Skin depth is often used to describe the behavior of current flow through circuit conductors, i.e., the copper on a PCB, especially at RF/microwave frequencies. Direct current (DC) may exhibit evenly distributed flow through a conductor such as copper, but high-frequency, sinusoidal current experiences changing energy density as it flows through a PCB’s conductors, with areas of lower and higher current density relative to the surface of the conductor. The skin depth of any conductor is measured from its surface to the point at which the current density drops below 1/e. It is a parameter of circuit laminates to consider when designing high-frequency transmission lines and circuits and a factor to include in any circuit-modeling software for a given RF/microwave circuit material.

In mathematical terms, skin depth, ð, is equal to

ð = (1/πfµσ)^0.5

where f is frequency, µ is permeability, and σ is conductivity. From this equality, and frequency in the denominator, one of the first things that is apparent about a conductor’s skin depth is that it is inversely proportional to frequency: the skin depth is not as deep at higher frequencies.  

Current Density

At DC, 100% of a conductor is used by the current. Any cross-sectional view of a conductor connected to DC would show the same amount of current throughout the conductor. But for current flowing with sinusoidal frequency the amount of current would be different throughout the conductor: the outer skin of the conductor would have greater current density than the inner, middle part of the conductor. As implied by the equality for skin depth, current density moves increasingly to the outer skin of the conductor with increasing frequency. At very high frequencies, there is little or no current density in the middle of a conductor and most of the current density can be found on the outer skin of the conductor. In fact, higher frequencies will result in lower skin depths for a conductor.

What are actual skin depths for an often-used conductor? For copper, with a value of µ approximately equal to 1 and σ of approximately 5.8 ×10⁷ S/m, the trend of shrinking skin depth with increasing frequency is clear. The skin depth for copper at 500 MHz is 2.95 µm (0.116 mils), at 1 GHz is 2.09 µm (0.082 mils), at 10 GHz is 0.66 µm (0.026 mils), at 50 GHz is 0.30 µm (0.012 mils), and at 80 GHz is 0.23 µm (0.009 mils). Clearly, most of the current density at millimeter-wave frequencies lies near the surface of a copper conductor.

At what point might copper on a circuit be too thin to be a good conductor? Some applications benefit from circuit materials with especially thin copper conductors, since thinner copper lends itself to better etching control of PCB circuit features, such as stripline and microstrip transmission lines. Since the size of circuit features is a function of wavelength, shrinking with increasing frequency, tight etching control of a PCB’s copper conductors is required for higher-frequency RF/microwave circuits, especially at millimeter wave frequencies. Quarter-ounce (0.25-oz.) copper is considered very thin copper, with a nominal thickness of 8.89 µm (0.35 mils). Compared to some of the example thicknesses, this thin copper conductor provides adequate skin depth for frequencies far below 500 MHz and well above.

Considering the higher surface current density for conductors at higher frequencies, perhaps as much a concern as a conductor’s skin depth is the surface roughness of the copper conductor, at the substrate-conductor interface. Because the current density increases towards the outer surfaces of a copper conductor at higher frequencies, a rougher copper conductor surface, especially at the substrate-conductor interface, can result in increased conductor circuit losses, especially at higher frequencies. 

In addition to adding to conductor loss, a rough surface for a copper conductor can degrade the phase response of a circuit, slowing the phase velocity of the circuit and making the circuit perform as if it was fabricated on a substrate with higher dielectric constant (Dk). A circuit with smooth copper conductor surface will exhibit a lower effective Dk than a circuit with the same dielectric substrate material with rougher copper conductor surface. The frequency at which the copper conductor surface roughness should be a concern is related to conductor skin depth. When the skin depth is the same dimension or thinner than the conductor surface roughness, the conductor surface roughness can have an impact on RF/microwave circuit performance. For example, electrodeposited (ED) copper typically has surface roughness of about 2 µm RMS and can affect RF performance around 1 GHz. Rolled copper has a much smoother surface roughness of about 0.35 µm RMS and will not impact RF/microwave circuit performance until about 40 GHz.

Not Finished Yet

When designing and modeling high-frequency circuits, skin depth must often be factored at many times a circuit material’s actual skin depth, as much as 5ð, for a meaningful simulation. From the equality equation for ð, a conductor’s conductivity (σ) plays a role in the determination of skin depth. However, the conductivity to be considered is not just that of copper, but also that of any final plated finish that protects a PCB’s copper conductors. Most of the plated PCB finishes are less conductive than copper, resulting in a lower composite conductivity and a thicker skin depth. For an electroless nickel immersion gold (ENIG) finish, the conductivity is a composite of the nickel, gold, and copper conductor. At lower frequencies, all three metal conductors are used for current flow and at higher frequencies, the skin depth decreases and only the nickel-gold gold serves as conductor, with mostly gold as the conductor at very high frequencies.

For an ENIG finish, the nickel is magnetic with an increased value of µ compared to copper (higher than 1), resulting in a decrease in skin depth for a copper conductor when an ENIG finish is applied. The use of such a finish with a copper conductor will result in an offset effect between the nickel’s magnetic properties reducing the skin depth and its lower conductivity increasing the skin depth of the PCB’s copper conductors. In contrast, immersion silver is also used as a final plated finish for copper conductors on PCBs. The silver has higher conductivity than copper and is not magnetic, so the copper conductor skin depth will decrease slightly when using immersion silver finish, although extremely thin immersion silver finishes are typically used so that performance benefits from such a finish may not be apparent except at higher millimeter-wave frequencies, such as 100 GHz and higher.

Skin depth is a circuit property to consider, especially at higher, millimeter-wave frequencies. While the final plated finish can also impact PCB performance, the weight/thickness and type of copper conductor can contribute to RF/microwave circuit performance as much as the quality of the dielectric material and the quality of the substrate. Smooth, thin copper such as rolled copper can provide the skin depth required for good high-frequency performance with low conductor and low overall PCB loss.

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