2
Assuming 1 GHz frequency and 10 S/m bulk conductivity, we have a wave impedance (|η|) of 28.06 Ω. This relatively low wave impedance
prevents the E-fields from penetrating into the silicon.
In conjunction with the very large capacitance (C), the signal also experiences a very large inductance (L), as the magnetic fields (H) are
able to easily penetrate the Si/SiO
2
boundary due to the relatively large skin depth (δ) associated with the H-fields.
The skin depth (δ) can be defined by Equation (4).
Equation (4)
Quantifying SWM Magnitude and Frequency Span with RLGC Equivalent Model
Now that we have a basic, fundamental understanding of the mechanism of SWM, we can construct an equivalent lumped resistance,
inductance, conductance, and capacitance (RLGC) model in Keysight ADS, which allows quick approximation of the SWM magnitude and
bandwidth.
Figure 2 depicts the SWM RLGC equivalent circuit model of a TSV signal/ground pair. Assuming 10 Ω/cm silicon bulk resistivity, 100 um
TSV height, 100 um TSV pitch, 30 um TSV diameter, and a SiO
2
thickness of 0.5 um.
Figure 2. SWM RLGC equivalent circuit model of a TSV signal/ground pair