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EE

Introduction to solid state electronics

Fall 2017

Chapter 2:

Atoms and Electrons

Atoms and Electrons

 Electronic structure of atoms

 Interaction of atoms and electrons with

excitation (absorption and emission of light)

 Quantum theory versus classical theory

Photoelectric Effect

 Escape energy Em = h f – q f

- Work function = q f = minimum energy required for an

electron to escape from material to vacuum

- Determination of f :

Measurement of escape energy Em

vs frequency of incident light n:

 Intercept = - q f

 Light energy is discretized (rather

than continuous) into packet

of energy (particle): The Photon

 Dual nature of light

Wave versus Particle

Atomic Spectra

 Atomic spectrum for a material (or gas) is obtained from analysis

of absorption / emission of light

Application: Electric discharge in a gas  Emission of light due

to electron transfer from higher to lower energy levels

D E = h f = hc/ l (c = speed of light = 10^8 m/s)

 Emitted quanta or photons have wavelengths ( l ) characteristic

of the electronic structure of the atoms of the gas

 Example: Spectral emission of hydrogen atom (established in

1900): Emitted wavelengths are organized in discrete series

following the rule

f = cR(1/ l^2 – 1/n^2 ) (R = Rydberg constant = 109.678 cm-1)

BOHR Model for the Hydrogen atom

 Emission spectra is based on mathematical model of planetary

systems

 Electrons can be excited to an outer orbit by gaining energy

 Electrons fall back to an inner orbit by loosing energy

 The energy lost is associated with a specific wavelength that is

represented by a line in the spectra

e-

E = h n

e-

Excited

BOHR Model for the Hydrogen atom

Postulates

1. Electrons exist in stable circular orbits about the nucleus.

Orbiting electron does not give off energy, else it would spiral

into the nucleus

2. The electron can move to an orbit of higher or lower energy by

gaining or loosing energy equal to the difference in the energy

levels (absorption or emission of a photon of energy h n )

3. The angular momentum P q of the electron in an orbit is an

integer multiple of h/2 p = ( h = Planck’s constant)

n = 1,2,3,4…

P q = n 

BOHR Model for the Hydrogen atom

Orbital radius

Electron stable in orbit  Force balance

 Electrostatic force = centripetal force  -q^2 / Kr^2 = -mv^2 /r

2

2 2

2

2 2 2

2

2

2

2 2 2 2

2

mq

K

r n

r mr

n

r

mv

Kr

q

r

m v n

P m r mvr n

n

n n n n

q = w = =

Postulate 3

Angular momentum P q = mvr = nh = m w r

2

BOHR Model for the Hydrogen atom

Energy of the electron in orbit n

Kinetic energy of the electron KE = ½ mv

2

Potential energy of the electron PE = =

Total Energy of the electron in n

th

orbit

En = KEn + PEn = + = -

Kn

q

Kn

n q

mr

n

v

n

2

2 2

2

2 2 2

4

K n 

mq



( )

( )

x q

x q

F dx

2 2 2

2 4

K n 

mq

Kr

q

n

2 2 2

4

K n 

mq

2 2 2

4

K n 

 mq

2 2 2

4

K n 

mq

BOHR Model for the Hydrogen atom

Theory confirms:

1. Quantization of Energy E = hc/ l

2. Establish relationship between photon energy and

electron transition between levels

Theory lacks:

1. Cannot explain experimental evidence of energy level

splitting

2. Cannot be extended to atoms more complicated than

hydrogen atom

Partial success led to development of Quantum Theory

Quantum Mechanics

Probability and Uncertainty Principle

 For events in the atomic scale:

No ABSOLUTE precision in position, momentum, energy

But MOST PROBABLE, EXPECTED, average values

Heisenberg Uncertainty Principle (HUP)

x is the particle position, px is its momentum

E is the particle energy, t is the time at which measurement

is done

 For large objects, uncertainty is negligible ( h/2 p range )

D D  

x

x. p

D E .D t  

Quantum Mechanics

Probability Density Function

The average value of function f(x) = <f(x)> is determined from the

PDF

If PDF is not normalized





 f ( x ) = f ( x ) P ( x ) dx

P x dx

f x P x dx

f x

Quantum Mechanics

The Shrödinger Wave equation

Apply quantum mechanics concept to classical mechanics

Basic Postulates

1. • Each particle in a system is described by a wave function Ψ(x,y,z,t).
   • Ψ(x,y,z,t) & its space derivatives ∂Ψ/ ∂x , ∂Ψ/ ∂y, ∂Ψ/ ∂z are continuous,

finite and single valued.

2. Each classical quantity has an equivalent quantum mechanical operator

Classical Quantum mechanical

x and/or (y,z) x and/or (y,z)

f(x) f(x)

p(x)

E

j  x

 

j  t

 



Quantum Mechanics

The Shrödinger Wave equation

Energy Equation KE + PE = Etotal

In classical mechanics:

In quantum mechanics:

In 3 dimensions: = Shrödinger Equation

p V E

m

2

( , ) ( ) ( , ) ( , ) 2

1

2

x t j t

x t V x x t m j x



   =  

  





  

j t

V x m (^) x 

2

2 2

j t

V m 

    =

 2 

2

2

( ) 2

2

2

2

2

2 2

x y  z

  

  

  =

   

Quantum Mechanics

( ) ( ) ( ) ( ) ( ) ( ) ( ) 2

1

2

x t j t

x t V x x t m j x

  

       =  

  





  

Solution of Shrödinger equation  wave mechanics

 Can be solved only in special simple cases

 Separation of variables:

2

x t j t

x t V x x t m j x

( x , t ) = ( x ).( t )

Term independent of t = E.  (x)

E = separation constant

2

2 2

x

t

t

j

V x x t

m x

Independent of x

( x , t )