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 asR = \rho \frac{L}{A} = \rho \frac{L}{W t},where • \rho is material
resistivity, • L is the length, • A is the cross-sectional area, which can be split into: • width W, • thickness t. Upon combining the resistivity with the thickness, the resistance can then be written asR = \frac{\rho}{t} \frac{L}{W} = R_\text{s} \frac{L}{W},where R_\text{s} is the sheet resistance. If the film thickness is known, the bulk resistivity \rho (in
Ω·m) can be calculated by multiplying the sheet resistance by the film thickness in m:\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 "\Omega\Box") or "ohms per square" (denoted "Ω/sq" or "\Omega/\Box"), 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, L = W, so 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 \overline{\rho} = 1 / \overline{\sigma} of the material: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):R_\text{s} = \frac{1}{\int_0^{x_\text{j}} \mu q N(x) \,dx},where x_\text{j} is the junction depth, \mu is the majority-carrier mobility, q is the carrier charge, and N(x) is the net impurity concentration in terms of depth. Knowing the background carrier concentration N_\text{B} and the surface impurity concentration, the
sheet resistance-junction depth product R_\text{s} x_\text{j} can be found using Irvin's curves, which are numerical solutions to the above equation. ==Measurement==