Materials & Substrates

CTE (Metal)

Q: What are the CTE values of common RF package metals?

Typical linear CTE values near room temperature: oxygen-free copper about 16.5 ppm per degree C, aluminum 6061 about 23.6 ppm per degree C, brass about 19 ppm per degree C, gold about 14.2 ppm per degree C, molybdenum about 4.8 ppm per degree C, tungsten about 4.5 ppm per degree C, Kovar (Fe-Ni-Co) about 5.5 ppm per degree C from 30 to 200 C, and Invar (Fe-36Ni) about 1.2 ppm per degree C up to its inflection near 230 C. CuMo and CuW composites are tailored between 6 and 11 ppm per degree C by adjusting the metal ratio. These values are why Kovar pairs with borosilicate sealing glass at about 5 ppm per degree C, while aluminum housings demand compliant solders or strain-relief features.

Q: How do I calculate thermal stress from a CTE mismatch in a soldered RF assembly?

For two bonded materials, the differential strain is delta-alpha times delta-T, where delta-alpha is the CTE difference and delta-T is the temperature swing from the stress-free (solidus) temperature. The shear stress at the joint scales roughly as tau = (delta-alpha times delta-T times G) divided by the bond-line thickness term, while the peak in-plane stress in the stiffer member approaches E times delta-alpha times delta-T for a fully constrained joint. For a GaAs MMIC (about 5.7 ppm per degree C) soldered to a copper carrier (16.5 ppm per degree C) with a 175 C reflow and 25 C operation, delta-alpha times delta-T is about 10.8e-6 times 150, near 1620 microstrain, enough to crack die or fatigue the solder over thermal cycling. Engineers reduce this with CuMo carriers, AuSn versus SnPb solder selection, and thicker compliant bond lines.

Q: Why is Invar used for dimensionally stable waveguide and antenna structures?

Invar (Fe-36Ni) exhibits an anomalously low CTE near 1.2 ppm per degree C between 0 and 100 C because magnetostriction nearly cancels normal lattice thermal expansion below the Curie point. A 1 meter Invar waveguide run drifts only about 60 micrometers over a 50 C swing, versus roughly 1180 micrometers for the same length of aluminum. That stability keeps electrical path length, resonant cavity frequency, and reflector geometry within tolerance for radar, satellite, and metrology systems. The trade-offs are higher density, lower thermal conductivity near 13 W/m-K, magnetic behavior, and a CTE that rises sharply above the Curie temperature, so Invar is reserved for the most dimension-critical members rather than general housings.

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Measured in parts per million per degree Celsius (ppm/°C), the coefficient of thermal expansion of a metal describes the fractional change in its linear dimension as temperature rises or falls. For RF hardware this single number quietly dictates whether a hermetic seal survives, whether a die cracks, and whether a waveguide holds its electrical length. Common housing metals range from Invar near 1.2 ppm/°C to aluminum at 23.6 ppm/°C, an almost 20:1 spread. Because mismatched metals expand at different rates, a CTE mismatch between a carrier, a substrate, and a solder joint generates shear stress over every temperature cycle. Matching CTE across the stack is therefore a core discipline in microwave package, carrier, and thermal management design.

Unit: ppm/°C (10^-6/K)

Cu / Al: 16.5 / 23.6 ppm/°C

Kovar / Invar: 5.5 / 1.2 ppm/°C

Why Metal CTE Drives RF Package Reliability

Every metal expands when heated because thermal energy increases the average spacing between atoms in its lattice. The coefficient of thermal expansion is simply the rate of that growth normalized to length and temperature. In a microwave assembly the consequences are mechanical rather than electrical at first glance: a copper carrier expands roughly three times faster than the GaAs or GaN die soldered on top of it, so as the part heats from a 175 °C reflow down to a 25 °C bench, the two materials try to shrink by different amounts and the solder bond absorbs the difference as shear strain. Over hundreds or thousands of thermal cycles that strain fatigues the joint, and in the worst case it fractures the semiconductor itself.

This is why package engineers do not pick metals for cost or machinability alone. A glass-to-metal feedthrough must use a sealing alloy whose CTE tracks the borosilicate glass across the full sealing temperature range, which is precisely why Kovar (an iron-nickel-cobalt alloy near 5.5 ppm/°C) became the standard hermetic lead material. Where dimensional stability dominates, Invar and its low-expansion cousins hold geometry to a few ppm. Where heat removal dominates, designers reach for high-conductivity copper or composites such as CuMo and CuW that trade some conductivity for a CTE tuned to silicon, GaAs, or alumina.

CTE is also mildly temperature dependent. Most handbook values are quoted as a mean over 20 to 100 °C, but the instantaneous coefficient rises with temperature for ordinary metals and behaves anomalously for Invar near its Curie point. For wide-temperature military and space hardware (often -55 to +125 °C), engineers integrate the curve rather than assuming a constant slope.

Governing Relationships

Linear thermal expansion:
ΔL = α_L × L₀ × ΔT

Volumetric expansion (isotropic):
α_V ≈ 3 × α_L

Differential (mismatch) strain:
ε = (α₁ − α₂) × ΔT = Δα × ΔT

Constrained peak stress in stiffer member:
σ ≈ E × Δα × ΔT

Where α_L = linear CTE (ppm/°C), L₀ = original length, ΔT = temperature change from the stress-free state, E = elastic modulus. Example: Cu (16.5) on GaAs (5.7) over ΔT = 150 °C gives ε ≈ 10.8 × 10^-6 × 150 ≈ 1620 µε.

CTE of Common RF Metals and Alloys

Metal / Alloy	CTE (ppm/°C, 20-100 °C)	Thermal Cond. (W/m·K)	Typical RF Use
Invar (Fe-36Ni)	1.2	~13	Dimension-critical waveguide, reflectors
Tungsten (W)	4.5	~170	Heat spreaders, CuW base metal
Molybdenum (Mo)	4.8	~138	CuMo carriers, flanges
Kovar (Fe-Ni-Co)	5.5	~17	Glass-to-metal hermetic seals, leads
CuW (85/15)	~6.5	~190	GaAs / GaN die carriers
CuMo (15/85)	~7.0	~160	Si / GaAs carriers, balanced CTE
Gold (Au)	14.2	~317	Bonding, plating, AuSn solder
Copper (OFC)	16.5	~390	Carriers, heat sinks, cavity walls
Brass (C260)	~19	~120	Connectors, machined fittings
Aluminum 6061	23.6	~167	Housings, chassis, heat sinks

Common Questions

Frequently Asked Questions

What are the CTE values of common RF package metals?

Near room temperature: OFC copper ≈ 16.5 ppm/°C, aluminum 6061 ≈ 23.6, brass ≈ 19, gold ≈ 14.2, molybdenum ≈ 4.8, tungsten ≈ 4.5, Kovar ≈ 5.5 (30 to 200 °C), and Invar ≈ 1.2 up to its inflection near 230 °C. CuMo and CuW composites are tailored between 6 and 11 ppm/°C by metal ratio, which is why Kovar pairs with borosilicate sealing glass while aluminum housings need compliant solders or strain relief.

How do I calculate thermal stress from a CTE mismatch in a soldered RF assembly?

Differential strain is ε = Δα × ΔT, measured from the solder solidus down to operating temperature. For a fully constrained joint the peak stress in the stiffer member approaches σ ≈ E × Δα × ΔT. A GaAs MMIC (5.7) on a copper carrier (16.5) over a 150 °C swing yields about 1620 µε, enough to fatigue solder or crack die. Designers mitigate with CuMo or CuW carriers, AuSn versus SnPb solder choice, and thicker compliant bond lines.

Why is Invar used for dimensionally stable waveguide and antenna structures?

Invar (Fe-36Ni) holds a CTE near 1.2 ppm/°C because magnetostriction nearly cancels normal lattice expansion below its Curie point. A 1 m Invar run drifts only about 60 µm over a 50 °C swing versus roughly 1180 µm for aluminum, keeping electrical path length and cavity resonant frequency in tolerance. The trade-offs are higher density, low thermal conductivity near 13 W/m·K, magnetic behavior, and a CTE that climbs above the Curie temperature.

Related Terms

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