Diode Equation

The Shockley diode equation mathematically describes the current-voltage (I-V) relationship of an ideal PN junction diode. It captures the exponential dependence of forward current on applied voltage and the near-constant reverse saturation current, making it the foundational equation of all junction device theory in electronics.

The Shockley Equation

The ideal diode current is described by: I = Is x (exp(V / n x VT) - 1), where Is is the reverse saturation current, V is the applied voltage (positive for forward bias), n is the ideality factor (also called emission coefficient), and VT = kT/q is the thermal voltage. This equation was derived by William Shockley from minority carrier diffusion theory and boundary conditions at the depletion region edges.

The derivation assumes that minority carrier injection at the depletion region edges is proportional to exp(V/VT), that minority carriers recombine only outside the depletion region (no generation-recombination inside the depletion region), and that the depletion region width is small compared to the minority carrier diffusion lengths. These are the ideal diode assumptions.

Physical Meaning of Each Term

The reverse saturation current Is represents the drift of thermally generated minority carriers across the junction. It depends on minority carrier diffusion constants, minority carrier lifetimes, and ni^2, making it extremely sensitive to temperature. For silicon, Is is typically in the picoampere to nanoampere range at 300 K. It roughly doubles for every 10 K increase in temperature.

The ideality factor n takes a value between 1 and 2. When n = 1, the current is dominated by minority carrier diffusion in the neutral regions (ideal Shockley behavior). When n = 2, the current is dominated by generation-recombination of carriers within the depletion region (Sah-Noyce-Shockley model). In real silicon diodes, n is typically between 1 and 2 depending on the operating point; at very low forward voltages, recombination dominates (n closer to 2), while at moderate voltages diffusion dominates (n closer to 1).

Forward Bias Approximation

For V greater than a few multiples of VT (about 100 mV for silicon), the exponential term is much larger than 1, so the equation simplifies to: I = Is x exp(V / n x VT). This shows that the current increases exponentially with voltage. A 60 mV (for n = 1) or 120 mV (for n = 2) increase in forward voltage causes a tenfold (decade) increase in current. This is the 60 mV per decade rule, which is fundamental to understanding transistor characteristics as well.

Reverse Bias Behavior

For V much less than -VT (reverse bias), the exponential term becomes negligibly small and the equation gives I = -Is. The current saturates at -Is, which is why it is called the reverse saturation current. In practice, real diodes show a slight increase in reverse current with increasing reverse voltage due to generation-recombination in the widening depletion region and surface leakage effects, but the ideal Shockley equation ignores these.

I-V Curve Analysis

The I-V characteristic has three practical regions. In the forward active region (V greater than approximately 0.5 V for silicon), the current rises steeply and exponentially. In the reverse saturation region (-V0 to large negative V), the current is approximately -Is, very small and nearly constant. At very large reverse voltages, breakdown occurs through either Zener tunneling or avalanche multiplication, causing a sharp increase in reverse current — this is not described by the Shockley equation.

Given:
Silicon diode with Is = 1 x 10^-12 A (1 pA)
Ideality factor n = 1
Temperature T = 300 K, VT = 0.02585 V
Applied forward voltage V = 0.65 V

Why this formula applies:
V >> VT so the (-1) term is negligible. Full Shockley equation used for precision.

Formula:
I = Is x (exp(V / n x VT) - 1)

Substitution:
I = 1 x 10^-12 x (exp(0.65 / 1 x 0.02585) - 1)
Exponent = 0.65 / 0.02585 = 25.14

Calculation:
exp(25.14) = 8.24 x 10^10
I = 1 x 10^-12 x (8.24 x 10^10 - 1)
I = 1 x 10^-12 x 8.24 x 10^10

Final Answer: I = 0.0824 A = 82.4 mA. The diode carries 82.4 mA at 0.65 V forward bias.

Exam Tip: In GATE, the Shockley equation is directly used in diode circuit problems. Remember: a 60 mV increase in VD multiplies the current by 10 (for n=1). If n=2, it requires 120 mV per decade. When two identical diodes are in series, the effective Is halves and effective n doubles — the turn-on voltage nearly doubles. Also, Is doubles roughly every 10 K, so at 350 K (77 C), Is is approximately 32 times its 300 K value.

• The Shockley equation I = Is(exp(V/nVT) - 1) describes the complete I-V behavior of an ideal PN junction diode.
• Is is the reverse saturation current governed by minority carrier properties and ni^2; it doubles approximately every 10 K.
• Ideality factor n = 1 when diffusion dominates; n = 2 when depletion-region recombination dominates.
• For n=1, current increases by a factor of 10 for every 60 mV increase in forward voltage (60 mV/decade rule).
• Breakdown is not described by the Shockley equation; it requires avalanche or Zener models.

Quick Revision

• Shockley diode equation: I = Is x (exp(V / n x VT) - 1).
• Thermal voltage VT = kT/q = 26 mV at 300 K.
• For n=1: 60 mV per decade; for n=2: 120 mV per decade.
• Reverse saturation current Is is in the pA to nA range for Si at room temperature.
• Is doubles approximately every 10 K increase in temperature.
• Trap: The (-1) in the Shockley equation gives the small reverse saturation current; for V much greater than VT it is negligible but for V near 0 it is important.
• Trap: n is not always 1; for GATE problems check whether diffusion-dominated (n=1) or recombination-dominated (n=2) behavior is specified.