Module 5 part A1
Overview of Semiconductor Current Generation
In this discussion, we focus on the generation and transport of current within semiconductors, highlighting the flow of charge carriers, specifically electrons and holes. We previously covered concepts such as the density of states, the distribution of charge carriers within energy bands, and the calculation of carrier concentrations in the conduction and valence bands. Today, we delve deeper into how current is generated within semiconductors and the mechanisms that influence this process.
Charge Carrier Movement and Transport Phenomena
The movement of charge carriers (like holes) within a semiconductor is termed transport. When examining this movement, it is important to consider the random nature of carrier movement due to thermal motion and Coulomb interactions with other charged particles and ions in the semiconductor. Initially, the movement of electrons and holes occurs randomly, resulting in no net displacement; carriers move both left and right without any overall direction.
Application of Electric Field
When an external electric field is applied, however, the behavior of the charge carriers changes significantly. The electric field exerts a force on the carriers: holes, which are positive, move in the direction of the electric field, while electrons move in the opposite direction. This force causes the carriers to experience a net motion in one direction, resulting in a phenomenon known as drift. The carriers acquire a drift velocity that adds to their thermal velocity, whereby drift velocity signifies the directed motion caused by the electric field.
Mean Free Path and Scattering
As charge carriers continue to move, they encounter scattering centers that interrupt their paths, causing them to lose momentum and energy. The mean time between these scattering events is referred to as the mean scattering time, while the average distance traveled by a carrier between collisions is known as the mean free path. The average thermal velocity of charge carriers can be derived from thermal energy, specifically expressed as:
[ v_{th} = rac{1}{2} m^{*} v_{th}^{2} ]
Where (k) is the Boltzmann constant and (T) is the absolute temperature. At room temperature, the average thermal velocity of carriers is approximately (10^5 m/s) or (10^7 cm/s).
Impact of Electric Field on Drift Velocity
In a semiconductor exposed to an electric field, charge carriers experience forces that depend on their effective mass (the mass that characterizes their response to applied forces). For holes, the force due to the electric field causes them to accelerate in the same direction as the field, while electrons accelerate in the opposite direction. The differing effective masses of electrons and holes result in varying accelerations and hence velocities in response to the same force.
Drift Velocity and Mobility
The expression for drift velocity can be modeled as: [ v_d = rac{e au}{m^{}} E ] Where the mobility (\mu = \frac{e \tau}{m^{}}) reflects how quickly a carrier can move through a semiconductor under an electric field. Here, (\tau) refers to the mean scattering time, and (E) is the electric field strength. The mobility of electrons and holes varies, influencing the conductance properties of the semiconductor.
Drift Current and Current Density
The movement of charge carriers due to drift results in a net current flow, termed drift current. Current density, which is the current per unit cross-sectional area, is a crucial determinant of the performance of semiconductor devices. We define current density (J) as: [ J = \rho v_d = q n v_d ] Where (q) represents the charge, (n) is the carrier concentration, and (v_d) is the drift velocity.
Calculating Charge Carrier Density
If we denote the concentration of electrons as (n) and the concentration of holes as (p), the total volume charge density (\rho) can be derived as: [ \rho = -e n + e p ] This relationship turns into expressions for drift current due to holes and electrons:
For Holes: (J_p = e p v_d = e p \mu_p E)
For Electrons: (J_n = -e n v_d = -e n \mu_n E)
Here, (\mu_p) and (\mu_n) represent the mobilities of holes and electrons respectively.
Total Current in Semiconductors
The total current is therefore the sum of both electron and hole currents: [ J = J_p + J_n = e p \mu_p E - e n \mu_n E ]This can also be expressed in terms of conductivity (\sigma), as described by: [ J = \sigma E ] Where conductivity is influenced by the concentrations of both carriers and their respective mobilities. Conductivity has units of (\text{(Ohm cm)}^{-1}). The resistivity, which is the inverse of conductivity, also provides relevant insight into how easily current flows through a semiconductor material under varying conditions.