Numericals on Diode Current

Diode Current-Voltage Relation:

The current-voltage relationship in a diode can be derived from semiconductor physics principles, specifically considering the behavior of charge carriers (electrons and holes) in a semiconductor junction. Here, I'll outline the derivation of the diode current equation, commonly known as the Shockley diode equation.

Assumptions:

1. Quasi-Fermi Levels: The derivation assumes that the electron and hole densities remain close to their equilibrium values, and therefore the quasi-Fermi levels for electrons and holes do not significantly change with applied voltage.

2. Low-Level Injection: The derivation assumes low-level injection, meaning that the majority carrier density (electrons in an n-type and holes in a p-type semiconductor) is much greater than the minority carrier density (holes in an n-type and electrons in a p-type semiconductor).

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1. 1. Electric Field and Drift Current: When a voltage is applied across the diode, it creates an electric field in the depletion region. This field accelerates the majority carriers (electrons in the n-type and holes in the p-type) towards the respective terminals, constituting a drift current.

2. 2. Diffusion Current: Due to the concentration gradient of minority carriers in the depletion region, a diffusion current arises.

3. 3. Total Current: The total current in the diode is the sum of the drift current and the diffusion current.

4. 4. Continuity Equation: By applying the continuity equation for the carrier densities (rate of change of carrier density equals the difference between generation and recombination rates), we can relate the electron and hole currents to each other.

5. 5. Einstein Relation: The diffusion current density is related to the carrier concentration gradient through the Einstein relation, which relates the diffusion coefficient to the mobility and thermal voltage.

6. 6. Ideal Diode Equation: Combining the expressions for drift and diffusion currents, along with the relationship between carrier densities and the applied voltage, leads to the diode equation.

Result:

The final Shockley diode equation is given by:

$�={�}_{�}({�}^{\frac{�}{�{�}_{�}}}-1)$

Where:

• $�$ is the diode current,
• ${�}_{�}$ is the reverse saturation current (leakage current),
• $�$ is the voltage across the diode,
• $�$ is the ideality factor (typically between 1 and 2),
• ${�}_{�}$ is the thermal voltage, approximately $\frac{��}{�}$, where $�$ is Boltzmann's constant, $�$ is the temperature in Kelvin, and $�$ is the elementary charge.

This equation describes the exponential relationship between the diode current and voltage, capturing both the exponential increase in current with forward voltage and the reverse saturation current behavior.