Materials & Substrates

Copper

Q: What is the skin depth of copper at common RF frequencies?

Skin depth in copper scales as 1/sqrt(frequency). Using sigma = 5.8x10^7 S/m, the skin depth is about 66 micrometers at 1 MHz, 2.1 micrometers at 1 GHz, 0.66 micrometers at 10 GHz, and roughly 0.30 micrometers at 50 GHz. This means RF current flows in an extremely thin surface layer, so only the outermost copper and its surface finish carry significant current. At 30 GHz the skin depth is about 0.38 micrometers, smaller than typical copper foil surface roughness, which is why roughness directly increases conductor loss at millimeter-wave frequencies.

Q: Why does electrodeposited copper have higher loss than rolled-annealed copper at mmWave?

Electrodeposited (ED) copper foil is grown on a drum and has a rough matte side, with RMS roughness commonly 2 to 6 micrometers for standard foil. Rolled-annealed (RA) copper is mechanically rolled to a much smoother surface, typically 0.3 to 0.7 micrometers RMS. Because RF current concentrates within one skin depth (about 0.66 micrometers at 10 GHz, 0.38 micrometers at 30 GHz), a rough profile forces the current along a longer path, raising the effective resistance. The Hammerstad model predicts a loss-multiplier ceiling of 2x as roughness exceeds skin depth. At 28 GHz the difference between rough ED and smooth RA or low-profile foil can be 0.3 to 0.5 dB/inch on a microstrip line.

Q: What copper conductivity should I use for RF simulation?

The IACS reference (100 percent) corresponds to 5.80x10^7 S/m for annealed copper at 20 C. Many simulators default to 5.8x10^7 S/m, but real plated or foil copper is often modeled at 4.7 to 5.6x10^7 S/m (80 to 97 percent IACS) to account for grain boundaries, oxidation, and plating additives. Conductivity falls about 0.39 percent per degree C rise, so at 85 C copper is roughly 80 percent IACS, dropping to about 75 percent near 105 C. For accurate insertion-loss prediction at microwave frequencies, combine a realistic conductivity with a surface-roughness model such as Hammerstad or Huray rather than assuming ideal bulk copper.

/KOP-er/ (Cu, atomic number 29)

The workhorse conductor of RF and microwave hardware, with a bulk conductivity of 5.8×10⁷ S/m (the 100% IACS reference) that is second only to silver. Because RF current obeys the skin effect and rides in a layer just a few micrometers deep, the copper surface, not its bulk, determines conductor loss. As copper foil cladding on PCB laminate, electroplated waveguide walls, and machined resonator cavities, its conductivity, plating quality, and surface roughness set the insertion loss and unloaded Q of nearly every passive RF structure from HF through millimeter-wave bands.

Conductivity: 5.8×10⁷ S/m (100% IACS)

Skin Depth @ 10 GHz: ≈ 0.66 μm

Temp Coefficient: +0.39% / °C resistivity

Why Copper Dominates RF Conductor Design

Among engineering metals only silver edges out copper for conductivity, and silver costs far more and tarnishes readily, so copper became the default conductor for everything from coaxial center pins to spacecraft cavity filters. At DC and low frequencies the full cross section of a copper trace carries current, but as frequency rises the field is expelled from the interior and current crowds into a thin surface sheet. By 1 GHz that sheet is only about 2.1 micrometers thick, and by 30 GHz it shrinks below 0.4 micrometers. The practical consequence is that thick copper buys you nothing electrically at RF; what matters is the conductivity, cleanliness, and smoothness of the outermost fraction of a micrometer.

This is why copper foil for high-frequency laminate comes in distinct grades. Electrodeposited (ED) foil is plated onto a rotating drum and develops a tooth-like matte side for adhesion, giving root-mean-square roughness of several micrometers. Rolled-annealed (RA) foil is mechanically rolled to a glassy finish near 0.3 micrometers, and reverse-treated and low-profile foils target a middle ground. When the roughness profile becomes comparable to or larger than the skin depth, RF current is forced to follow the contoured surface, lengthening its path and raising the effective sheet resistance. The widely used Hammerstad correction caps this penalty near a factor of two, while the Huray "snowball" model better fits the steep millimeter-wave behavior of modern foils.

Copper conductivity is also temperature dependent, dropping about 0.39 percent per degree Celsius of temperature rise. A power amplifier output network or a high-power filter that heats to 85 C therefore sees copper conductivity drop to roughly 80 percent of its 20 C IACS rating (and near 75 percent by about 105 C), which must be folded into thermal and loss budgets. Plated copper used to metallize waveguide and connectors carries additional caveats: plating additives, grain structure, and a barrier or final finish of nickel or gold all reduce the effective conductivity below the textbook bulk value.

Skin Depth and Conductor Loss

Skin Depth:
δ = 1 / √(π × f × μ × σ)

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

Roughness-Corrected Loss (Hammerstad):
K_SR = 1 + (2/π) × arctan[1.4 × (Δ/δ)²], R_eff = K_SR × R_s

Where f = frequency, μ ≈ 4π×10^-7 H/m, σ = 5.8×10⁷ S/m, Δ = RMS roughness. Example: at 10 GHz, δ ≈ 0.66 μm and R_s ≈ 0.026 Ω/square; with Δ = 1 μm roughness, K_SR ≈ 1.8.

Copper Conductor Materials and Finishes

Material / Finish	Conductivity (% IACS)	RMS Roughness	Typical Use	RF Note
Silver	105%	Plating dependent	Cavity walls, high-Q filters	Lowest loss; tarnishes
Copper (annealed, IACS ref)	100%	Bulk reference	Baseline conductor	Best cost vs. loss
Rolled-annealed (RA) foil	97 to 100%	0.3 to 0.7 μm	mmWave laminate cladding	Lowest foil loss
Electrodeposited (ED) foil	95 to 100%	2 to 6 μm	Standard PCB cladding	High loss above 20 GHz
Copper + ENIG (Ni/Au)	Ni-P ~3 to 5%	Finish dependent	Solderable RF pads	Resistive Ni-P adds skin loss
Aluminum (for reference)	61%	Machining dependent	Lightweight housings	~30% more loss than Cu

Common Questions

Frequently Asked Questions

What is the skin depth of copper at common RF frequencies?

Skin depth scales as 1/√f. With σ = 5.8×10⁷ S/m, copper skin depth is about 66 μm at 1 MHz, 2.1 μm at 1 GHz, 0.66 μm at 10 GHz, 0.38 μm at 30 GHz, and roughly 0.30 μm at 50 GHz. RF current rides in this thin layer, so only the surface copper and its finish carry meaningful current. Above 20 to 30 GHz the skin depth drops below typical foil roughness, which is exactly why surface profile dominates loss at millimeter wave.

Why does electrodeposited copper have higher loss than rolled-annealed copper at mmWave?

ED foil is grown on a drum with a rough matte side, commonly 2 to 6 μm RMS, while RA foil is rolled to about 0.3 to 0.7 μm. Since current concentrates within one skin depth (0.66 μm at 10 GHz, 0.38 μm at 30 GHz), a rough profile lengthens the current path and raises effective resistance, with the Hammerstad model capping the penalty near 2x. At 28 GHz the gap between rough ED and smooth low-profile foil can reach 0.3 to 0.5 dB/inch on microstrip.

What copper conductivity should I use for RF simulation?

The 100% IACS reference is 5.80×10⁷ S/m for annealed copper at 20 °C. Solvers often default to that, but real plated or foil copper is better modeled at 4.7 to 5.6×10⁷ S/m (80 to 97% IACS) to account for grain boundaries, oxidation, and plating additives. Conductivity falls about 0.39% per °C, so copper at 85 °C is near 80% IACS (and about 75% by 105 °C). Pair a realistic σ with a Hammerstad or Huray roughness model for trustworthy insertion-loss predictions.

Related Terms

← Coplanar Waveguide (TL) Copper Adhesion →