# Sheet resistance

Resistor based on the sheet resistance of carbon film

Sheet resistance is a measure of resistance of thin films that are nominally 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).

## Calculations

Geometry for defining resistivity (left) and sheet resistance (right). In both cases, the current is parallel to the direction of the double-arrow near the letter "L".

Sheet resistance is applicable to two-dimensional systems in which thin films are considered as 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: $R = \rho \frac{L}{A} = \rho \frac{L}{W t}$ where $\rho$ is the resistivity, $A$ is the cross-sectional area and $L$ is the length. The cross-sectional area can be split into the width $W$ and the sheet thickness $t$.

Upon combining the resistivity with the thickness, the resistance can then be written as:

$R = \frac{\rho}{t} \frac{L}{W} = R_s \frac{L}{W}$

where $R_s$ is the sheet resistance. If the film thickness is known, the bulk resistivity $\rho$ (in ohm cm) can be calculated by multiplying the sheet resistance by the film thickness in cm.

$\rho = R_s \cdot t$

### Units

Sheet Resistance is a special case of resistivity for a uniform sheet thickness. Commonly, resistivity (also known as bulk resistance, specific electrical resistance, or volume resistivity) is in units of Ω∙cm, which is more completely stated in units of Ω∙cm2/cm (Ω∙Area/Length). When divided by the sheet thickness, ∙1/cm, the units are Ω∙cm∙(cm/cm)∙1/cm = Ω. The term "(cm/cm)" cancels, but represents a special "square" situation yielding an answer in ohms. An alternate, common unit is "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 thusly 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 $R_S=R$.) The unit can be thought of as, loosely, "ohms ∙ aspect ratio". Example: A 3 unit wide by 1 unit tall (aspect ratio = 3) sheet made of material having a Sheet Resistance of 7 Ω/sq would measure 21 Ω, 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 $\overline{\rho}=\frac{1}{\overline{\sigma}}$ of the material:

$R_s = \overline{\rho} / x_j = (\overline{\sigma} x_j)^{-1} = \frac{1}{ \int_0^{x_j} \sigma(x)dx }$

which in materials with majority-carrier properties can be approximated by (neglecting intrinsic charge carriers):

$R_s = \frac{1}{\int_0^{x_j} \mu q N(x) dx}$

where $x_j$ 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 $N_B$ and the surface impurity concentration the sheet resistance-junction depth product $R_s x_j$ can be found using Irvin's curves, which are numerical solutions to the above equation.

## Measurement

A four point probe is used to avoid contact resistance, which can often be the same magnitude as the sheet resistance. Typically a constant current is applied to two probes and the potential on the other two probes is measured with a high impedance voltmeter. A geometry factor needs to be applied according to the shape of the four point array. Two common arrays are square and in-line. For more details see Van der Pauw method.

Measurement may also be made by applying high conductivity buss bars to opposite edges of a square (or rectangular) sample. Resistance across a square area will equal Ω/sq. For a rectangle an appropriate geometric factor is added. Buss bars must make ohmic contact.

Inductive measurement is used as well. This method measures the shielding effect created by eddy currents. In one version of this technique a conductive sheet under test is placed between two coils. This non-contact sheet resistance measurement method also allows to characterize encapsulated thin-films or films with rough surfaces.[1]

A very crude two point probe method is to measure resistance with the probes close together and the resistance with the probes far apart. The difference between these two resistances will be the order of magnitude of the sheet resistance.