particles can be mathematically described by the same

equation.

In the case of waves, = 0 and k= 0In the case of particles, we set and kto be very large.

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Semi-Conducting & Magnetic Materials-Week 1-Jan 9-2012