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• Introduction to quantum mechanics (Chap.2)
• Quantum theory for semiconductors (Chap. 3)
•Allowed and forbidden energy bands (Chap. 3.1)
What Is An Energy Band And How Does It Explain The
Operation Of Semiconductor Devices?
To Answer These Questions, We Will Study:
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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
Quantum Mechanics: describing behavior of systems at atomic length scales and smaller .
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Photoelectric Effect
Tm
ax
0
ννo
• 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.
Metal Plate
Incident light with frequency ν
Emitted electron kinetic energy = T
The photoelectric effect ( year1887 by Hertz) Experiment results
Concept of “energy quanta”
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Planck’s constant: h = 6.625×10-34 J-s
Photon energy = hν
Work function of the metal material = hνo
Maximum kinetic energy of a photoelectron: Tmax= h(ν-νo)
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
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Electron’s Wave Behavior
Electron beam
Nickel sample
Detector
Scattered beam
θ
θ =0
θ =45º
θ =90º
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:h
p
The wavelength of a particle is:p
h
λ is called the de Broglie wavelength
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The Uncertainty PrincipleThe Heisenberg Uncertainty Principle (year 1927):
• It is impossible to simultaneously describe with absolute accuracy the position and momentum of a particle
• It is impossible to simultaneously describe with absolute accuracy the energy of a particle and the instant of time the particle has this energy
xp
tE
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 in the semiconductor?
• Mathematical description of motion of electrons in quantum mechanics ─ Schrödinger’s Equation
• Solution of Schrödinger’s Equation energy band structure and probability of finding a electron at a particular position
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Schrӧdinger’s Equation
One dimensional Schrӧdinger’s Equation:
:),( tx Wave function
:)(xV Potential function
:m Mass of the particle
t
txjtxxV
x
tx
m
),(
),()(),(
2 2
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dxtx2
),( , the probability to find a particle in (x, x+dx) at time t
2),( tx , the probability density at location x and time t