Formula and Method for the Shockley Diode Equation
Shockley Diode Equation:
I = I_S × (e^(V / (n × V_T)) − 1)
I_S = reverse saturation current; V = voltage across the diode; n = ideality factor; V_T = thermal voltage = kT/q ≈ 25.7mV at 25°C
The Shockley diode equation (also called the ideal diode law) describes how current flows through a p-n junction diode as a function of the voltage across it. Enter the diode's reverse saturation current, the voltage across the junction, the ideality factor, and the operating temperature, and this calculator returns the resulting current, the thermal voltage, the power dissipated in the diode, and its small-signal (dynamic) resistance at that operating point.
Reading the terms in the equation
- Saturation current (IS): the tiny reverse-bias leakage current a real diode would carry if V were very negative. It depends on the semiconductor material, junction area, and doping, and roughly doubles for every 10°C rise in temperature. Typical values range from about 1 pA to 1 nA for small-signal silicon diodes.
- Thermal voltage (VT): V_T = kT/q, where k is Boltzmann's constant (1.380649×10⁻²³ J/K), q is the elementary charge (1.602176634×10⁻¹⁹ C), and T is the absolute temperature in kelvin. At 25°C (298.15 K), V_T ≈ 25.7 mV — the widely used "26 mV at room temperature" figure comes from rounding to 300 K.
- Ideality factor (n): a dimensionless correction, typically between 1 and 2, that accounts for how closely the junction follows ideal diffusion behavior. n = 1 for a pure diffusion-current diode; n approaches 2 when recombination current in the depletion region becomes significant, which is common in real silicon diodes at low forward current.
Forward bias, reverse bias, and dynamic resistance
- Forward bias (V > 0): once V exceeds a few times n·V_T, the "−1" term becomes negligible and current grows exponentially with voltage — this is why diodes appear to have a fairly sharp "turn-on" voltage in practice, even though the underlying curve is smooth.
- Reverse bias (V < 0): the exponential term collapses toward zero and the current saturates at approximately −I_S — this is the origin of the equation's name.
- Dynamic resistance (rd): found by differentiating the equation with respect to V, r_d = n·V_T / (I + I_S). It is the small-signal AC resistance the diode presents at a given DC operating point, used when modeling a diode as a resistor for small-signal circuit analysis.
Frequently Asked Questions
What is the Shockley diode equation?
The Shockley diode equation, I = I_S × (e^(V / (n·V_T)) − 1), models the current through an ideal p-n junction diode. I_S is the reverse saturation current, V is the voltage across the diode, n is the ideality factor, and V_T is the thermal voltage (kT/q).
What is thermal voltage and how is it calculated?
Thermal voltage V_T = kT/q, where k is Boltzmann's constant (1.380649×10⁻²³ J/K), T is the absolute temperature in kelvin, and q is the elementary charge (1.602176634×10⁻¹⁹ C). At room temperature (25°C / 298.15 K), V_T ≈ 25.7 mV, which is why 25-26 mV is commonly used as a rule of thumb.
What does the ideality factor (n) represent?
The ideality factor describes how closely a real diode follows the ideal diffusion-current model. n ≈ 1 when diffusion current dominates, and n approaches 2 when recombination current in the depletion region dominates, which is common in silicon diodes at low forward current.
How is the diode's dynamic (small-signal) resistance found?
Dynamic resistance is the derivative of the I-V curve at the operating point: r_d = n·V_T / (I + I_S). It approximates how much the diode's voltage changes for a small change in current, and is commonly used to model a diode as a resistor in small-signal AC circuit analysis.