Skin Effect & Frequency-Dependent
=====================================

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

.. list-table::
   :header-rows: 1

   * - #
     - 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