Download BasicQuantumMechanics18And20January2017

Basic Quantum Mechanics
18 and 20 January 2017
What Is An Energy Band And How Does It Explain The
Operation Of Classical Semiconductor Devices and
Quantum Well Devices?
To Address These Questions, We Will Study:




Introduction to quantum mechanics (Chap.2)
Quantum theory for semiconductors (Chap. 3)
Allowed and forbidden energy bands (Chap. 3.1)
Also refer to Appendices: Table B 2 (Conversion Factors), Table B.3
(Physical Constants), and Tables B.4 and B.5 Si, Ge, and GaAs key
attributes and properties.
 We will expand the derivation of Schrӧdinger’s Wave Equation as
summarized in Appendix E.
 Will use a transmission line analogy for discussing and interpretaing
Schrӧdinger’s wave equation solutions
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Classical Mechanics and Quantum Mechanics
Mechanics: the study of the behavior of
physical bodies when subjected to forces or
displacements
Classical Mechanics: describing
the motion of macroscopic objects.
Macroscopic: measurable or
observable by naked “eyes” or by
optical microscopy
Quantum Mechanics: describing
behavior of systems at atomic
length scales and smaller .
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Incident light with
frequency ν
Emitted electron
kinetic energy = T
Tmax
Photoelectric Effect
0
Metal Plate
The photoelectric effect ( year1887 by Hertz)
νo
ν
Experiment results
• Inconsistency with classical light theory
According to the classical wave theory, maximum kinetic energy of the photoelectron
is only dependent on the incident intensity of the light, and independent on the light
frequency; however, experimental results show that the kinetic energy of the
photoelectron is dependent on the light frequency.
Concept of “energy quanta”
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Energy Quanta
• Photoelectric experiment results suggest that the energy in
light wave is contained in discrete energy packets, which are
called energy quanta or photon
• The wave behaviors like particles. The particle is photon
Planck’s constant: h = 6.625×10-34 J-s
Photon energy = hn
for frequency)
(Greek lower case “nu”, Could also use “f”
Work function of the metal material = hno
Maximum kinetic energy of a photoelectron: Tmax= h(n-no)
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Electron’s Wave Behavior
Nickel sample
θ =0
Electron beam
θ
Scattered
beam
θ =45º
θ =90º
Detector
Davisson-Germer experiment (1927)
Electron as a particle has wave-like behavior
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Wave-Particle Duality
Particle-like wave behavior
(example, photoelectric effect)
Wave-like particle behavior
(example, Davisson-Germer experiment)
Wave-particle duality
Mathematical descriptions:
The momentum of a photon is:
The wavelength of a particle is:
p
h

h

p
λ is called the de Broglie wavelength
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The Uncertainty Principle
The Heisenberg Uncertainty Principle (year 1927):
• It is impossible to simultaneously describe with absolute accuracy the
position and momentum of a particle
p x  
• It is impossible to simultaneously describe with absolute accuracy the
energy of a particle and the instant of time the particle has this energy
E t  
The Heisenberg uncertainty principle applies to electrons and states
that we can not determine the exact position of an electron. Instead, we
could determine the probability of finding an electron at a particular
position.
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Quantum Theory for Semiconductors
How to determine the behavior of electrons )and
holes) in the semiconductor?
• Mathematical description of motion of electrons in quantum
mechanics ─ Schrödinger’s Wave Equation
• Solution of Schrödinger’s Wave Equation energy band
structure and probability of finding a electron at a particular
position
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Schrӧdinger’s Wave Equation
Text Equation 2.6 which we will now derive
One dimensional Schrӧdinger’s Wave Equation:
  2  2  ( x, t )
 ( x, t )
 V ( x )  ( x , t )  j
2
2m
x
t
 ( x, t ) :
Wave function
 ( x, t ) dx
2
 ( x, t )
V (x) :
m:
, the probability to find a particle in (x,
x+dx) at time t
2
, the probability density at location
x and time t
Potential function
Mass of the particle
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