Download CHAPTER 2 Introduction to Quantum Mechanics

CHAPTER 2
Introduction to Quantum Mechanics
• The behavior and characteristics of electrons in a semiconductor
can be described by the formulation of quantum mechanics
called wave mechanics. The essential elements of this wave
mechanics, using Schrodinger’s wave equation.
• Discuss a few basic principles of quantum mechanics that apply to
semiconductor device physics.
• State Schrodinger’s wave equation and discuss the physical
meaning of the wave function.
• Consider the application of Schrodinger’s wave equation to
various potential functions to determine some of the
fundamental properties of electron behavior in a crystal.
• Apply Schrodinger’s wave equation to the one-electron atom. The
result of this analysis yields the four basic quantum numbers, the
concept of discrete energy bands, and the initial buildup of the
periodic table.
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2.1 | PRINCIPLES OF QUANTUM MECHANICS
2.1.1 Energy Quanta
At a constant incident intensity, the maximum kinetic energy of
the photoelectron varies linearly with frequency with a limiting
frequency v = v0, below which no photoelectron is produced.
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thermal radiation is emitted from a heated surface in discrete
packets of energy called quanta. The energy of these quanta is
given by E = hv, where v is the frequency of the radiation and h is a
constant now known as Planck’s constant (h = 6.625 x 10-34 J-s).
2.1.2 Wave–Particle Duality
The hypothesis of de Broglie was the existence of a wave– particle
duality principle.
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The existence of a peak in the density of scattered electrons can be
explained as a constructive interference of waves scattered by the
periodic atoms in the planes of the nickel crystal.
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2.1.3 The Uncertainty Principle
• The first statement of the uncertainty principle is that it is
impossible to simultaneously describe with absolute
accuracy the position and momentum of a particle.
• The second statement of the uncertainty principle is
that it is impossible to simultaneously describe with
absolute accuracy the energy of a particle and the instant of time the particle has this energy.
One consequence of the uncertainty principle is that
we cannot, for example, determine the exact position
of an electron.
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2.2 | SCHRODINGER’S WAVE EQUATION
2.2.1 The Wave Equation
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2.2.2 Physical Meaning of the Wave Function
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2.2.3 Boundary Conditions
In Figure 2.5a, the potential
function is finite everywhere.
The wave function as well as
its first derivative is
continuous.
In Figure 2.5b, the potential
function is infinite for x < 0 and
for x > a. The wave function is
continuous at the boundaries,
but the first derivative is
discontinuous.
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2.3 | APPLICATIONS OF SCHRODINGER’S WAVE EQUATION
2.3.1 Electron in Free Space
The first term, with the coefficient A, is a wave traveling in the +x direction, while
the second term, with the coefficient B, is a wave traveling in the -x direction.
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2.3.2 The Infinite Potential Well
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The free electron was represented by a traveling wave, and
now the bound particle is represented by a standing wave.
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the energy of the particle is quantized. That is, the energy of
the particle can only have particular discrete values.
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2.3.3 The Step Potential Function
• assume that a flux of particles is
incident on the potential barrier.
• A particularly interesting result is
obtained for the case when the
total energy of the particle is
less than the barrier height, or E
< V0.
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• A1 A1* is the probability density function of the incident
particles
• vi A1 A1* is the flux of incident particles in units of #/cm2-s.
• vr B1 B1* is the flux of the reflected particles,
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there is a finite probability that the incident particle
will penetrate the potential barrier and exist in
region II.
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2.3.4 The Potential Barrier and Tunneling
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there is a finite probability that a particle impinging a
potential barrier will penetrate the barrier and will appear
in region III. This phenomenon is called tunneling
• Comment of EXAMPLE 2.5
The tunneling probability may appear to be a small value,
but the value is not zero. If a large number of particles
impinge on a potential barrier, a significant number can
penetrate the barrier.
•
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2.4 | EXTENSIONS OF THE WAVE THEORY TO ATOMS
2.4.1 The One-Electron Atom
where again n is the principal quantum number. The negative
energy indicates that the electron is bound to the nucleus and
we again see that the energy of the bound electron is quantized.
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2.4.2 The Periodic Table
• The first concept needed is that of electron spin.
• The second concept needed is the Pauli exclusion principle.
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