Sheet resistance

The utility of sheet resistance as opposed to resistance or resistivity is that it is directly measured using a four-terminal sensing measurement (also known as a four-point probe measurement) or indirectly by using a non-contact eddy-current-based testing device. Sheet resistance is invariable under scaling of the film contact and therefore can be used to compare the electrical properties of devices that are significantly different in size.

Calculations

Sheet resistance is applicable to two-dimensional systems in which thin films are considered two-dimensional entities. When the term sheet resistance is used, it is implied that the current is along the plane of the sheet, not perpendicular to it.

In a regular three-dimensional conductor, the resistance can be written as$${\displaystyle R=\rho \frac{L}{A}=\rho \frac{L}{Wt},}$$where

• ${\displaystyle \rho }$ is material resistivity,
• ${\displaystyle L}$ is the length,
• ${\displaystyle A}$ is the cross-sectional area, which can be split into:
  • width ${\displaystyle W}$,
  • thickness ${\displaystyle t}$.

Upon combining the resistivity with the thickness, the resistance can then be written as$${\displaystyle R=\frac{\rho }{t}\frac{L}{W}=R_{\text{s}}\frac{L}{W},}$$where ${\displaystyle R_{\text{s}}}$ is the sheet resistance. If the film thickness is known, the bulk resistivity ${\displaystyle \rho }$ (in Ω·m) can be calculated by multiplying the sheet resistance by the film thickness in m:$${\displaystyle \rho =R_s\cdot t.}$$

Units

Sheet resistance is a special case of resistivity for a uniform sheet thickness. Commonly, resistivity (also known as bulk resistivity, specific electrical resistivity, or volume resistivity) is in units of Ω·m, which is more completely stated in units of Ω·m²/m (Ω·area/length). When divided by the sheet thickness (m), the units are Ω·m·(m/m)/m = Ω. The term "(m/m)" cancels, but represents a special "square" situation yielding an answer in ohms. An alternative, common unit is "ohms square" (denoted "${\displaystyle \mathrm{\Omega }◻}$") or "ohms per square" (denoted "Ω/sq" or "${\displaystyle \mathrm{\Omega }/◻}$"), which is dimensionally equal to an ohm, but is exclusively used for sheet resistance. This is an advantage, because sheet resistance of 1 Ω could be taken out of context and misinterpreted as bulk resistance of 1 ohm, whereas sheet resistance of 1 Ω/sq cannot thus be misinterpreted.

The reason for the name "ohms per square" is that a square sheet with sheet resistance 10 ohm/square has an actual resistance of 10 ohm, regardless of the size of the square. (For a square, ${\displaystyle L=W}$, so ${\displaystyle R_{\text{s}}=R}$.) The unit can be thought of as, loosely, "ohms · aspect ratio". Example: A 3-unit long by 1-unit wide (aspect ratio = 3) sheet made of material having a sheet resistance of 21 Ω/sq would measure 63 Ω (since it is composed of three 1-unit by 1-unit squares), if the 1-unit edges were attached to an ohmmeter that made contact entirely over each edge.

For semiconductors

For semiconductors doped through diffusion or surface peaked ion implantation we define the sheet resistance using the average resistivity ${\displaystyle \overline{\rho }=1/\overline{\sigma }}$ of the material:$${\displaystyle R_{\text{s}}=\overline{\rho }/x_{\text{j}}=(\overline{\sigma }{x}_{\text{j}}{)}^{-1}=\frac{1}{{\int }_0^{x_{\text{j}}}\sigma (x)dx},}$$which in materials with majority-carrier properties can be approximated by (neglecting intrinsic charge carriers):$${\displaystyle R_{\text{s}}=\frac{1}{{\int }_0^{x_{\text{j}}}\mu qN(x)dx},}$$where ${\displaystyle x_{\text{j}}}$ is the junction depth, ${\displaystyle \mu }$ is the majority-carrier mobility, ${\displaystyle q}$ is the carrier charge, and ${\displaystyle N(x)}$ is the net impurity concentration in terms of depth. Knowing the background carrier concentration ${\displaystyle N_{\text{B}}}$ and the surface impurity concentration, the sheet resistance-junction depth product ${\displaystyle R_{\text{s}}{x}_{\text{j}}}$ can be found using Irvin's curves, which are numerical solutions to the above equation.