The diode I–V law
Treat the diode first as a two-terminal nonlinear constitutive element, then connect its exponential I–V relation to the minimum p–n junction physics needed to understand where it comes from. Define voltage/current references, derive the ideal Shockley form under stated assumptions, separate the ideality-factor compact extension from the ideal law, introduce thermal voltage and logarithmic slope, explain reverse-bias and breakdown limits, reject a universal 0.7 V threshold, and solve a resistor–diode operating point by load-line and numerical intersection.
- Define diode voltage and current references using the passive sign convention before writing any I–V equation.
- Explain why a diode is nonlinear by testing homogeneity/superposition against its constitutive law.
- State the physical assumptions behind the ideal p–n junction diffusion-current law rather than presenting the exponential as an empirical magic formula.
- Write the ideal Shockley relation iD=IS[exp(vD/VT)−1] and define VT=kT/q.
- Distinguish the ideal Shockley n=1 law from the practical compact extension iD≈IS[exp(vD/(nVT))−1] with ideality factor n.
- Derive the forward-bias logarithmic slope ΔvD=nVT ln 10 per current decade when iD≫IS.
- Explain why a fixed '0.7 V diode drop' is an approximation, not a constitutive law.
- Describe reverse saturation in the ideal model and explain why breakdown, leakage and generation/recombination require additional physics/models.
- Find the DC operating point of a source–resistor–diode circuit as the intersection of the resistor load line and diode I–V curve.
- Validate a nonlinear operating point using KVL, current consistency, power sign and limiting/model-scope checks.
A resistor has one constant slope R in its v–i relation. A diode does not. Before using rectifiers, clamps or a memorized “0.7 V drop,” we need the actual constitutive law and a disciplined way to combine it with KCL/KVL.
01.1Define the two-terminal variables first
Figure 1.1 — Diode electrical references
The one question is: which iD and vD references are used by every equation?
The reference current enters the anode, the positive terminal of vD. A negative solved value simply means the actual current is opposite to the arrow.
We define vD from anode to cathode and iD entering the anode. The diode law is then a relation between these two signed variables. Changing a reference changes signs in the equations, not the physical component.
Unlike an ideal resistor, the diode cannot be characterized by one constant resistance over all operating points.
01.2Why the diode is nonlinear
A linear one-port would obey homogeneity: doubling v would double i. For an exponential diode law,
Superposition therefore does not apply directly to the complete diode circuit. KCL and KVL remain valid conservation statements; what changes is the constitutive equation used for the diode branch.
01.3Where the exponential law comes from
In an ideal p–n junction under low-level injection, an applied junction voltage changes minority-carrier concentrations at the edges of the depletion region by a Boltzmann factor. Those excess carriers diffuse through the quasi-neutral regions. Solving the steady diffusion problem makes both electron and hole diffusion currents proportional to the same exponential factor.
Combining the two diffusion-current contributions yields the ideal diode form
This derivation assumes an idealized p–n junction: nondegenerate statistics, low-level injection, quasi-neutral diffusion regions, a well-defined junction temperature and transport conditions for which depletion-region recombination/generation and high-injection effects are neglected or secondary.
01.4Thermal voltage and the ideal Shockley equation
Define thermal voltage
At T=300 K, VT≈25.85 mV. The ideal diffusion diode law becomes
IS is a scale/saturation current set by device area and semiconductor transport parameters; it is not one universal constant for “a silicon diode.”
A widely used practical compact extension writes
where n is an ideality factor. The ideal Shockley diffusion result has n=1. Using n≠1 is a compact way to represent additional mechanisms over a chosen operating range; it should not be confused with a first-principles statement that every real diode has one constant n at every current.
01.5The logarithmic slope replaces the idea of a sharp threshold
In forward bias with iD≫IS,
Increasing current by one decade requires
At 300 K this is about 59.5n mV per decade. Current therefore changes continuously and very rapidly with voltage; there is no exact universal voltage at which a real diode suddenly turns on.
Figure 1.2 — A continuous law, not a magic threshold
The one question is: how does slope change from reverse bias through weak to strong forward bias?
The ideal model tends toward −IS in reverse bias and grows exponentially in forward bias. Real breakdown and series resistance are outside this ideal curve.
01.6Reverse bias and breakdown are model boundaries
For sufficiently negative vD within the ideal model, exp(vD/VT) becomes negligible and
Real reverse current can include generation, surface leakage and other mechanisms. At sufficiently large reverse field, Zener or avalanche breakdown produces a qualitatively different branch of the I–V curve.
The basic Shockley expression is therefore not a breakdown model. Likewise, at large forward current, series resistance, high-level injection and self-heating can make the observed I–V depart strongly from a single exponential.
01.7Temperature changes more than VT
The explicit thermal voltage increases linearly with absolute temperature:
But a real diode’s IS also depends strongly on temperature through intrinsic carrier concentration and transport parameters. Therefore one must not predict the temperature coefficient of forward voltage by varying VT alone while holding every other device parameter fixed.
At a fixed forward current, many silicon p–n diodes show a decreasing forward voltage with increasing temperature over common operating ranges. The exact coefficient is device- and current-dependent.
01.8The operating point is an intersection, not a guessed diode voltage
Figure 1.3 — Source–resistor–diode operating point
The one question is: where do Vs, R, iD and vD live in the load-line equation?
With the drawn references, KVL requires Vs=RiD+vD. The diode simultaneously imposes its own I–V law; their intersection fixes the operating point Q.
KVL gives the resistor load line
The diode simultaneously imposes its I–V relation. The physical DC operating point Q is the pair (vD,iD) satisfying both equations.
Pedagogical example: Vs=5 V, R=1 kΩ, IS=1 pA, n=1 and T=300 K. Solving
gives approximately
The result is not 0.7 V because no such constant was assumed. Changing IS, n, temperature or load shifts Q.
01.9Newton iteration and local slope
Define the operating-point residual
Newton’s method updates
For the example above, starting at 0.55 V converges through about 0.58971, 0.57800, 0.57442 and finally 0.57415 V.
The derivative also previews chapter 02: at one bias point, the local incremental conductance is diD/dvD. A nonlinear device can therefore be locally linearized without pretending its full large-signal law is linear.
01.10Mastery exercises
- At 300 K and n=1, by what factor does forward current change for +30 mV when i≫IS?
- Derive the voltage change required for a ×100 current change.
- At vD=0, what current does the ideal Shockley law predict?
- For vD≪−VT but before breakdown, what current does the ideal law predict?
- Why is n=1 special in the ideal Shockley derivation?
- For Vs=3 V, R=2 kΩ, IS=1 pA, n=1, T=300 K, write the nonlinear operating-point equation even if you do not solve it analytically.
- Give three physical effects that can invalidate a single-exponential forward model.
- Explain why KVL remains valid even though superposition fails in a diode circuit.
Solutions
1. exp(30/25.85)≈3.19.
2. Δv=nVT ln100=2nVT ln10≈119n mV at 300 K.
3. iD=IS(exp0−1)=0.
4. iD≈−IS in the ideal model.
5. n=1 follows from the ideal low-level diffusion-current p–n junction derivation; n≠1 is a compact extension for additional mechanisms.
6. IS[exp(vD/VT)−1]=(3−vD)/2000.
7. Series resistance, high-level injection, depletion-region recombination, self-heating and frequency-dependent charge storage are examples.
8. KVL is a conservation/topology relation in the lumped circuit model; superposition additionally requires linear constitutive relations, which the diode does not satisfy.
Chapter 02 will derive ideal, constant-drop, piecewise-linear and small-signal diode models as controlled approximations to this I–V law, with explicit validity ranges and error checks.