Ideality factor

The ideality factor (or quality factor) $n$ is the dimensionless parameter in the diode equation:

$$I \;=\; I_0 \left[ \exp\!\left( \frac{qV}{n kT} \right) - 1 \right],$$

where $I_0$ is the saturation current, $V$ is the forward voltage, $T$ is temperature, $k$ is Boltzmann's constant, and $q$ is the elementary charge.

For an ideal diode with only diffusion-driven carrier transport (Shockley's original derivation), $n = 1$. Deviations from $n = 1$ reveal additional recombination or transport mechanisms.

Common values of $n$ and their interpretation:

Ideality factor	Dominant mechanism	Where observed
$n = 1.0$	Diffusion-limited recombination in neutral regions	Ideal silicon p-n junction at moderate bias
$n = 2.0$	Recombination in the depletion region (Shockley-Read-Hall)	Defect-rich junctions, wide-gap semiconductors
$n = 1.0 - 2.0$	Combination of diffusion + depletion-region recombination	Most real diodes
$n > 2.0$	Trap-assisted tunneling or barrier inhomogeneity	Heterojunctions, defective material
$n < 1.0$	Apparent only — usually instrumentation artifact (series-resistance effect on the curve fitting)	Improper extraction

Extraction from sub-threshold I-V data. Below threshold (or in the forward-bias regime where no significant series resistance dominates), the ideality factor is extracted from the local slope of $\ln(I)$ versus $V$:

$$n \;=\; \frac{q}{kT} \cdot \frac{dV}{d\ln I}.$$

At room temperature ($kT/q \approx 25.85$ mV), this becomes:

$$n \;\approx\; \frac{1}{25.85 \text{ mV}} \cdot \frac{dV}{d\log_{10} I} \cdot \frac{1}{\ln 10} \;=\; \frac{dV}{d\log_{10} I} \cdot 16.8 \text{ V}^{-1}.$$

A clean diode at room temperature with $n = 1$ shows a slope of 59.6 mV per decade of current; $n = 2$ shows 119 mV/decade.

Critical for laser characterization. Below threshold, the ideality factor of a semiconductor laser indicates the dominant recombination mechanism in the active region:

• $n \approx 1$: radiative recombination dominates — excellent epitaxy, low defect density
• $n \approx 1.5 - 2$: significant SRH (defect-mediated) recombination — moderate material quality
• $n > 2$: trap-assisted tunneling — poor material; expect short device lifetime

A new laser process with $n > 2$ is a flag for fundamental epitaxy problems and indicates that further development is needed before product qualification.

Why $n$ matters above threshold. Above threshold, the carrier density is clamped (because additional injection goes into stimulated emission rather than carrier population), so the diode equation no longer applies directly. The terminal voltage above threshold is approximately:

$$V(I) \;\approx\; V_\text{th} + R_d (I - I_\text{th}),$$

dominated by differential resistance, not ideality. Ideality factor characterizes sub-threshold (LED-regime) behavior.

Practical complications.

• Series resistance introduces an additional $IR_s$ voltage drop that masks the true junction voltage. At high currents this dominates and makes $n$ appear larger than its true value. Standard practice is to fit $n$ only in the moderate-current regime where $IR_s \ll kT/q$.
• Shunt resistance dominates at very low currents, where leakage paths around the junction add to the measured current. Fit $n$ at currents above the leakage floor.
• Temperature dependence: $n$ is approximately temperature-independent for SRH-dominated junctions; for diffusion-dominated, it stays at 1.0 over a wide range. Strong temperature variation of extracted $n$ suggests a more complex (tunneling) mechanism.
• Multi-junction stacks: tandem solar cells or cascaded laser diodes have multiple junctions in series; the extracted $n$ is roughly the sum of the individual ideality factors.

LED applications. Ideality factor of an LED determines the "knee" sharpness of the LED's I-V curve and influences the optimal drive scheme. White-light LEDs (InGaN-on-sapphire) typically show $n = 2 - 4$ at the operating point due to defects in the active region — this is a known and accepted feature of the GaN material system, not a fault.

Solar cell applications. Solar cell efficiency depends on ideality factor: lower $n$ corresponds to better-quality material and higher achievable open-circuit voltage. High-efficiency cells (HIT, IBC, perovskite) target $n = 1.0 - 1.2$.

References: Sze, Physics of Semiconductor Devices (3rd ed., 2007), Ch. 2 for the diode equation derivation; Pierret, Semiconductor Device Fundamentals (1996), Ch. 6 for the recombination-mechanism treatment.