Materials & Substrates

Conductivity (Metal)

Q: Why does measured RF conductor loss exceed the value predicted from DC conductivity?

Two effects raise loss above the DC prediction. First, skin effect confines current to a thin surface layer whose depth shrinks with the square root of frequency, so at 10 GHz copper carries current in only about 0.66 micrometers of its thickness instead of the full cross section. Surface resistance Rs equals 1 divided by sigma times skin depth and rises with the square root of frequency. Second, real plated or machined surfaces have roughness comparable to the skin depth, and the current must follow the longer path over the peaks and valleys. The Hammerstad correction multiplies the smooth-surface loss by a factor that can reach 1.4 to 2.0 when RMS roughness equals one to two skin depths, which is why low-loss microwave laminates specify rolled or very low profile copper foils.

Q: Is silver plating worth the cost compared with bare copper at microwave frequencies?

Silver has the highest conductivity of any metal at about 6.3e7 S/m versus 5.8e7 S/m for annealed copper, so its surface resistance is roughly 4 percent lower. By itself that improvement is marginal, on the order of a few hundredths of a dB per connector or filter resonator. The real reason silver plating is used on waveguide flanges, cavity filters, and high-Q resonators is that silver oxide remains conductive, whereas copper oxide is a poor conductor that degrades surface resistance and solderability over time. Silver also provides a clean, repeatable contact surface. For the most demanding cavity filters, silver plating on an Invar or copper body can lift unloaded Q by 5 to 10 percent over bare copper once aging is considered.

Q: How does temperature change the conductivity of copper used in RF hardware?

Metal conductivity falls as temperature rises because lattice vibrations scatter the conduction electrons more strongly. For copper the temperature coefficient of resistivity is about 0.0039 per degree C near room temperature, so heating from 20 to 85 degrees C raises resistivity by roughly 25 percent and lowers conductivity by a similar amount. Because surface resistance scales with the square root of resistivity, RF conductor loss rises by about 12 percent over that span. Cooling has the opposite effect; near 4 kelvin oxygen-free copper conductivity can climb by one to two orders of magnitude, which is one reason cryogenic receiver front ends and superconducting-adjacent hardware use high purity copper for their feed lines and shields.

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Symbolized by σ and expressed in siemens per meter (S/m), the electrical conductivity of a metal measures how freely it carries current and is the reciprocal of resistivity. In RF and microwave hardware it directly sets conductor ohmic loss: higher σ lowers surface resistance, but the skin effect means only a micrometer-thin surface layer actually carries the current at gigahertz frequencies. Annealed copper sits at 5.8×10⁷ S/m and silver at about 6.3×10⁷ S/m, while aluminum (3.5×10⁷ S/m) and gold (4.1×10⁷ S/m) trade some conductivity for corrosion resistance or bondability. These values, not the bulk DC resistance, govern insertion loss in transmission lines, the unloaded Q of cavity filters, and the heating in high-power waveguide runs.

Symbol: σ (S/m)

Copper: 5.8×10⁷ S/m

Silver: 6.3×10⁷ S/m

Why Metal Conductivity Drives RF Conductor Loss

At DC, the resistance of a conductor depends on its full cross-sectional area and the bulk conductivity of the metal. At RF this picture changes completely. Time-varying fields force the current toward the conductor surface, and by the low gigahertz range it flows in a layer only a fraction of a micrometer thick. Because the cross section actually carrying current shrinks with frequency, the loss that matters is not bulk resistance but surface resistance R_s, which is governed jointly by the metal's conductivity and the operating frequency. A metal with higher σ produces a thinner skin depth and a proportionally lower R_s, so conductor choice and plating quality become first-order design parameters rather than afterthoughts.

The practical ranking of common conductors is set by their conductivity relative to the International Annealed Copper Standard (IACS), where copper is defined as 100 percent. Silver edges copper at roughly 108 percent IACS, gold sits near 71 percent, and aluminum near 60 percent. Designers rarely chase the last few percent of conductivity for its own sake; instead they weigh it against oxidation behavior, mass, cost, and how the surface bonds or solders. Silver and gold plating are specified mostly because their oxides stay conductive, preserving low loss and stable contact resistance over the life of a connector, filter, or waveguide flange.

Surface roughness interacts strongly with conductivity at millimeter-wave frequencies. When the RMS roughness of a plated or machined surface approaches the skin depth, the current is forced to follow the longer path over the surface profile, and the effective loss can rise well above the smooth-metal prediction even though the bulk σ is unchanged. This is why high-frequency laminates and precision waveguide use rolled or very-low-profile copper and tightly controlled plating.

Skin Depth and Surface Resistance

Skin Depth:
δ = 1 / √(π × f × μ × σ) (m)

Surface Resistance:
R_s = 1 / (σ × δ) = √(π × f × μ / σ) (Ω/square)

Conductivity from Resistivity:
σ = 1 / ρ (ρ in Ω·m)

Where f = frequency, μ = permeability (≈ μ₀ for non-magnetic metals), σ = conductivity, ρ = resistivity. Example: copper at 10 GHz, σ = 5.8×10⁷ S/m → δ ≈ 0.66 μm and R_s ≈ 0.026 Ω/square.

Conductivity of Common RF Metals

Metal	σ (S/m, 20°C)	% IACS	Skin Depth @ 10 GHz	R_s @ 10 GHz	Typical RF Use
Silver	6.3×10⁷	~108%	0.63 μm	0.025 Ω/sq	Cavity filters, flange plating
Copper (annealed)	5.8×10⁷	100%	0.66 μm	0.026 Ω/sq	Microstrip, waveguide, traces
Gold	4.1×10⁷	~71%	0.79 μm	0.031 Ω/sq	Wire bonds, MMIC pads, connectors
Aluminum	3.5×10⁷	~60%	0.85 μm	0.034 Ω/sq	Lightweight waveguide, housings
Brass	1.5×10⁷	~26%	1.30 μm	0.051 Ω/sq	Connector bodies (plated)

Common Questions

Frequently Asked Questions

Why does measured RF conductor loss exceed the value predicted from DC conductivity?

Two effects raise loss above the DC prediction. Skin effect confines current to a layer that thins with the square root of frequency, so at 10 GHz copper conducts in only about 0.66 μm, and R_s = 1/(σδ) rises with √f. On top of that, surface roughness comparable to the skin depth forces a longer current path; the Hammerstad correction can multiply smooth-metal loss by 1.4 to 2.0 when RMS roughness equals one to two skin depths, which is why low-loss laminates use rolled or very-low-profile copper.

Is silver plating worth the cost compared with bare copper at microwave frequencies?

Silver's conductivity (~6.3×10⁷ S/m) is only about 4 percent above annealed copper, so the direct R_s gain is marginal, hundredths of a dB per part. The real benefit is that silver oxide stays conductive while copper oxide degrades surface resistance and solderability with age. On high-Q cavity filters, silver plating can preserve 5 to 10 percent more unloaded Q than bare copper once aging is included.

How does temperature change the conductivity of copper used in RF hardware?

Conductivity falls as temperature rises because phonon scattering increases. Copper's resistivity coefficient is about 0.0039 per °C near room temperature, so heating from 20 to 85°C raises resistivity roughly 25 percent. Since R_s scales with √ρ, conductor loss climbs about 12 percent over that range. Near 4 K, high-purity copper conductivity can rise one to two orders of magnitude, a key reason cryogenic front ends use oxygen-free copper.

Related Terms

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