Shockley Diode Equation

The Shockley diode equation is the master equation of every semiconductor p-n junction. Every SPICE simulation of a 1N4007 rectifier, every solar cell model, and every BJT analysis starts from this one formula.

Core Concept

William Shockley derived this equation in 1949 to describe current flow in a p-n junction by solving the diffusion equations for minority carriers. The equation says: current is proportional to the exponential of applied voltage divided by the thermal voltage. Without any applied voltage, the two exponential terms cancel and net current is zero, as expected at equilibrium.

The term thermal voltage VT equals kT/q, where k = 1.38 × 10⁻²³ J/K (Boltzmann constant) and q = 1.6 × 10⁻¹⁹ C. At T = 300K (room temperature), VT = 26 mV. This quantity appears constantly in semiconductor device physics and BJT small-signal analysis.

The ideality factor n accounts for recombination in the depletion region. For an ideal junction with only diffusion current, n = 1. Real silicon diodes have n between 1 and 2 because recombination inside the depletion region adds extra current. LEDs typically have n closer to 2. Schottky diodes like the BAT43 have n close to 1 and switch faster as a result.

Key Equations

Shockley diode equation: I = IS × (e^(V / nVT) - 1)

Thermal voltage: VT = kT / q At 300K: VT = (1.38×10⁻²³ × 300) / 1.6×10⁻¹⁹ = 25.875 mV ≈ 26 mV.

Reverse saturation current: IS = A × q × (Dp×pn0/Lp + Dn×np0/Ln) where A = junction area, pn0 and np0 are minority carrier equilibrium concentrations, Lp and Ln are diffusion lengths.

For large forward bias (V >> VT): I ≈ IS × e^(V / nVT) The -1 term is negligible.

Dynamic (small-signal) resistance: rd = dV/dI = nVT / I At I = 26 mA and n = 1: rd = 26 mV / 26 mA = 1Ω.

Effect of temperature on IS: IS doubles for every 10°C rise (silicon) and VT increases by 0.086 mV per kelvin.

Given:
  IS = 2 × 10⁻¹⁴ A
  n = 1
  VT = 26 mV = 0.026 V
  Forward voltage V = 0.6 V
  Task: find forward current, then find rd at this current

Why this formula:
  The Shockley equation is the exact model.
  rd gives the small-signal resistance at the operating point.

Formula:
  I = IS × (e^(V/nVT) - 1)
  rd = nVT / I

Substitution:
  I = 2×10⁻¹⁴ × (e^(0.6/0.026) - 1)

Calculation:
  Exponent = 0.6 / 0.026 = 23.077
  e^23.077 = e^23 × e^0.077
           = 9.74×10⁹ × 1.080
           = 1.052 × 10¹⁰
  I ≈ 2×10⁻¹⁴ × 1.052×10¹⁰
    = 2.104 × 10⁻⁴ A = 0.21 mA

  rd = 0.026 / (2.104×10⁻⁴)
     = 123.6 Ω

Final Answer:
  Forward current I = 0.21 mA at V = 0.6V
  Dynamic resistance rd = 123.6 Ω at this operating point

Exam Tip: GATE asks three things about the Shockley equation. First, the value of VT at 300K (26 mV or 25 mV; use 26 mV unless told otherwise). Second, the dynamic resistance formula rd = nVT/ID; students forget the n factor and use just VT/ID. Third, IS doubles every 10°C for silicon; a 30°C rise multiplies IS by 8 (2³). Also note: for V >> VT, write I ≈ IS×e^(V/nVT) and drop the -1 to simplify calculations.

Key Properties

• VT = kT/q = 26 mV at 300K (room temperature). It increases linearly with absolute temperature.
• IS is extremely small (10⁻¹⁵ to 10⁻¹⁰ A). A larger junction area gives a larger IS.
• Ideality factor n = 1 for diffusion-dominated current (ideal). n = 2 for recombination-dominated current (common in real Si diodes and LEDs at low current).
• Dynamic resistance rd = nVT/I decreases as current increases. At 26 mA with n=1, rd = 1Ω.
• IS doubles every 10°C for silicon, meaning reverse leakage grows significantly at high temperatures.
• For V << 0 (reverse bias), I ≈ -IS. The diode acts as a current source of magnitude IS.
• The Schottky diode BAT43 has n ≈ 1.04 and VF ≈ 0.3V, making it faster and lower-drop than the 1N4148 silicon diode.

Quick Revision

• I = IS × (e^(V/nVT) - 1) is the Shockley diode equation.
• VT = kT/q = 26 mV at 300K.
• n = 1 for ideal diffusion; n = 2 for recombination-dominated (LEDs, low-current Si).
• Dynamic resistance: rd = nVT / ID.
• IS doubles every 10°C (Si); a 30°C rise multiplies IS by 8.
• For V >> VT: drop the -1 and write I ≈ IS×e^(V/nVT).
• 60 mV increase in V at n=1 increases I by a factor of 10 at 300K.
• Exam trap: forgetting to include n in the dynamic resistance formula. Using rd = VT/ID instead of rd = nVT/ID gives the wrong answer when n = 2 (LEDs or low-current Si diodes).