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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