Thermal resistance

Thermal resistance is the steady-state temperature difference between two points in a heat path divided by the heat flowing between them:

$$R_\text{th} = \frac{\Delta T}{P_\text{diss}},$$

in K/W (equivalently °C/W). For a laser diode the two points are usually the active region and the heatsink or case, and $P_\text{diss}$ is the electrical input power minus the optical output. A telecom DFB laser mounted on a submount typically has 30–60 K/W, a high-power 980 nm pump diode on a copper heatsink 5–15 K/W, and an unmounted edge-emitter die clamped between contacts can exceed 100 K/W. These are representative ranges; each assembly has its own value, which is why it is measured.

Worked example for a laser diode

A DFB laser driven at 100 mA and 1.5 V that emits 20 mW dissipates

$$P_\text{diss} = 0.150 - 0.020 = 0.130\ \text{W}.$$

With $R_\text{th} = 50$ K/W its junction runs 6.5 K above the heatsink. At the typical DFB temperature coefficient of about 0.1 nm/K, that rise shifts the emission wavelength by about 0.65 nm relative to an isothermal (pulsed) measurement at the same heatsink temperature. The threshold also rises with the junction temperature, exponentially with the characteristic temperature, and at high current the heating feeds back on itself until the output peaks and falls in thermal rollover.

Series stack

Heat from the junction crosses the chip, the die-attach solder, the submount, the submount attach and the package base before reaching the thermoelectric cooler or heatsink. For layers in series the resistances add,

$$R_\text{th} = R_\text{chip} + R_\text{attach} + R_\text{sub} + \ldots$$

and are often grouped as junction-to-case and case-to-heatsink. Each interface also adds a contact resistance, which for a loose mechanical mount or a voided solder joint can dominate the total.

For a uniform slab of thickness $t$, thermal conductivity $k$ and area $A$, with heat flowing straight through it,

$$R_\text{th} = \frac{t}{k A}.$$

A 100 µm thick AlN submount ($k \approx 170$ W/(m·K); commercial ceramics range up to about 220) under a 300 µm × 250 µm chip footprint gives $R = 7.8$ K/W if the heat stayed within the footprint. Silicon ($k \approx 150$ W/(m·K)) gives 8.9 K/W and pure copper ($k \approx 400$ W/(m·K)) 3.3 K/W for the same slab. Over the full area of a 2 mm × 1 mm submount the AlN figure falls to 0.29 K/W. The true value lies between these limits because heat spreads laterally as it descends; the footprint figure is a conservative estimate for thin layers and the full-area figure an optimistic one. A 5 µm AuSn die attach ($k \approx 57$ W/(m·K)) over the same footprint adds about 1.2 K/W.

In a narrow-ridge laser mounted epitaxial side up, the largest single contribution is usually the chip itself: heat generated in a stripe a few micrometers wide must spread through the InP or GaAs substrate, whose conductivity (about 68 W/(m·K) for InP) is well below that of the submount. Mounting the chip epitaxial side down, as in a flip-chip bonded laser, places the active region within a few micrometers of the solder and removes most of that spreading resistance.

Measurement

$R_\text{th}$ is obtained from the junction temperature rise at a known dissipated power. The two common bench methods use a temperature-sensitive parameter calibrated under pulsed, self-heating-free conditions: the lasing wavelength of a DFB, or the forward voltage at a small sense current. Comparing the CW value with the calibration gives $\Delta T$, and dividing by $P_\text{diss}$ gives $R_\text{th}$. The procedure and its corrections are described in Measuring Laser Diode Junction Temperature and Thermal Resistance, and the junction temperature calculator applies the result.

Pitfalls

• Using electrical input power instead of dissipated power; for a laser with a high wall-plug efficiency the difference is large.
• Quoting a value without its reference points: junction-to-case and junction-to-ambient can differ by an order of magnitude.
• Treating $R_\text{th}$ as constant: thermal conductivities of semiconductors fall as temperature rises, so the resistance grows at high dissipation.
• Applying the steady-state value to pulses. For pulses shorter than the thermal time constants, the transient thermal impedance $Z_\text{th}(t)$, which rises toward $R_\text{th}$ over microseconds to seconds as heat reaches successive layers, is the relevant quantity.

Common questions

What is a typical thermal resistance for a laser diode?

About 30–60 K/W for a telecom DFB on a submount with active cooling, 5–15 K/W for a high-power pump diode on a copper heatsink, and more than 100 K/W for poorly mounted chips.

How is junction temperature calculated from thermal resistance?

$T_j = T_\text{case} + R_\text{th} P_\text{diss}$, where $R_\text{th}$ is the junction-to-case value and $P_\text{diss}$ the electrical power not emitted as light.

Why does flip-chip mounting reduce thermal resistance?

It puts the heat-generating active region next to the solder and submount, so the heat no longer has to cross the roughly 100 µm thick substrate of the chip.

References: L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012); T. L. Bergman, A. S. Lavine, F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass Transfer, 7th ed. (Wiley, 2011).