Differential resistance

Differential resistance is the local slope of a diode's current-voltage relationship at a specific operating point:

$$R_d\equiv {\frac{dV}{dI}\mid }_{I_{\text{op}}}\mathrm{.}$$

Unlike static resistance $V\mathrm{/}I$, which conflates the diode's nonlinear turn-on with its bulk resistive behavior, differential resistance isolates the device's response to a small current change at the operating point.

Why $R_d$ matters for laser diodes. Above threshold, the laser's terminal voltage is approximately:

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

where $V_{\text{th}}$ is the voltage at threshold. The first term is dominated by the junction physics (carrier-density-clamped Fermi-level separation, set by the lasing condition); the second is the Ohmic drop across all bulk resistance contributors.

Components of $R_d$:

Total $R_d$ for a well-designed telecom DFB: 4 – 8 Ω. For research or aggressive geometries: 8 – 25 Ω.

Extraction from LIV.

$$R_d\approx \frac{V(I_2)-V(I_1)}{I_2-I_1},$$

evaluated over an interval $I_2>I_1>I_{\text{th}}$. Choosing the interval well above threshold (e.g., $2I_{\text{th}}$ to $3I_{\text{th}}$) ensures the lasing carrier-density clamp is in effect throughout, so the slope reflects bulk-resistance contributions only.

Why $R_d$ matters for high-speed modulation. Direct modulation of a laser at high frequencies requires driving signal current into the device's effective electrical impedance. The differential resistance determines how much voltage swing is needed to produce a given modulation current. For RF drive:

$$v_{\text{swing}}=I_{\text{mod}}{R}_d\mathrm{.}$$

A lower $R_d$ (5 Ω) allows efficient RF coupling from a 50 Ω driver via a matching network. A higher $R_d$ (20 Ω) is closer to 50 Ω and easier to match, but allows less current swing per volt. The optimal design depends on the driver architecture.

Bandwidth implications. Combined with package parasitic capacitance $C_p$ (typically 0.5 – 2 pF), $R_d$ sets an RC bandwidth limit:

$$f_{RC}=\frac{1}{2\pi R_d{C}_p}\mathrm{.}$$

For $R_d=5$ Ω and $C_p=1$ pF, $f_{RC}\approx 32$ GHz. Modern 50+ GHz directly-modulated lasers require careful design to minimize both $R_d$ and $C_p$.

Thermal implications. Power dissipated in the resistance, rather than in the junction, becomes heat. Above threshold the junction voltage is clamped, so $R_d$ is essentially the series resistance, and the series resistance carries the whole drive current, not only the part above threshold:

$$P_{\text{ohmic}}\approx R_d{I}^2\mathrm{.}$$

For a laser with a 10 mA threshold, a threshold voltage of 1.0 V and $R_d=5$ Ω, run at 10 times threshold (100 mA), this is 50 mW of the 145 mW of electrical input, about a third. Lower $R_d$ directly reduces self-heating, improving thermal performance and increasing the maximum operating current before thermal rollover.

Diagnostic value. Changes in $R_d$ over aging or temperature reveal specific failure modes:

• Increase in $R_d$ with aging: contact degradation, wire bond fatigue, or metallization issues
• Reversible change in $R_d$ with temperature: expected, not a fault, since the heterojunction barriers, the contacts and the doped layers all conduct differently as the temperature changes; compare readings taken at the same heatsink temperature
• Sudden $R_d$ jump: catastrophic contact damage or wire bond break
• $R_d$ different from design value at first measurement: process variation; may indicate doping or epitaxy nonuniformity

References: Coldren, Corzine, Mašanović, Diode Lasers and Photonic Integrated Circuits, Ch. 2 for the analytic LIV treatment.