Silicon resistivity and doping concentration converter
Find the doping concentration from the resistivity on a wafer spec sheet, or the resistivity from a concentration. The calculation covers n-type (phosphorus, arsenic) and p-type (boron) and includes the drop in mobility as doping increases.
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Results
Enter a positive value that gives a concentration between 1e10 and 1e21 cm⁻³.
Doping concentration N
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Resistivity ρ
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Majority-carrier mobility μ
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Conductivity σ = 1/ρ
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Both axes are logarithmic. The solid line is the selected conductivity type and the dashed line is the other type. The red dot is the current value (shown for 10¹³–10¹⁹ cm⁻³).
Assumes 300 K, full ionization of the dopants, and a negligible minority-carrier contribution. Mobility uses the Caughey–Thomas form μ = μmin + (μmax − μmin)/(1 + (N/Nref)^α) with representative fit values widely used in textbooks (electrons: μmax 1414, μmin 68.5 cm²/V·s, Nref 9.2×10¹⁶ cm⁻³, α 0.711; holes: 470.5, 44.9, 2.23×10¹⁷, 0.719). Compensation (material containing both donors and acceptors) and incomplete ionization at concentrations above 10¹⁹ cm⁻³ are not included.
See it as a graph
Other inputs stay at their current values while the chosen input varies from 1/4 to 4 times its current value. The red dot is the current value.
Resistivity and doping formulas
The conductivity of a semiconductor is the product of how many carriers there are and how easily they move: σ = q(nμn + pμp). In n-type material with fully ionized dopants, n ≈ ND and the minority-carrier contribution is negligible, so the resistivity is ρ = 1/(q·ND·μn). Likewise for p-type, ρ = 1/(q·NA·μp). Because the mobility μ itself depends on concentration, however, resistivity is not simply inversely proportional to doping. This tool calculates mobility with the Caughey–Thomas form μ(N) = μmin + (μmax − μmin)/(1 + (N/Nref)^α), and when converting resistivity to concentration it solves this equation for the concentration numerically.
Assumptions and valid range
The calculation assumes silicon at 300 K containing a single dopant species that is fully ionized, and ignores the minority-carrier contribution. As a guide, it is intended for 10¹³–10¹⁹ cm⁻³. In compensated crystals that contain both donors and acceptors, the carrier density is set by the net concentration while the mobility is reduced by scattering from the total concentration, so the resistivity is higher than this tool gives. Above 10¹⁹ cm⁻³, incomplete ionization and band-gap narrowing come into play, and the deviation from empirical conversion tables grows.
Worked example
With the default p-type, NA = 10¹⁵ cm⁻³, the hole mobility is about 462 cm²/V·s, so ρ = 1/(1.602×10⁻¹⁹ × 10¹⁵ × 462) ≈ 13.5 Ω·cm. For n-type at the same concentration, the electron mobility is about 1362 cm²/V·s, roughly three times larger, giving ρ ≈ 4.58 Ω·cm. Going the other way, a wafer specified as "p-type 10 Ω·cm" works out to about 1.36×10¹⁵ cm⁻³, and "n-type 1 Ω·cm" to about 4.9×10¹⁵ cm⁻³. Raising the concentration 100-fold to 10¹⁷ cm⁻³ gives a p-type resistivity of about 0.197 Ω·cm, not the 0.135 Ω·cm that a simple 1/100 scaling would suggest, because Coulomb scattering from the dopants lowers the mobility to about 317 cm²/V·s.
Values used and their sources
The coefficients of the mobility formula are representative fit values matched to measured silicon mobility, and are widely used in semiconductor device textbooks and tutorials. The elementary charge is q = 1.602×10⁻¹⁹ C. For real wafers, resistivity is usually measured by methods such as the four-point probe and converted to concentration using a conversion standard such as SEMI MF723 (formerly ASTM F723). Treat this tool's results as estimates, and check against a conversion standard or measurements when you need accurate values.
Conductivity is σ = qnμ = 1.6×10⁻¹⁹ × 1×10¹⁶ × 1000 = 1.6 S/cm, and resistivity is its inverse, ρ ≈ 0.63 Ω·cm. ρ ≈ 1/(qnμ) is the basic formula for estimating dopant concentration from wafer resistivity. In practice, conversion tables (such as ASTM standards) that include the doping dependence of mobility are used; for example, a 1–10 Ω·cm p-type wafer corresponds to boron at about 10¹⁵–10¹⁶ cm⁻³.
AThe higher the doping, the higher the resistivity
BDoping and resistivity are unrelated
CThe higher the doping, the lower the resistivity
Show answer and explanation
Answer: C. The higher the doping, the lower the resistivity
More dopant means more carriers and lower resistivity, so wafer resistivity is specified as a proxy for doping concentration: 1–10 Ω·cm p-type corresponds to boron at about 10¹⁵–10¹⁶ cm⁻³, and 0.01 Ω·cm to about 10¹⁹ cm⁻³. At high concentrations, ionized-impurity scattering lowers mobility, so the relationship is not simply inverse; standard conversion tables such as ASTM's are used.
CResistivity is constant regardless of temperature
Show answer and explanation
Answer: A. Resistivity decreases as temperature rises
In an intrinsic semiconductor, the number of carriers excited across the band gap rises exponentially with temperature, so resistivity falls (a negative temperature coefficient). In metals the carrier count is nearly fixed and lattice-vibration scattering increases, so resistivity rises. NTC thermistors exploit this behavior. Note that in doped silicon near room temperature the carrier count is roughly constant, so there is a range where resistance actually rises as mobility drops.
An AC magnetic field induces eddy currents in a wafer whose magnitude depends on its conductivity (sheet resistance); detecting them with a coil measures resistivity without contact. Unlike four-point probing, it neither scratches nor contaminates the surface, so it is used for incoming wafer inspection and for metal-film thickness (derived from sheet resistance), and mapping reveals in-wafer distributions.
FAQ
Why do n-type and p-type silicon have different resistivities at the same concentration?
Because in silicon the electron mobility is about three times the hole mobility. At low concentrations electrons have about 1400 cm²/V·s and holes about 470 cm²/V·s, so at the same concentration n-type has roughly one third the resistivity of p-type.
To make the resistivity 10 times higher, should I reduce the concentration to one tenth?
At low concentrations of 10¹⁵ cm⁻³ or below, mobility is nearly constant, so roughly one tenth is enough. At high concentrations, lowering the doping raises the mobility, which slows the increase in resistivity, so you need to reduce the concentration by more than a factor of ten. Check the change in slope on the chart.
Can I use this to find the doping of a real wafer from its resistivity?
Yes, as an estimate. However, compensation from both donors and acceptors, or thermal donors that form when CZ silicon is annealed around 450 °C, change the relationship between resistivity and concentration. For accurate values, check against a conversion standard or a concentration measurement such as SIMS.
