# RF Sheet Resistance

### Key Takeaways

• RF sheet resistance varies with frequency due to the skin effect, impacting the performance of high-frequency circuits.

• Accurate measurements and calculations of RF sheet resistance are achieved through methods like the Four-Point Probe and Terahertz Time-Domain Spectroscopy, which are crucial for optimizing RF component design.

• In RF design, increasing metal thickness beyond five skin depths offers minimal reduction in RF resistance, guiding efficient material usage and system performance.

Diagram showing the percentage of RF sheet resistance vs. skin depth distance

RF sheet resistance is also known as surface resistance or surface impedance. Sheet resistance is influenced by the inherent resistivity of a metallic film and its specific thickness.

RSH, or sheet resistance, is measured in ohms per square, a unique measure where the square represents a ratio of length to width, devoid of units. In the context of RF, sheet resistance isn't just a static value. It changes with frequency, which is a critical consideration in RF applications.

## Important Equations for RF Sheet Resistance

 Value Equation DC sheet resistance Incremental RF conductivity Maximum RF sheet conductance

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 Directly Measuring RF Sheet Resistance Four-Point Probe Method This is a common method for measuring sheet resistance. It involves using four equally spaced probes to apply a current and measure the voltage drop in the material. Terahertz Time-Domain Spectroscopy This is a more advanced method suitable for high-frequency applications. It measures the response of the material to a terahertz pulse. Microstrip Line Method This method uses a microstrip transmission line patterned on the material whose sheet resistance is to be measured.

## Sheet Resistance Basics for DC

When dealing with direct current, this sheet resistance is derived by considering both the metal's resistivity and thickness.

### DC Sheet Resistance Equation

rho is the bulk resistivity measured in ohm-meters, and t is the metal thickness measured in meters.

If a material has a standard sheet resistance of 10 ohms/square and is cut into a square shape such that the length and width are the same, then the measured resistance from one edge to the other will be 10 ohms no matter what size the square is.

## Sheet Resistance as a Function of Frequency

It's commonly overlooked that sheet resistance varies with frequency. In radio frequency (RF), relying solely on the definition of direct current (DC) sheet resistance can lead to inaccuracies. The idea that sheet resistance remains constant holds true only for conductors whose thickness is less than their skin depth, a function of the skin effect. While this is often true for thin-film resistors, it does not apply to transmission lines.

### The Skin Effect

As frequency increases, the effective depth at which current flows (skin depth) decreases, concentrating the current near the surface of the conductor. This effect causes the sheet resistance to vary with frequency. At higher frequencies, the skin depth becomes so small that the resistance can significantly differ from the DC value.

## Sheet Conductivity

Sheet conductivity is the inverse of sheet resistance, and is measured in Siemens per square or mhos per square. This measure becomes particularly useful when dealing with conductors made up of multiple layers. In such cases, the conductivities of the different layers are summed in parallel, and then this total is inverted to yield a collective sheet resistance.

## Calculating Conductivity

The skin effect is accounted for by incorporating a decreasing exponential factor. This factor’s exponent is inversely related to a parameter known as skin depth.

The depth-wise variation of metal conduction in relation to RF is described by a negative exponential function of the depth, measured in units of skin depth. At the metal's surface, conduction is at its peak, with the metal exhibiting its full resistivity (rho) as it would in a DC scenario. As depth increases, the conductivity diminishes: at one skin depth, it drops to 36.8% of the RF Sheet Resistance; at two skin depths, it's just 13.5%; and so forth. By the time the depth reaches five skin depths, the conductivity is reduced to a mere 0.7% of its original value.

This drastic reduction at five skin depths forms the basis for the rule of thumb in RF design, where adding significantly more metal beyond this depth yields minimal reduction (only about 0.7%) in RF resistance, rendering it largely unnecessary.

### Maximum Incremental RF Conductivity

Incremental RF Conductivity =

rho is bulk resistivity (in ohm-meters), t is the depth into the metal, and delta_s is the skin depth. This incremental RF conductivity value is measured in Siemens/meter.

Maximum Sheet Conductance

The incremental RF conductivity can be integrated to get the maximum RF sheet conductance, measured in Siemen-squares:

## RF Sheet Resistance Applications

Accounting for RF sheet resistance is particularly crucial in evaluating the attenuation in strip conductors like microstrips. High-frequency signals are sensitive to the resistance characteristics of the materials they pass through. The sheet resistance affects signal attenuation, power loss, and the overall efficiency of the RF component or circuit.

## Cadence AWR Software and RF Sheet Resistance

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