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Drexel University
ECE Department
ECEE 302 Electronic
Devices
30 September 2002
ECEE 302: Electronic Devices
Lecture 2. Physical Foundations of
Solid State Physics
30 September 2002
BMF-Lecture 3-093002-Page -1
Copyright © 2002 Barry Fell
Drexel University
ECE Department
Outline
ECEE 302 Electronic
Devices
• Characteristics of Quantum Mechanics
–
–
–
–
–
Black Body Radiation
Photoelectric Effect
Bohr’s Atom and Spectral Lines
de Broglie relations
Wave Mechanics
•
•
•
Probability
Schrodinger Equation
Uncertainty Relations
• Applications of Wave Mechanics
– Infinite Well
– Finite Barrier (Tunneling)
– Hydrogen Atom
• Periodic Table
– Pauli Exclusion Principle
– Minimum Energy
– Bohr’s Aufbauprinzip (Building-Up Principle)
• Atomic Structure
30 September 2002
BMF-Lecture 3-093002-Page -2
Copyright © 2002 Barry Fell
Drexel University
ECE Department
What is Quantum Mechanics?
ECEE 302 Electronic
Devices
• Classical Mechanics
–
–
–
–
Newton’s Three Laws
applies to particles (localized masses) or mass distributions
Well defined deterministic trajectories
Initial Conditions and Equations of Motion determine particle behavior for all
time
• Classical Optics
– Light is wave (non-localized, spread over region)
– Interference and Diffraction effects are seen
• Quantum Mechanics
– Laws based on Geometrical Optics (short wave-length region)
– Describes system behavior in terms of “Wave Function”
•
“square of wave function” provides probabilistic information about state of system
– Evolution of Wave Function in Time is deterministic
– An observation based on the wave function
•
•
possible outcomes of observation
probability of each outcome
30 September 2002
BMF-Lecture 3-093002-Page -3
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Black Body Radiation
• Black Body is perfect absorber (and re-radiator)
• Based on classical mechanics, the black body should have infinite
energy
• Planck found an empirical formula to fit Black Body experimental
curve
• To derive this formula from first principles he had to assume light
energy (electromagnetic energy) was not spread out in space but
came in small packages or bundles (quanta - german word for
“dose”). E=hn
Classical Equi-partition of Energy Model
Planck' s Law for Black Body Radiation
8 n2
Un   3
c
Thermal
(Black
Body)
Energy
hn
hn
kT
e 1
Un   Black Body spectral Radiant Energy
per unit volume per Hertz
Frequency
30 September 2002
BMF-Lecture 3-093002-Page -4
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Planck’s Black Body Radiation Law
Consider Planck' s Law for Black Body Radiation
8 n2
Un   3
c
hn
hn
kT
e 1
Un   Black Body spectral Radiant Energy per unit volume per Hertz
n  frequency of radiation
c  velocity of light
h  Planck' s constant  6.62  10- 27 erg  sec (unit of action)
k  Boltzmann' s constant  1.38  10-16 erg /  K  energy per degree Kelvin
T  Temperatur e of the Black Body Radiator in degrees Kelvin
Consider hn  kT
8 n2
Un   3
c
hn
e
hn
kT
8 n2
hn
8 n2
 3
 3
c 1  hn      1
c
1
kT
hn
8 n2
 3 kT (classical limit)
c
 hn 


 kT 
Consider hn  kT
8 n2
Un   3
c
30 September 2002
hn
e
hn
kT
hn

8 n2 hn 8 n2
kT
 3

h
n
e
(Boltzmann Factor - Wien' s Law)
hn
3
c
c
1
e kT
BMF-Lecture 3-093002-Page -5
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Photo-Electric Effect
• When light of frequency n is incident on a metal, electrons are
emitted from the metal
– Emission is instantaneous
– kinetic energy of the emitted electrons are dependent on the frequency of the
light (n)
• Einstein (1905) used Planck’s idea of bundle of light to explain this
effect
– (1/2) mv2=hn-F
– Einstein called this particle of light a “photon”
• Einstein had shown that Planck’s hypothesis could be interpreted
as showing that waves can exhibit particle like properties
hn
Ekinetic 
F=Work Function
30 September 2002
Ekinetic 
1
mv 2
2
1
mv 2  hn  F
2
F  Work Function
BMF-Lecture 3-093002-Page -6
of the metal Copyright © 2002 Barry Fell
Drexel University
ECE Department
The Crisis in Atomic Theory
ECEE 302 Electronic
Devices
• Ernst Rutherford determined the atom has a small positvely
charged, solid nucleus which is surrounded by a swarm of
negatively charged electrons
• Classical Electrodynamics predicted that an accelerating particle
(such as an electron moving around the nucleus of an atom) should
radiate continuously, reduce its radius until it spirals into the
nucleus
• Observation shows us that
– Atoms are stable
– Radiation from atoms is with discrete spectral lines that follow a geometric
series
• Niels Bohr resolved these issues with a new atomic model based on
the work of Einstein and Planck
30 September 2002
BMF-Lecture 3-093002-Page -7
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
The Atom of Neils Bohr
• Postulates
– Atoms are stable
•
•
Electrons can exist in well defined orbits around a nucleus (Atomic State)
Electrons do not radiate when in the orbit (Contrary to Classical Electromagnetism)
– Discrete Spectral lines
•
•
•
Atoms radiate (emit a photon) or absorb energy (absorb a photon) when an electron makes a
transition from one fixed orbit (initial state) to another orbit (final state)
The frequency of the light emitted or absorbed is given by Planck’s formula n=DE/h
The discrete orbits are determined by “quantization” of the orbital angular momentum. This is
determined by the relation mvrn=nh
• This theory successfully reproduced the spectral line pattern seen
in H, He+, and Li++, single electron atoms
30 September 2002
BMF-Lecture 3-093002-Page -8
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Bohr’s Atom (1 of 3)
For circular motion, Newton' s law for circular motion is
mv 2
Ze 2

r
4 0r 2
The orbital angular momentum, L, is
nh
 n
2
Where we have invoked the quantum
Momentum of the Atom
L  mvr 
h 

 

2



condition on the Orbital Angular
e
and
 mv 2
F
r
nh
mv 
2r
and
mv    nh 
 2r 
2
Ze
2
FElectrostatic
30 September 2002
Ze 2

4 0r 2
BMF-Lecture 3-093002-Page -9
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Bohr’s Atom (2 of 3)
So
m 2 v 2 mZe 2

r
4 0r 2
and
2
mZe 2
 nh  1
  3 
4 0r 2
 2  r
and
4 0  nh 
rn 
 
mZe 2  2 
2
Hence
v
nh
2r
and
2
nh
nh mZe 2  2 
mZe 2  2 
vn 

  
 
2rn 2 4 0  nh 
4 0  nh 
30 September 2002
BMF-Lecture 3-093002-Page -10
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Bohr’s Atom (3 of 3)
Kinetic Energy (KE n ) of Electon in the Orbit
1
mZ 2e 4
h 

2
KE n  mv n 




2
2
2 
24 0  n 2  2 
The Potential Energy PE n  of the electon in the Orbit is
Ze 2
mZ 2e 4
PE n   2  
rn
4 0 2 n 2 2
so
mZ 2e 4
En  KE n  PE n  
2
24 0  n 2  2
and
mZ 2e 4  1
1 
En  Em 

 hn n ,m

2 2
2
2 
24 0    m n 
and
30 September 2002
n n ,m
mZ 2e 4  1
1 



2 2
2
2 
m
n
24 0   h 

BMF-Lecture 3-093002-Page -11
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
The atom of Bohr Kneels
(“The Strange Story of the Quantum” - Banesh Hoffman)
• Bohr’s Theory failed to predict the spectral behavior of more
complex atoms with two or more electrons
• Spectral line splitings (fine structure) was not predicted by Bohr’s
Theory
• The anomalous Zeeman effect (splitting of spectral lines in
Magnetic Fields) was also not predicted successfully
• This required a more fundamental theory
30 September 2002
BMF-Lecture 3-093002-Page -12
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Davision-Germer Experiment and the de Broglie
Hypothesis
• In 19XX Davision and Germer of the Bell Telephone Laboratories
showed that high energy electrons could be diffracted by crystals
• Particles had shown characteristics of waves
• Louis de Broglie characterized the wave nature of particles by the
expression l=h/p
• localized particle properties: Energy & Momentum
• non-local wave properties: frequency & wavelength
Characteristics
mass
energy
momentum
0
E=hn
p=hk=h/l=E/c
Particles (~electron) m
E=hn
p=hk=h/l=(2mE)1/2
Wave (light)
30 September 2002
BMF-Lecture 3-093002-Page -13
Copyright © 2002 Barry Fell
Drexel University
ECE Department
Wave Mechanics (1 of 2)
ECEE 302 Electronic
Devices
• Schrodinger used classical geometrical optics to formulate a new
mechanics he called wave mechanics

 x 
j 2   nt    
 l 

Start with a " wave" function,  x, t   e
Make use of the Einstein (E  hn ), de Broglie p  h/ l  relations
h
2E
E  hn 
 ,   2 n 
2
h
h hk
2p
p 
 k , k  2 l 
l 2
h
E p 
j t  x 
  
Transform  x, t   e j2  t  kx   e
 x, t 
 x, t 
Note that E x, t   j
, and p x, t    j
t
x
30 September 2002
BMF-Lecture 3-093002-Page -14
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Wave Mechanics (2 of 2)
The Schrodinger Wave Equation
The Schrodinge r wave equation is determined from
the classical relationsh ip
p2
Ep, x  
 V( x)
2m
Substitute the relations E x, t   j
 x, t 
 x, t 
, and p x, t    j
t
x
into the above equation. We find
 x, t 
 2  2  x, t 
j

 V( x) x, t 
2
t
2m x
What is the physical significan ce of  x, t ?
30 September 2002
BMF-Lecture 3-093002-Page -15
Copyright © 2002 Barry Fell
Physical Interpretation of y
Drexel University
ECE Department
ECEE 302 Electronic
Devices
• Schrodinger initially believed that  was a guiding wave
• Max Born introduced a probability interpretation for 
E x, t  is a probabilit y amplitude
Px, t   E x, t   E x, t  E x, t   probabilit y that
2

a particle of energy E is at the position
x to x  dx at the time t to t  dt
We can show that

 
Px, t 
 div Sx, t   0
t
 



Sx, t  
E x, t  grad E x, t   grad E x, t  E x, t 
2 jm

where


which is called the " probabilit y current"
The change in probability within a volume V
is due to the “flow” of propability across the
bounding surface A
30 September 2002



P x, t 
t

 
Sx, t 
BMF-Lecture 3-093002-Page -16
Copyright © 2002 Barry Fell
Drexel University
ECE Department
Solutions of the Schrodinger Equation
ECEE 302 Electronic
Devices
• The Schrodinger equation is a second order differential equation
• It can be split into the product of a time solution and a spatial
solution by the method of separation of variables
– Q(r,t)=A( r) B(t)
– Show equations for t and for r
• The spatial equation is called a sturm-louisville equation.
Solutions exist only for certain values of the separation constant.
These are called eigen (proper) solutions and eigen (proper) values
• These possible solutions are called quantum levels and the specific
values are called quantum values (or quantum numbers)
• The corresponding functions are called quantum state functions
30 September 2002
BMF-Lecture 3-093002-Page -17
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Boundary Conditions
• Solutions must obey the following conditions called boundary
conditions
– Q must be bounded (finite) everywhere
– At boundaries between multiple solutions the magnitude and the derivative of
the solutions must be continuous
30 September 2002
BMF-Lecture 3-093002-Page -18
Copyright © 2002 Barry Fell
Drexel University
ECE Department
Normalization of the Wave Function
ECEE 302 Electronic
Devices
• Since Q is a probability amplitude and QQ* is a probability we
must have
– integral of QQ* over all space = 1
– This is called normalization of the wave function
• Examples
– Particle in a box
– Hydrogen Equation
30 September 2002
BMF-Lecture 3-093002-Page -19
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Applications
•
•
•
•
•
•
Potential Well - (Stationary States)
Potential Barrier - (Tunneling)
Hydrogen Atom (quantum numbers)
Electron Spin
Hydrogen Molecule
Bonds
– Ionic Bond
– Valence Bond
30 September 2002
BMF-Lecture 3-093002-Page -20
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Potential Well (in 1-dimension) and Bound States
•
•
•
•
•
Schrodinger Equation
Boundary Conditions
Solutions
Quantization of Energy Levels
Uncertainty Principle
30 September 2002
BMF-Lecture 3-093002-Page -21
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Potential Barrier in 1 dimension (Tunneling)
•
•
•
•
Schrodinger Equation
Boundary Conditions
Solution
Uncertainty Principle
30 September 2002
BMF-Lecture 3-093002-Page -22
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Hydrogen Atom
•
•
•
•
•
Schrodinger Equation in 3 dimensions
Schrodinger Equation in spherical coordinates
Separation of Variable for r, q, h
Solution for h = magnetic quantum number
Legendre polynomials for q = angular momentum quantum
number
• Legarre polynomials for r = principle (orbital) quantum number
• Electron spin s=+/- 1/2
• Designation of a quantum state n,l,m,s
30 September 2002
BMF-Lecture 3-093002-Page -23
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Uncertainty Principle
• Introduced by Werner Heisenberg
• Statement of Uncertainty Relations
– Uncertainty in position x uncertainty in momentum > h
– Uncertainty in energy x uncertainty in time > h
30 September 2002
BMF-Lecture 3-093002-Page -24
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Hydrogen Molecule
•
•
•
•
Schrodinger Equation
Solution
Interpretation
Electron Spin
30 September 2002
BMF-Lecture 3-093002-Page -25
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Ionic Bond
• Ionic Bonding
– exchange of an electron between two atoms so each acheives a closed shell
– result is a positive (electron donor) and negative (electron acceptor) ion
– ions attract forming a bond
• Examples: NaCl, KCl, KFl, NaFl
30 September 2002
BMF-Lecture 3-093002-Page -26
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Valance Bond
• Valance Bond: Bonding due to two atoms of complementary
valance combining chemically
– Valance Band 4 (and 4): C, Si, Ge, SiC
– Valance Band 3 and 5: GaAs, InP,
– Valance Band 2 and 6: CdS, CdTe
• Examples
– Face Centered Cubic: Diamond (C), Silicon (Si)
30 September 2002
BMF-Lecture 3-093002-Page -27
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Periodic Table of Elements
• History
– Developed by Medelaev in 1850 based on chemical properties of atoms
– Understood initially in terms of chemical affinities (Valence)
– Quantum Mechanics provides the Physical Theory of Valence
• Atoms are arranged in 8 basic columns related to the valence of
each atom
• Transition elements build up their electronic structure
• Periodic Table can be understood in terms of two principles
– Pauli Exclusion Principal
– Minimum Energy Principal
30 September 2002
BMF-Lecture 3-093002-Page -28
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Periodic Table
30 September 2002
BMF-Lecture 3-093002-Page -29
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Pauli Exclusion Principle
• No two electrons can be in the same quantum state at the same
time
• Fundamental in understanding the structure of the periodic table
30 September 2002
BMF-Lecture 3-093002-Page -30
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Bohr’s Building up (Aufbauprinzip) Principle
• Determines the basic structure of atoms
• Based on the Pauli Exclusion Principle
• Minimum Energy Principal
30 September 2002
BMF-Lecture 3-093002-Page -31
Copyright © 2002 Barry Fell
Drexel University
ECE Department
ECEE 302 Electronic
Devices
Minimum Energy Criteria
• Over-arching principle is minimum energy
• explains electronic structure of rare earth elements
30 September 2002
BMF-Lecture 3-093002-Page -32
Copyright © 2002 Barry Fell
Drexel University
ECE Department
Atomic Structure
ECEE 302 Electronic
Devices
• Quantum Numbers
– n-principal quantum number signifies the electron orbit
– l-orbital quantum number signifies the angular momentum in orbit n
(l=0,1,2,…,n-1)
– m - magnetic quantum number signifies the projection of the angular
momentum quantum number on a specific axis (z), (m=-l,-(l-1), -(l-2), …,1,0,1,…,(l-1),l)
– s - electron spin signifies and internal state of the electron (s=+1/2, -1/2)
• Atom is described by the set of quantum numbers that describe
each electron state
–
–
–
–
Hydrogen
Helium
Lithium
Boron
30 September 2002
= 1s1
= 1s2
= 1s2,11p1
= 1s2, 1p2
X
Carbon
Nitrogen
Neon
1s2, 1p4
Potassium 1s2,1p6,2s1
1s2,1p6,2s2
1s2, 1p6
BMF-Lecture 3-093002-Page -33
Copyright © 2002 Barry Fell