Theoretical understanding and properties of individual properties of nanoparticles - quantum well states detailed study

1. QUANTUM MECHANICS OF LOW
DIMENSIONAL SYSTEMS
QUANTUM WELL STATES

3. QUANTUM WELL
A quantum well is a potential well that confines particles, which were originally free to
move in three dimensions, to two dimensions, forcing them to occupy a planar region. The
effects of quantum confinement take place when the quantum well thickness becomes
comparable at the de Broglie wavelength of the carriers (generally electrons and holes),
leading to energy levels called "energy sub-bands", i.e., the carriers can only have
discrete energy values.
The conduction electrons would be delocalized in the plane of the plate, but confined in the
narrow dimension, a configuration referred to as a quantum well

4. Preparation of quantum nanostructure
• Bottom-up method
• Top-down method (lithography)

5. Properties
Size : Large percentage of atoms on the surface for small n is one of the principal factors
that differentiates properties of nanostructures from bulk material.
𝑁𝑠 = 12 𝑛2
𝑁𝑇 = 8𝑛3 + 6𝑛2 + 3𝑛
𝑑 = 𝑛𝑎 = 0.565 𝑛

8. Standard quantum-mechanical texts show that for an infinitely deep square potential well of width a in
one dimension, the coordinate 𝑥 has the range of values −
1
2
𝑎 ≤ 𝑥 ≤
1
2
𝑎 inside the well, and the energies
there are given by the expressions
𝐸 =
𝜋2
ℎ2
2𝑚𝑎2 𝑛2
=𝐸0𝑛2
which are plotted in Fig. 9.1 1, where 𝐸0=
𝜋2ℎ2
2𝑚𝑎2 is the ground-state energy and the quantum number 𝑛
assumes the values 𝑛 = 1,2,3, . . . . The electrons that are present fill up the energy levels starting
from the bottom, until all available electrons are in place. An infinite square well has an infinite number
of energy levels, with ever-widening spacing as the quantum number 𝑛 increases. If the well is finite,
then its quantized energies 𝐸 , all lie below the corresponding infinite well energies, and there are only
a limited number of them. Figure 9.12 illustrates the case for a finite well of potential depth 𝑉0 = 7𝐸0
which has only three allowed energies. No matter how shallow the well, there is always at least one
bound state 𝐸.

11. The electrons confined to the potential well move back and forth along the direction 𝑥,
and the probability of finding an electron at a particular value of 𝑥 is given by the square
of the wave function 𝜓𝑛 𝑥 2 for the particular level 𝑛 where the electron is located.
There are even and odd wave functions 𝜓𝑛 𝑥 that alternate for the levels in the one-
dimensional square well, and for the infinite square well we have the un normalized
expressions
𝜓𝑛=cos
𝑛𝜋𝑥
𝑎
𝑛=1,3,5, … … even parity
𝜓𝑛 = sin
𝑛𝜋𝑥
𝑎
𝑛 = 2,4,6 … … odd parity
These wave functions are sketched in Fig. 9.1 1 for the infinite well. The property called
parity is defined as even when 𝜓𝑛 −𝑥 = 𝜓𝑛(𝑥), and it is odd when 𝜓𝑛 −𝑥 = −𝜓𝑛(𝑥)