Linearly Graded PN-Junction Calculator
Calculate the space charge (depletion) width of a linearly graded PN junction.
Formula: \( W = \left( \frac{4 \varepsilon_s (V_{bi} – V)}{q \, a} \right)^{\frac{1}{3}} \)
Default for Silicon: 11.7ε₀ ≈ 1.035e-10 F/m
Enter the doping gradient (dN/dx) in m-4.
Space-Charge Width of a Linearly Graded PN Junction
In semiconductor physics, the space-charge region (or depletion region) in a PN junction is the zone around the interface where free carriers are depleted, leaving behind charged donor and acceptor ions. For a linearly graded PN junction, the doping concentration changes linearly with position, rather than being abrupt or uniform on each side. This graded doping profile influences how wide the depletion region extends and how it responds to applied bias.
Linearly Graded Doping Profile
In a linearly graded junction, the doping concentration $N(x)$ varies linearly with distance from the junction. Often, we approximate this as:
Here, $\alpha$ (alpha) is the doping gradient (units often $\text{cm}^{-4}$ or similar), and $N(0)$ is the doping concentration at the metallurgical junction.
Because of this continuously varying doping concentration, the usual depletion approximation changes compared to a step-junction (where doping abruptly changes from $N_A$ to $N_D$). For linearly graded profiles, the space-charge region width $W$ typically depends on the cube root of the applied voltage, rather than the square root (as in the abrupt junction case).
Space-Charge Width Formula
For a one-sided linearly graded junction under the depletion approximation, a common expression for the depletion width $W$ is:
- $\varepsilon_s$: the semiconductor permittivity $(\varepsilon_s = \varepsilon_r \varepsilon_0)$
- $q$: the electronic charge $\approx 1.602 \times 10^{-19}\,\text{C}$
- $\alpha$: the doping concentration gradient
- $V_0$: the built-in potential of the junction
- $V_{\text{bias}}$: the externally applied voltage (forward or reverse)
Notice that the exponent is $1/3$. In contrast, for an abrupt (step) junction the width goes like $\sqrt{V}$. This difference arises from how the doping profile changes across the junction.
Example: Depletion Width Calculation
Suppose a linearly graded PN junction has:
- $\alpha = 1 \times 10^{21}\,\text{cm}^{-4}$
- $\varepsilon_s \approx 1.05 \times 10^{-12}\,\text{F/cm}$ (typical for silicon)
- Built-in potential $V_0 = 0.7\,\text{V}$
- Reverse bias $V_{\text{bias}} = 5\,\text{V}$
The total voltage across the depletion region is $(V_0 + V_{\text{bias}}) = 0.7 + 5 = 5.7\,\text{V}$.
Using the formula:
To estimate roughly (with approximate unit conversions), let’s consider:
Numerator in the bracket:
- $2 \times 1.05 \times 10^{-12} \times 5.7 \approx 1.2 \times 10^{-11}$
Denominator in the bracket:
- $(1.6 \times 10^{-19}) \times (1 \times 10^{21}) = 1.6 \times 10^2 = 160$
So inside the brackets we have approximately:
Taking the cube root:
This yields a fraction of a micron (on the order of a few hundred nanometers). The exact arithmetic may vary, but the key point is that $W$ depends on $(V_0 + V_{\text{bias}})^{1/3}$.
- In a linearly graded junction, doping varies continuously instead of abruptly.
- The space-charge width scales with $\bigl(V_0 + V_{\text{bias}}\bigr)^{1/3}$, different from the square-root dependence in an abrupt junction.
- Accurate numeric results require consistent units and careful handling of $\alpha$, $\varepsilon_s$, $V_0$, and $V_{\text{bias}}$.
- Linearly graded junctions often appear in epitaxial processes or specialized semiconductor designs.
Understanding the space-charge region width in a linearly graded PN junction is crucial for predicting junction capacitance, breakdown characteristics, and other electrical behaviors in semiconductor devices. Although less common than abrupt or step junctions, linearly graded junctions demand this distinctive cubic-root relationship with voltage.