Reverse Applied Bias & Capacitance
A Reverse Bias (\( V_R \)) is applied when a positive potential relative to the p-region is connected to the n-region. Because the electric fields in the neutral p and n regions are essentially zero, nearly all of the applied voltage drops across the space charge (depletion) region.
This external bias shifts the Fermi levels, increasing the total potential barrier across the junction.
\( V_{total} \) = total potential barrier
\( \phi_{Fn}, \phi_{Fp} \) = position of Fermi levels relative to intrinsic Fermi level
\( V_R \) = magnitude of reverse applied voltage
\( V_{bi} \) = built-in potential barrier
Effects of Reverse Bias
- Depletion Width Increases: To uncover more fixed ionic charge and support the higher voltage, the space charge width (\( W \)) expands.
- Electric Field Increases: The magnitude of the maximum electric field (\( E_{max} \)) at the metallurgical junction grows significantly.
Space Charge Width
\( W \) = space charge (depletion) width
\( \varepsilon_s \) = permittivity of the semiconductor
\( e \) = elementary charge (magnitude of electron charge)
\( N_a, N_d \) = acceptor and donor doping concentrations
Maximum Electric Field
\( E_{max} \) = maximum electric field occurring at the metallurgical junction (\( x = 0 \)).
Junction Capacitance (\( C' \))
As charges are separated across the space charge region, it acts like a parallel plate capacitor. A small change in applied voltage (\( dV_R \)) causes a small change in the uncompensated charge (\( dQ' \)) at the depletion edges.
\( C' \) = junction (or depletion) capacitance per unit area (F/cm²)
\( dQ' \) = differential change in charge density per unit area
\( dV_R \) = differential change in reverse bias voltage
Interactive Biasing Simulator
Adjust the applied voltage (\( V_a \)) to observe band bending and depletion width changes.
Junction Breakdown Mechanisms
A reverse bias voltage cannot be increased indefinitely. At a critical reverse voltage, the reverse-biased current increases extremely rapidly, signaling the onset of junction breakdown. There are two primary physical mechanisms for this phenomenon.
Zener Effect
Occurs primarily in highly doped pn junctions. High doping results in a very narrow depletion region. A moderate reverse bias creates a massive electric field across this thin barrier, allowing valence electrons on the p-side to quantum-mechanically tunnel directly into the empty conduction band states on the n-side.
Avalanche Effect
The predominant mechanism in most moderately doped pn junctions. Carriers crossing the space charge region are violently accelerated by the high electric field. If they gain sufficient kinetic energy, they can collide with host atoms, breaking bonds and creating new electron-hole pairs (Impact Ionization), which then accelerate and cause a runaway chain reaction.
Forward Bias & Carrier Injection
"Under forward bias, the massive floodgates holding back the majority carriers are lowered. Electrons spill into the p-side and holes spill into the n-side, drastically altering the landscape of minority carrier populations."
Applying a positive potential to the p-region relative to the n-region establishes a Forward Bias (\( V_a \)). The external electric field opposes the built-in field, reducing the net potential barrier to \( V_{bi} - V_a \).
The Boundary Conditions
Because the barrier is lowered, majority carriers diffuse easily across the junction. This injection means that the minority carrier concentration exactly at the edges of the space charge region deviates exponentially from its thermal-equilibrium value:
\( p_n(x_n), n_p(-x_p) \) = minority carrier concentrations at depletion edges
\( p_{n0}, n_{p0} \) = thermal-equilibrium minority carrier concentrations
\( V_a \) = forward applied bias voltage
\( k \) = Boltzmann's constant
\( T \) = absolute temperature (K)
Ideal I-V Assumptions
- The abrupt depletion layer approximation applies.
- Maxwell-Boltzmann statistics apply to the carriers.
- Low injection applies (majority carrier concentration doesn't change significantly).
- Total current is constant throughout the entire device.
- Individual electron and hole currents are constant across the narrow depletion region (no recombination inside the space charge region).
Minority Carrier Distribution
Once excess minority carriers are injected across the space charge boundaries, how deep do they penetrate into the neutral bulk regions?
Assuming zero electric field in the neutral regions and steady-state conditions, the ambipolar transport equation simplifies to a pure diffusion equation (shown here for holes on the n-side):
\( D_p \) = hole diffusion coefficient
\( \delta p_n \) = excess minority hole concentration
\( \tau_{p0} \) = minority hole lifetime
The solution yields an exponential decay. They recombine with the abundant majority carriers as they diffuse away from the junction.
\( L_p = \sqrt{D_p \tau_{p0}} \) = minority hole diffusion length
Minority Carrier Profile Simulator
Adjust Forward Bias to exponentially increase the injection levels at the junction edges.
The Ideal I-V Relationship
The total current flowing through the PN junction is the sum of the electron and hole diffusion currents evaluated at the edges of the space charge region.
The Shockley Diode Equation
By taking the derivative of the carrier distribution equations to find diffusion current (\( J = -eD \frac{dp}{dx} \)), we arrive at the foundational ideal current-voltage characteristic equation:
\( J \) = total current density
\( J_s \) = reverse saturation current density
Reverse Saturation Current Density (\( J_s \)):
\[ J_s = \frac{e D_p p_{n0}}{L_p} + \frac{e D_n n_{p0}}{L_n} \]
\( D_n, D_p \) = electron/hole diffusion coefficients
\( L_n, L_p \) = electron/hole minority carrier diffusion lengths
Curve Parameters
Adjust the physical properties to see their effect on the diode characteristics.
Hover your mouse over the graph to track the precise exponential growth of the current density.
Lecture 8 Knowledge Check
Put your understanding of Biased PN Junctions, Capacitance, Breakdown Mechanisms, and Carrier Injection to the test.