Series resistance

Series resistance $R_s$ is the total Ohmic resistance in the current path of a diode, including:

• Contact metal traces and wirebonds
• Metal-semiconductor contact (specific contact resistivity × contact area)
• Doped cladding layers above and below the junction
• Series-coupled passive elements in the package

It excludes the junction itself, whose I-V relationship is exponential (described by the ideality factor equation), and any shunt-leakage paths in parallel (described by the shunt resistance).

Where it appears in the diode equation. The modified diode equation including series resistance:

$$V \;=\; \frac{nkT}{q} \ln\!\left( \frac{I}{I_0} + 1 \right) + I R_s.$$

At low currents, $I R_s \ll nkT/q$ and the equation reduces to the ideal diode form. At high currents, $I R_s$ dominates and the I-V curve becomes a straight line with slope $1/R_s$.

Distinction from differential resistance. Differential resistance $R_d = dV/dI$ at a specified operating point. For a laser above threshold (where carrier density is clamped and the junction voltage is constant), $R_d = R_s$ in the limit. Below threshold, $R_d$ includes the junction's exponential contribution, while $R_s$ is the linear (Ohmic) portion only.

Components of $R_s$ in a typical InP DFB laser.

Component	Typical value (Ω)
Top contact metallization (Ti/Pt/Au)	0.1 – 0.5
Top contact specific resistivity	0.1 – 1
p-InP cladding (3 μm thick, $5 \times 10^{17}$ cm$^{-3}$)	2 – 5
Active region (MQW) sheet resistance	0.5 – 2
n-InP cladding (3 μm, $5 \times 10^{18}$ cm$^{-3}$)	0.3 – 1
Bottom contact + submount metallization	0.1 – 0.5
Wire bonds	0.05 – 0.2
Total $R_s$	3 – 10 Ω

The p-cladding typically dominates because hole mobility ($\sim 200$ cm²/V·s in p-InP) is much lower than electron mobility ($\sim 5000$ cm²/V·s in n-InP) at typical doping levels, and the activation energy for Zn or C acceptors is non-negligible, limiting the active carrier density.

Why $R_s$ matters.

1. Self-heating. Power dissipated in $R_s$ is $P = I^2 R_s$ — not delivered to the optical output. For a laser at $5 \times I_\text{th}$ ($\sim 100$ mA) with $R_s = 8$ Ω: $P = 80$ mW dissipated as heat, comparable to the optical output power. Higher $R_s$ requires more current to reach a given output power and creates more self-heating.

2. Modulation bandwidth. Combined with package capacitance, $R_s$ limits the laser drive bandwidth: $f_{RC} = 1/(2\pi R_s C_p)$. For $R_s = 5$ Ω, $C_p = 1$ pF: $f_{RC} \approx 32$ GHz. Modern 50 GHz directly-modulated lasers require $R_s < 3$ Ω and $C_p < 0.5$ pF.

3. Impedance matching to drivers. RF drivers typically have 50 Ω output impedance. Lasers with very low $R_s$ ($\sim 1$ Ω) require step-down impedance matching networks, which add insertion loss and are bandwidth-limited. Lasers with $R_s \approx 50$ Ω can be directly driven without matching, but they are too lossy for high-efficiency operation. The compromise is typically $R_s \approx 4 - 10$ Ω with a quarter-wave or distributed-element matching network.

4. Operating voltage at high current. Voltage required is $V = V_\text{th} + I \cdot R_s$. For a laser with $V_\text{th} = 1$ V, $I = 100$ mA, $R_s = 8$ Ω: $V = 1.8$ V. This sets the drive voltage budget for the entire system; CMOS driver chips typically have 1.8 – 3.3 V supply, leaving limited margin for the laser drop.

Reducing $R_s$.

• Increase contact area: lower current density through the contact reduces specific contact resistivity
• Lower contact resistivity: use lower-barrier metal (e.g., Pt instead of Ti for p-contact on InP)
• Reduce cladding resistance: shorter clad path or higher doping
• Use higher-mobility p-doped material: e.g., AlGaInAs instead of InGaAsP

Measurement. From an LIV curve well above threshold, the slope $dV/dI$ approximates $R_s + R_d$ (which for a laser is approximately $R_s$). Best practice: fit a line to the high-current portion of the V-I curve, where $I R_s \gg V_\text{junction}$, and the slope equals $R_s$ to good approximation.

References: Coldren, Corzine, Mašanović, Diode Lasers, Ch. 2; Sze, Physics of Semiconductor Devices, Ch. 5 (specific contact resistivity treatment).