Diode I-V Characteristic: Shockley Equation

The diode is the simplest nonlinear circuit element, but its exponential I-V law shows up everywhere else too: BJT base-emitter junctions, LED drivers, rectifier loss calculations, temperature sensors, and even MOSFET body diodes. Understanding the Shockley equation quantitatively — not just "current flows one way" — is what lets you compute forward voltage, estimate power dissipation, and predict how a diode's operating point shifts with temperature and current.

The Shockley Diode Equation

The ideal diode current-voltage relationship is:

I = I_S · (exp(V / (n·V_T)) − 1)

Where:

• I — diode current (A), positive in the forward direction
• I_S — reverse saturation current (A), typically 1 pA–1 nA for small silicon diodes; strongly temperature-dependent
• V — voltage across the junction (V), positive when forward biased
• n — emission (ideality) coefficient, ≈1 for an ideal junction, 1.5–2 for real diodes at low current where recombination current dominates
• V_T — thermal voltage, V_T = kT/q ≈ 25.85 mV at T = 300 K (often rounded to 25 mV or 26 mV)

At room temperature, V_T ≈ 0.02585 V, and for n = 1 this gives the well-known rule that current increases roughly 10× for every ~60 mV of forward voltage increase (since ln(10)·V_T ≈ 59.5 mV).

Reading the Three Regions

The "−1" term matters only near V = 0 and in reverse bias — in forward conduction (V > ~100 mV), exp(V/nV_T) ≫ 1 and it's dropped without meaningful error.

Worked Example: Finding Forward Voltage at a Given Current

A silicon diode has I_S = 5 nA, n = 1, at T = 300 K (V_T = 25.85 mV). Find the forward voltage that produces I = 10 mA.

Step 1 — Solve for V:

I = I_S · exp(V/(n·V_T))
V = n·V_T · ln(I / I_S)
V = 1 · 0.02585 · ln(0.01 / 5e-9)
V = 0.02585 · ln(2×10^6)
V = 0.02585 · 14.51
V ≈ 0.375 V

Step 2 — Verify by plugging back in:

I = 5×10⁻⁹ · exp(0.375 / 0.02585)
  = 5×10⁻⁹ · exp(14.50)
  = 5×10⁻⁹ · 1.983×10⁶
  ≈ 9.91×10⁻³ A ≈ 9.9 mA

Close to the target 10 mA (small rounding from truncating the exponent) — the result checks out. Note this 0.375 V is lower than the "0.6–0.7 V" rule of thumb often quoted for silicon; that rule assumes a larger I_S (roughly 10⁻¹²–10⁻¹⁴ A) typical of small-signal diodes at datasheet test currents. I_S varies by orders of magnitude between diode types, which is why forward voltage at a given current differs so much between a small-signal diode, a power rectifier, and a Schottky diode.

Temperature Dependence

Two competing temperature effects act on the diode:

• V_T increases with T (directly proportional): higher T → exponential term more gradual.
• I_S increases roughly exponentially with T (doubling every ~5–10°C for silicon, driven by the bandgap-dependent intrinsic carrier concentration).

The I_S effect dominates. Net result: at constant current, forward voltage decreases with rising temperature, typically by about −2 mV/°C for silicon diodes near typical operating currents. This is the basis for diode-based temperature sensors and for the need for thermal compensation in bias circuits (e.g., V_BE compensation networks in BJT biasing).

From I_S to a Practical Model: Series Resistance

The ideal Shockley equation predicts current rising without bound as V increases, which is unphysical — real diodes have bulk and contact resistance. A more complete model:

V_total = n·V_T·ln(I/I_S) + I·R_S

where R_S is the series (bulk) resistance, typically 0.1–1 Ω for power diodes. At high current this resistive drop dominates and the I-V curve becomes closer to linear than exponential — visible on a datasheet curve as the characteristic "knee then near-straight rise."

Small-Signal (Dynamic) Resistance

Differentiating the Shockley equation gives the diode's incremental resistance at a DC bias point:

r_d = dV/dI = n·V_T / I_D

At I_D = 10 mA, n = 1: r_d = 0.02585 / 0.01 ≈ 2.6 Ω. This small-signal resistance is what matters for AC analysis — ripple on a rectifier output, or the diode's contribution to gain in a diode-connected transistor bias network — while the large-signal exponential governs DC operating point and switching behavior.

Practical Design Implications

• Never assume a fixed "0.7 V" for precision work. It's a convenient approximation for quick hand analysis at typical currents (1–10 mA on small-signal diodes); real V_F depends on I_S, current level, and temperature — verify against the datasheet I-V curve or SPICE model at your actual operating point.
• Log-linear datasheet curves are the norm because the exponential relationship becomes a straight line on a semi-log I-V plot — read the slope to extract n·V_T, and the intercept to extract I_S.
• Matched diode pairs (or diode-connected transistors) rely on I_S tracking between devices — used in temperature sensing, bandgap references, and translinear circuits (current-mode multipliers/dividers) where the exponential law is the whole point, not a nonideality to avoid.
• SPICE diode models (.model D D(IS=... N=... RS=...)) implement exactly this equation plus junction capacitance and breakdown — matching hand calculations to simulation is a good sanity check before trusting a larger circuit simulation.

Key Takeaways

• The Shockley equation I = I_S·(exp(V/(n·V_T)) − 1) governs diode forward conduction, reverse leakage, and the transition near V = 0.
• Forward voltage rises logarithmically with current: V = n·V_T·ln(I/I_S), so ~60 mV/decade of current change for n ≈ 1 at room temperature.
• I_S spans orders of magnitude across diode types, which is why fixed "0.6–0.7 V" rules only roughly apply and always need datasheet or SPICE verification for precision work.
• Forward voltage drops with temperature (≈ −2 mV/°C) because I_S grows faster with T than V_T does — the basis of diode temperature sensing.
• Small-signal resistance r_d = n·V_T/I_D describes AC behavior at a bias point, distinct from the large-signal exponential DC relationship; real diodes also need a series resistance term at high current.