Diode Equation (Shockley Equation)

Introduction

The Shockley diode equation mathematically describes the current-voltage relationship of an ideal PN junction diode. This exponential equation is fundamental to understanding semiconductor device behavior and is the basis for modeling all bipolar devices including BJTs.

The Shockley Equation

Thermal Voltage

Ideality Factor (n)

The ideality factor accounts for non-ideal recombination mechanisms:

Analysis of the Equation

Forward Bias (V >> nVT)

When V is much greater than nVT (say V > 100 mV):

Reverse Bias (V << -nVT)

When V is negative and |V| >> nVT:

Small Voltage (|V| ≈ nVT)

Reverse Saturation Current (Is)

Graphical Analysis

Logarithmic Plot

Decade Change

For n = 1 at 300K, a voltage change of:

Numerical Examples

Example 1: Forward Current Calculation

Problem: A silicon diode has Is = 2 × 10⁻¹⁴ A and n = 1. Calculate the forward current at V = 0.6V and V = 0.7V at 27°C.

Solution:

At V = 0.6V:

At V = 0.7V:

Note: A 100 mV increase caused current to increase by 46× (almost 2 decades).

Example 2: Finding Is from Measurements

Problem: A diode carries 5 mA at VF = 0.72V and 50 mA at VF = 0.78V. Find n and Is at 300K.

Solution:

Step 1: Find n using ratio of currents

Step 2: Find Is

Example 3: Temperature Effect on Forward Voltage

Problem: A diode carries a constant current of 1 mA. At 25°C, VF = 0.65V. Find VF at 75°C given that Is doubles every 10°C.

Solution:

Step 1: Find Is at 75°C

Step 2: At constant current I:

Using the practical rule: ΔV/ΔT ≈ -2 mV/°C

Small-Signal Model

For small AC signals superimposed on DC bias:

Interview Questions

1. Derive the expression for dynamic resistance from the diode equation.

Starting with I = Is×exp(V/nVT), differentiate: dI/dV = Is×exp(V/nVT)/(nVT) = I/(nVT). Therefore rd = dV/dI = nVT/I. At ID = 1mA with n=1: rd = 26mV/1mA = 26Ω.

1. Why does forward voltage decrease with increasing temperature at constant current?

Is increases exponentially with temperature (doubles per 10°C). To maintain the same current I = Is×exp(V/nVT), as Is increases, V must decrease. The rate is approximately -2mV/°C. VT also increases but this effect is smaller.

1. What physical mechanism causes the ideality factor to differ from 1?

n > 1 occurs due to recombination of carriers in the depletion region (generation-recombination current). At low currents, depletion region recombination dominates (n→2). At higher currents, diffusion current dominates (n→1).

1. How accurate is the Shockley equation at high currents?

At high currents, the equation underestimates voltage because it ignores series resistance (bulk resistance of semiconductor regions and contact resistance). The modified equation includes: V = nVT×ln(I/Is) + I×Rs, where Rs is series resistance.

1. Explain the decade-per-60mV rule for diode current.

From I = Is×exp(V/nVT), a voltage increase of nVT×ln(10) = 60mV (for n=1, T=300K) increases current by a factor of 10. This logarithmic relationship is used in dB-linear conversion and analog computation circuits.

Summary

The Shockley equation I = Is×(exp(V/nVT) - 1) completely describes ideal diode behavior. The thermal voltage VT sets the temperature scale, the ideality factor n accounts for recombination, and Is provides the baseline current level. This equation is fundamental to all semiconductor device analysis and forms the basis for transistor models.