Sheet resistance

Sheet resistance is a measure of resistance of thin films that are namely 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 flowing 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. It is analogous to resistivity as used in three-dimensional systems. When the term sheet resistance is used, it is implied that the current flow 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 ρ 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 clubbing the resistivity with the thickness, the resistance can then be written as:

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

R_s is then the sheet resistance.

Units

Because the bulk resistance is multiplied with a dimensionless quantity to obtain sheet resistance, the units of sheet resistance are 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 1Ω could be taken out of context and misinterpreted as bulk resistance of 1 ohm, whereas sheet resistance of 1Ω/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 per aspect ratio".

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, μ 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_sx_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.

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.

References

General references

• Van Zant, Peter (2000). Microchip Fabrication. New York: McGraw-Hill. pp. 431–2. ISBN 0-07-135636-3. 

• Jaeger, Richard C. (2001). Introduction to Microelectronic Fabrication. New Jersey: Prentice Hall. pp. 81–88. ISBN 0-201-44494-7. 

• Schroder, Dieter K. (1998). Semiconductor Material and Device Characterization. New York: J Wiley & Sons. pp. 1–55. ISBN 0-471-24139-3. 

External links

• Sheet resistivity measurement Measuring sheet resistivity of thin conductive coatings e.g. metallization, wafers, ITO, TCO, CVD, PVD,etc.