Forward Bias: Exponential Current, Knee Voltage

Every diode-based circuit — rectifiers, clamps, reference generators, ESD protection, BJT base-emitter junctions — is set by how forward current relates to voltage across the junction. That relationship isn't linear, isn't a fixed 0.7 V switch, and getting it wrong by even 50 mV in a bias calculation can put a BJT into the wrong operating region or make a "diode drop" assumption fail badly in a low-current or high-current design. Understanding the exponential law and where the practical "knee voltage" comes from is what lets you size bias resistors, predict thermal drift, and read a datasheet I–V curve correctly.

The Shockley Diode Equation

The ideal diode law gives forward current as a function of applied voltage:

I = I_S · (exp(V/(n·V_T)) − 1)

Where:

• I_S — reverse saturation current (typically 10⁻¹² to 10⁻¹⁵ A for small-signal silicon diodes), strongly temperature-dependent
• V_T — thermal voltage, V_T = kT/q ≈ 25.85 mV at 300 K (commonly rounded to 25 mV or 26 mV)
• n — ideality factor, ≈1 for diffusion-current-dominated conduction (typical at moderate-to-high forward current); ≈2 where recombination current in the depletion region dominates (typically at low forward current). This is independent of whether the material is wide-bandgap or not (typical small-signal diodes: n ≈ 1.0–1.8)

For any forward voltage more than a few V_T above zero (i.e., V > ~100 mV), the −1 term is negligible and the exponential dominates:

I ≈ I_S · exp(V/(n·V_T))

This means every ~60 mV (n=1) to ~120 mV (n=2) increase in V_F multiplies current by 10× at room temperature — the diode's I–V curve is a straight line on a semi-log plot, not a hard threshold.

Why It Looks Like a "Knee" on a Linear Plot

On a linear I–V axis, this exponential curve appears almost flat near zero, then bends sharply upward over a narrow voltage range, then rises almost vertically. That bend is the so-called knee voltage — the informal threshold (≈0.6–0.7 V for silicon, ≈0.2–0.3 V for germanium, ≈1.8–3.3 V for various LED colors) where the diode "appears" to start conducting on a linear scale.

The knee is not a physical discontinuity — it's a consequence of plotting an exponential on linear axes combined with the fact that useful circuit currents (mA range) require several hundred mV, while at low voltage the current is immeasurably small (nA–µA). The "knee" location depends entirely on what current you call "on."

This table itself is the exponential law: each decade of current costs roughly n·V_T·ln(10) ≈ 60 mV (n=1) of additional forward voltage.

Worked Example: Two Diodes in a Current Mirror-Like Bias

Suppose a silicon diode has I_S = 5×10⁻¹⁵ A, n = 1, T = 300 K (V_T ≈ 25.85 mV). Find V_F at I = 2 mA, then check how V_F shifts if the current is increased to 20 mA (a decade higher).

Step 1 — solve for V_F at 2 mA:

V = n·V_T·ln(I/I_S)
V = 1 × 0.02585 × ln(2×10⁻³ / 5×10⁻¹⁵)
V = 0.02585 × ln(4×10¹¹)
V = 0.02585 × 26.71
V ≈ 0.690 V

Step 2 — verify by plugging back into the Shockley equation:

I = I_S · exp(V/V_T) = 5×10⁻¹⁵ · exp(0.690/0.02585)
  = 5×10⁻¹⁵ · exp(26.69)
  = 5×10⁻¹⁵ · 3.93×10¹¹
  ≈ 1.97×10⁻³ A ≈ 1.97 mA  ✓ (matches 2 mA within rounding)

Step 3 — at 20 mA (10× current):

ΔV = V_T·ln(10) = 0.02585 × 2.303 ≈ 0.0595 V ≈ 60 mV
V_F(20 mA) ≈ 0.690 + 0.060 = 0.750 V

Check: I = 5×10⁻¹⁵ · exp(0.750/0.02585) = 5×10⁻¹⁵ · exp(29.01) = 5×10⁻¹⁵ · 3.99×10¹² ≈ 19.9 mA ✓ — closes to within 0.5%.

This is the practical rule engineers use on the bench: a decade of current change moves V_F by about 60 mV (at n≈1, room temperature) — useful for estimating how a diode's drop shifts across a load range without resolving the full exponential each time.

Temperature Dependence

I_S roughly doubles every 5–10 °C rise (dominated by the exponential dependence of intrinsic carrier concentration on temperature), while V_T increases linearly with T. The net effect on a silicon diode held at constant current is a forward voltage temperature coefficient of about −2 mV/°C. This is why:

• Diode-based bandgap references must cancel this drift deliberately (combining a −2 mV/°C V_BE term with a PTAT term).
• Two "matched" diodes at different temperatures will not track unless thermally coupled — relevant for BJT base-emitter junctions used as temperature sensors.
• A forward-biased diode used as a simple voltage reference or level-shifter drifts noticeably over an automotive or industrial temperature range (−40 °C to +125 °C ⇒ up to ~±0.33 V shift from a 165 °C span at −2 mV/°C).

Series Resistance: Where the Exponential Stops

Real diodes deviate from the pure exponential at higher currents because of bulk and contact series resistance r_S. The terminal voltage becomes:

V_terminal = n·V_T·ln(I/I_S) + I·r_S

At low-to-moderate current the log term dominates and the I–V curve looks exponential; at high current the I·r_S term dominates and the curve becomes linear again (resistive), with a slope of r_S. This is visible on any datasheet forward-characteristic plot as the curve "straightening out" at high current — it's not a change in physics, just the ohmic drop across the undepleted silicon and lead/bond-wire resistance overtaking the junction's log behavior.

Practical Design Implications

• Don't use a fixed 0.7 V for precision work. For rough biasing (BJT base-emitter estimate, rectifier drop) it's a fine approximation; for reference circuits, current mirrors, or low-voltage rail budgets, use the log relationship or datasheet curve at your actual operating current.
• Bias current sets V_F, not the other way around in most circuits — a diode/BJT junction is normally driven by a current source or resistor-limited current, and V_F is the resulting (weakly-varying) output.
• Small-signal (dynamic) resistance falls directly out of the exponential: r_d = dV/dI = n·V_T/I. At 1 mA, n=1, r_d ≈ 25.85 Ω; at 10 mA, r_d ≈ 2.6 Ω. This is the resistance seen by small AC signals riding on a DC bias — central to diode-based attenuators, log amps, and BJT small-signal models (r_π, r_e).
• Check your knee assumption against current level. A "0.7 V" LED-driver assumption is wrong for an LED — knee voltages for LEDs range from ~1.8 V (red/IR) to >3 V (blue/white) because of the larger bandgap.
• Temperature compensation is mandatory in any circuit using diode/BJT V_F as a reference or comparator threshold across a wide operating temperature range.

Key Takeaways

• Forward diode current follows I ≈ I_S·exp(V/(n·V_T)) — an exponential law, not a fixed threshold.
• The "knee voltage" is a visual artifact of plotting this exponential on a linear axis; its apparent value depends on the current level you consider "on" (~0.6 V at 1 mA, ~0.75 V at 10 mA for silicon).
• Every decade of forward current change shifts V_F by roughly n·V_T·ln(10) ≈ 60 mV (n≈1) to ~120 mV (n≈2) at room temperature.
• V_F has a temperature coefficient of about −2 mV/°C at constant current — critical for bandgap references and diode-based temperature sensing.
• At high current, series resistance r_S dominates and the I–V curve becomes linear (ohmic) again, departing from the pure exponential.
• Small-signal resistance r_d = n·V_T/I falls directly from the exponential law and underlies diode attenuators, log amps, and BJT small-signal models.