This chapter describes a number of models to understand the implications
of quantized energy levels and wavefunctions that describe the behavior
of small particles, such as electrons, in nanoscale regions of space. We will
discover that these models allow us to make powerful predictions about
the properties of simple nanosystems. As we will discover in later chapters,
the widespread application of quantization has important implications in
how energy is stored and distributed in nanosystems. It also provides the
tools by which we can use light to probe particles and molecular systems.
We begin by providing an introduction to quantum theory and the fundamental equation of quantum mechanics (the Schrödinger equation).
The application of this equation will allow us solve for energy in particles
confined to nanoscale regions of space.
4.1 QUANTUM CONFINEMENT
IN NANOMATERIALS
We begin by describing the significance of placing geometrical constraints
on materials to create nanosystems of different dimensionalities. We will
then explain these consequences using simple quantum mechanical
models in the proceeding sections. Quantum confinement describes the
changes that occur in atomic structure due to very small length scales in
particles, usually on the order of nanometers. The changes occur because
electrons are trapped in regions in which they interact with the boundaries
of the system. Quantum-confined structures can be classified into three
different types as determined by the number of dimensions restricting the
motion of electrons. Macroscopic bulk materials are not confined at all,
with three free dimensions. However, a thin two-dimensional film has two
free dimensions (say, on the yz plane) and motion is entirely confined in
the x-direction. These two-dimensional planes are sometimes referred to
as quantum well superlattices. A quantum wire has only one free dimension. An electron on this wire, for example, may be free to move along the
x-direction. The electron is not allowed to move along the other two axes.
Restricting motion in all three dimensions gives us a confined structure
similar to an individual atom, where the electron is essentially confined
to a “dot.” Quantum dots fall into this category. One consequence of
quantum confinement is that the optical and electronic properties of these
materials differ substantially from the bulk phase. This is because the
discrete electronic energy levels of an electron in a confined material are
size-dependent. We now describe in detail the origin of this quantum size
effect.
CHAPTER 4: Quantum Effects at the Nanoscale
96
of quantized energy levels and wavefunctions that describe the behavior
of small particles, such as electrons, in nanoscale regions of space. We will
discover that these models allow us to make powerful predictions about
the properties of simple nanosystems. As we will discover in later chapters,
the widespread application of quantization has important implications in
how energy is stored and distributed in nanosystems. It also provides the
tools by which we can use light to probe particles and molecular systems.
We begin by providing an introduction to quantum theory and the fundamental equation of quantum mechanics (the Schrödinger equation).
The application of this equation will allow us solve for energy in particles
confined to nanoscale regions of space.
4.1 QUANTUM CONFINEMENT
IN NANOMATERIALS
We begin by describing the significance of placing geometrical constraints
on materials to create nanosystems of different dimensionalities. We will
then explain these consequences using simple quantum mechanical
models in the proceeding sections. Quantum confinement describes the
changes that occur in atomic structure due to very small length scales in
particles, usually on the order of nanometers. The changes occur because
electrons are trapped in regions in which they interact with the boundaries
of the system. Quantum-confined structures can be classified into three
different types as determined by the number of dimensions restricting the
motion of electrons. Macroscopic bulk materials are not confined at all,
with three free dimensions. However, a thin two-dimensional film has two
free dimensions (say, on the yz plane) and motion is entirely confined in
the x-direction. These two-dimensional planes are sometimes referred to
as quantum well superlattices. A quantum wire has only one free dimension. An electron on this wire, for example, may be free to move along the
x-direction. The electron is not allowed to move along the other two axes.
Restricting motion in all three dimensions gives us a confined structure
similar to an individual atom, where the electron is essentially confined
to a “dot.” Quantum dots fall into this category. One consequence of
quantum confinement is that the optical and electronic properties of these
materials differ substantially from the bulk phase. This is because the
discrete electronic energy levels of an electron in a confined material are
size-dependent. We now describe in detail the origin of this quantum size
effect.
CHAPTER 4: Quantum Effects at the Nanoscale
96