Calculator

Sheet resistance and resistivity calculator (four-point probe, doping)

Device physics and crystals. Everything is calculated in your browser; nothing you enter is sent to a server.

KEY POINTS

With a four-point probe, sheet resistance is Rs = 4.532 × V/I for a sufficiently large thin film, and multiplying by layer thickness t gives resistivity ρ. Resistivity ρ = 1/(qNμ) depends on both doping N and mobility μ; because mobility falls as doping rises, a tenfold increase in doping lowers resistivity by less than a factor of ten.

Four-point probe: sheet resistance and resistivity

Measured voltage V (mV)Applied current I (mA)Probe spacing s (mm)Sample shapeFilm or layer thickness t (µm)

Sheet resistance Rs9.971Ω/sq

Resistivity ρ = Rs·t·F4.986×10⁻⁴Ω·cm

Geometry correction factor C4.5324

Thickness correction factor F (t/s)1.0000

For an infinitely large, thin sample, Rs = (π/ln2)·V/I ≈ 4.532·V/I. A finite circular sample gets Smits's diameter correction, and a layer whose thickness is not negligible relative to the probe spacing gets the thickness correction (for an insulating substrate). When measuring a diffused layer isolated by a pn junction, measure under conditions where the junction acts as an insulator (reverse bias, low current).

Doping concentration ⇄ resistivity (silicon, 300 K)

Conductivity typeDoping concentration N (cm⁻³)Layer thickness (for Rs) (µm)Resistivity to convert ρ (Ω·cm)

Resistivity ρ = 1/(qNμ)4.583Ω·cm

Mobility μ1362cm²/V·s

Sheet resistance of a uniformly doped layer45,830Ω/sq

Concentration for ρ = 10 Ω·cm4.51×10¹⁴cm⁻³

Mobility at that concentration1384cm²/V·s

Assumes all dopants are ionized at room temperature. Above 10¹⁹ cm⁻³, activation and degeneracy make results drift from measurements, and differences of tens of percent from ASTM F723 (the Thurber tables) can occur.

Mobility and resistivity by concentration

Calculated for silicon (300 K) with the Caughey–Thomas model.
Concentration [cm⁻³]	μn [cm²/Vs]	ρ n-type [Ω·cm]	μp [cm²/Vs]	ρ p-type [Ω·cm]
1.0×10¹⁴	1404	44.5	469	133
1.0×10¹⁵	1362	4.58	462	13.5
1.0×10¹⁶	1184	0.527	429	1.45
1.0×10¹⁷	721	0.0865	317	0.197
1.0×10¹⁸	277	0.0225	153	0.0408
1.0×10¹⁹	115	5.44×10⁻³	71	8.81×10⁻³
1.0×10²⁰	78	8.03×10⁻⁴	50	1.25×10⁻³

Sheet resistance, resistivity and concentration

Sheet resistance Rs [Ω/sq] is resistivity ρ divided by layer thickness t. The resistance between opposite edges of a square film equals Rs regardless of the square's size, hence the unit "ohms per square." It can be measured directly with a four-point probe even when the thickness of a diffused, silicide or metal layer is uncertain, which makes it a basic process control metric.

Resistivity ρ = 1/(qNμ) depends on both doping concentration N and mobility μ. Mobility itself falls as concentration rises (impurity scattering), so a tenfold increase in concentration lowers resistivity by less than a factor of ten. Resistivity alone cannot separate N and μ; Hall effect measurements are used when both are needed.

FAQ

Where does 4.532 come from?

It is π/ln2, the correction factor for an infinitely large, thin sample measured with an in-line four-point probe. Finite samples need further geometry corrections.

What does Ω/sq mean?

The resistance between opposite edges of a square of the film. It is the same for any size of square, so sheet resistance is quoted in ohms per square.

Related equipment and companies

More calculators

Device physics and crystals

Last updated: October 23, 2026