Module 3 part C1

In quantum mechanics, electrons occupy specific energy states in systems like semiconductors.
Example: An electron confined in a two-dimensional potential well demonstrates quantization of energy levels.
Current flow in conductors is due to the movement of electrons, while semiconductors utilize both electrons and holes as charge carriers.

Two types of charge carriers generating current:
- Electrons: Carry negative charge.
- Holes: Represent the absence of an electron and carry a positive charge.
The current in semiconductors depends on the quantity of these carriers, necessitating knowledge of their numbers to understand current performance under applied voltage.

Electrons and holes cannot occupy arbitrary locations in semiconductors; they fill certain discrete energy states.
These states aggregate into energy bands, encompassing many quantum states.
The number of available energy states per band directly influences the number of electrons the conduction band can accommodate.
Pauli Exclusion Principle: States that no two electrons can occupy the same quantum state simultaneously, making the density of states critical for understanding electron distribution.

In one-dimensional potential wells, energy states are quantized, influenced by an integer quantum number (n).
The quantized energy levels correspond to the confinement of electrons in parts of a semiconductor known as potential wells, leading to unique electronic properties.

Definition: An extension of the one-dimensional treatment to three dimensions involves calculating energy states within a cubic box (potential well).
The potential is characterized as infinite outside the box and zero within the dimensions (x,y,z).
Schrodinger Equation: Solutions for three dimensions involve three principal quantum numbers (n_x, n_y, n_z), each of which is a positive integer.

K-Space: The vector space used to describe wave functions and energy states in quantum mechanics.
Energy states are organized into a spherical structure in k-space where the distance between k-states in three dimensions is characterized as
- Volume of a single quantum state = (( \frac{\pi}{a} ))^3.
By determining the volume of the sphere in k-space (4/3 (\pi k^3)), the density of states can be calculated, accounting for spin states.
Density of quantum states in k space is expressed as a relation involving k, H, and system volume, providing insight into the available energy states in a given energy band range.

Relation between k and electron energy allows transformation of density of states from k space to real energy space.
Calculation of energy density states involves the integral of energy range and volume considerations, resulting in the total density of states per unit energy and volume.
General form obtained demonstrates density of states within conduction and valence bands.

Effective mass concepts (m* sub n for electrons and m* sub p for holes) modify calculations within the energy bands to account for interactions within the crystal lattice compared to free electron models.
The practical implications highlight changes to density states as structures scale down in size.

Generating energy states varies as systems transition from three-dimensional to two-dimensional and one-dimensional structures.
2D Systems: Characterized by energy levels that represent constant, stepped functions based on quantum state occupancy.
1D Systems: Display discrete energy levels resembling delta functions, with rapid changes in density states indicating sharp energy states.

Advances in nanotechnology necessitate an understanding of energy states in consistently smaller devices (e.g., nanowires, thin films).
New insights into 2D and 1D systems inform future semiconductor technology, particularly for CMOS designs that operate at the nano-scale.