Skin Effect & Frequency-Dependent

Frequency-dependent checks show how conductor behavior changes with microwave frequency, penetration depth, and kinetic inductance.

#	Parameter	Symbol / Unit	Extraction Method	Typical Q3D Value	Ideal / Optimal	Good Range	Worst Case	Why It Matters	Key Design Note
45	Skin Depth at 5 GHz	delta_s / μm	delta_s = √(2ρ/ωμ); Q3D skin-effect mode at frequency	0.9 μm (Al at RT)	0.5 – 2 μm	0.5 – 3 μm	> 5 μm (film < δ_s)	If metal thickness < δ_s entire cross-section carries current and R_ac ≈ R_dc (good for thin films)	Al at 4K is superconducting so delta_s concept replaced by London penetration depth λ_L: bulk Al ~16–55 nm; thin-film Al (50–200 nm film) typically 60–163 nm — increases as film thickness decreases
46	AC/DC Resistance Ratio	R_ac/R_dc	Q3D frequency sweep; skin-effect solver comparison at DC vs 5 GHz	1.0 – 1.05 (thin film)	≈ 1.0 (thin film < δ_s)	1.0 – 2.0	> 5	Thin-film qubits (t ~ 100–200 nm) operate below skin-depth limit so R_ac ≈ R_dc	Normal-metal (Cu, Au) transmission lines show R_ac/R_dc ~ 3–5 at 5 GHz; use SC lines at mK
47	Propagation Constant (gamma)	α / dB/m, β / rad/m	Q3D RLGC → gamma = √((R+jωL)(G+jωC))	α < 0.1 dB/m (SC CPW)	α < 0.1 dB/m; β = ω√(L’C’)	α 0.1 – 1 dB/m	α > 10 dB/m	α sets transmission line attenuation; β sets phase velocity; both from RLGC per unit length	For long interconnects (> 10 mm) even 0.1 dB/m causes measurable signal loss; use SC Al/Nb
48	Phase Velocity (v_ph)	v_ph / ×10⁸ m/s	Q3D RLGC → v_ph = ω/β = 1/√(L'C’)	1.2 – 1.4 ×10⁸ m/s	1.2 – 1.4 ×10⁸ m/s (CPW on Si)	1.0 – 1.6 ×10⁸ m/s	< 0.8 or > 2.0	Sets resonator physical length for target frequency; L = v_ph/(4f_r) for λ/4 resonator	v_ph = c/√varepsilon_eff; on Si varepsilon_eff ≈ 6.3 → v_ph ≈ 1.19×10⁸ m/s; λ/4 at 7 GHz ≈ 4.25 mm
49	Per-Unit-Length Resistance (R’)	R’ / mΩ/mm	Q3D frequency-dependent R matrix; RLGC R’ vs frequency	< 0.1 mΩ/mm (SC Al)	< 0.1 mΩ/mm	0.1 – 2 mΩ/mm	> 10 mΩ/mm	Distributed series resistance determines attenuation α ≈ R’/(2Z₀); critical for long interconnects	At 4K Al becomes superconducting: R’ → 0 below T_c; use R’ to identify non-SC regions
50	Per-Unit-Length Inductance (L’)	L’ / nH/mm	Q3D RLGC magnetostatic solve	0.3 – 0.5 nH/mm	0.3 – 0.5 nH/mm (50 Ω CPW on Si)	0.2 – 0.8 nH/mm	< 0.1 or > 2 nH/mm	Distributed inductance per mm; with C’ sets Z_0 = √(L’/C’) and v_ph = 1/√(L'C’)	L’ includes both geometric and kinetic contributions; L'_kinetic small for Al (~0.01–0.05 nH/mm)
51	Per-Unit-Length Capacitance (C’)	C’ / pF/mm	Q3D RLGC electrostatic solve	0.1 – 0.2 pF/mm	0.1 – 0.2 pF/mm (50 Ω CPW on Si)	0.05 – 0.3 pF/mm	< 0.02 or > 0.5 pF/mm	Distributed capacitance per mm; with L’ sets Z_0 and varepsilon_eff; narrow gap increases C’ (lowers Z_0)	Check: Z_0 = √(L’/C’) ≈ 50 Ω; v_ph = 1/√(L’×C’) ≈ 1.2×10⁸ m/s; these are consistency checks