Sheet resistance is the resistance of a square piece of a thin material with contacts made to two opposite sides of the square. It is usually a measurement of electrical resistance of thin films that are uniform in thickness. It is commonly used to characterize materials made by semiconductor doping, metal deposition, resistive paste printing, and glass coating. Examples of these processes are: doped semiconductor regions (e.g., silicon or polysilicon), and the resistors that are screen printed onto the substrates of thick-film hybrid microcircuits. 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 R = ρ L A = ρ L W t , {\displaystyle R=\rho {\frac {L}{A}}=\rho {\frac {L}{Wt}},} where
ρ {\displaystyle \rho } is material resistivity,
L {\displaystyle L} is the length,
A {\displaystyle A} is the cross-sectional area, which can be split into: width W {\displaystyle W} , thickness t {\displaystyle t} . Upon combining the resistivity with the thickness, the resistance can then be written as R = ρ t L W = R s L W , {\displaystyle R={\frac {\rho }{t}}{\frac {L}{W}}=R_{\text{s}}{\frac {L}{W}},} where R s {\displaystyle R_{\text{s}}} is the sheet resistance. If the film thickness is known, the bulk resistivity ρ {\displaystyle \rho } (in Ω·m) can be calculated by multiplying the sheet resistance by the film thickness in m: ρ = R s ⋅ t . {\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 Ω/sq. An alternative, common unit is "ohms square" (denoted " Ω ◻ {\displaystyle \Omega \Box } ") or "ohms per square" (denoted "Ω/sq" or " Ω / ◻ {\displaystyle \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 {\displaystyle L=W} , so R s = R {\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.
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