How this instrument works
The Fermi level marks the energy at which an electron in a semiconductor has even odds of occupying a state, and in an undoped crystal it sits at mid-gap, E_i, by symmetry between electrons and holes. Add donor or acceptor atoms and that symmetry breaks: majority carriers pile up, and the level moves toward whichever band gained carriers. E_f − E_i = kT ln(N ⁄ n_i) is Boltzmann statistics dressed as an engineering formula — N is the net dopant concentration doing the work, n_i is the intrinsic carrier concentration the same crystal would have with no dopant at all, and a process engineer setting an ion-implant dose, or a device physicist checking a diode's built-in potential, reads the shift straight off this ratio.
The logarithm is not decorative. Carrier concentration in a non-degenerate semiconductor grows exponentially with how close the Fermi level sits to a band edge — n = n_i·exp((E_f − E_i) ⁄ kT) is the defining relation this formula inverts. Because that relation is exponential, solving it for the shift necessarily produces a logarithm: multiply the dopant concentration by ten and the shift grows by only kT·ln(10), about 0.060 eV at room temperature, not ten times as much. That single order-of-magnitude step is why doping charts are drawn on log axes and why device engineers think in decades of concentration rather than raw numbers.
The formula assumes non-degenerate doping — the level still sits comfortably inside the band gap, several kT away from either band edge, so Boltzmann statistics stand in for full Fermi-Dirac statistics. Push the dopant concentration high enough — roughly above 10^18 to 10^19 cm⁻³ in silicon, depending on which band — and the level crosses into the band itself; the material turns degenerate, the simple logarithm overstates the shift, and Fermi-Dirac integrals are needed instead. Across the doping ranges used in ordinary diodes and MOSFET channels, though, this logarithmic form stays accurate to a few millivolts.
- Enter the Dopant concentration in cm⁻³ — the net donor or acceptor density from your process specification or doping profile.
- Enter the Intrinsic carrier concentration in cm⁻³; the default 1.5×10¹⁰ is silicon at room temperature, so swap it for germanium or GaAs when needed.
- Set Temperature in kelvin — 300 K for room conditions, or the junction's actual operating temperature if you're checking a hot device.
- Read Ef − Ei in eV: positive means the level sits above mid-gap toward the conduction band (n-type); negative means it sits toward the valence band (p-type).
Worked example — silicon doped to 10¹⁶ cm⁻³
Take a silicon wafer implanted to a dopant concentration of 1×10¹⁶ cm⁻³ — a typical level for an n-type MOSFET source or drain region — held at room temperature, 300 K, with silicon's standard intrinsic carrier concentration, 1.5×10¹⁰ cm⁻³. The ratio N ⁄ n_i works out to 6.667×10⁵, and its natural log is 13.410.
Multiply by kT at 300 K — 8.617×10⁻⁵ eV/K × 300 K = 0.025852 eV — and the formula gives E_f − E_i = 0.025852 × 13.410 = 0.3467 eV. The Fermi level sits about 0.347 eV above mid-gap, roughly 0.21 eV short of silicon's conduction band edge at 300 K — a comfortable margin that keeps the material non-degenerate while still setting unambiguous n-type behavior for the device.
Questions
What does a positive Ef − Ei value mean?
A positive shift places the Fermi level above the intrinsic mid-gap level, toward the conduction band, which is the signature of n-type doping. A negative shift places it toward the valence band, signalling p-type material. The magnitude tells you how strongly doped the crystal is — a larger shift means the majority carrier concentration sits further above the intrinsic level.
Why does the formula use a natural log instead of log base 10?
Because it comes directly from Boltzmann statistics, where carrier concentration depends on energy through a natural exponential, n = n_i·exp((E_f − E_i) ⁄ kT). Inverting an exponential produces a natural logarithm, not a base-10 one; converting would only fold a factor of ln(10) ≈ 2.303 into the constant, which is unnecessary bookkeeping.
What intrinsic carrier concentration should I use for germanium or GaAs?
Use whatever figure matches your material and temperature. Silicon's is about 1.5×10¹⁰ cm⁻³ at 300 K; germanium runs far higher, near 2.4×10¹³ cm⁻³, because its smaller band gap makes carriers easier to excite thermally; gallium arsenide is far lower, around 2×10⁶ cm⁻³, thanks to its wider gap. Swap the field's default for the correct value before reading the result.
Why does raising temperature change the shift even at fixed doping?
Two effects pull opposite ways. The prefactor kT grows linearly with temperature, which alone would widen the shift. But the intrinsic carrier concentration n_i grows exponentially with temperature too, shrinking the ratio N ⁄ n_i and its logarithm faster than kT grows. Net result: the Fermi level shift shrinks and drifts back toward mid-gap as a doped semiconductor heats up — why hot devices lose their sharp doping-defined behavior.
Does this formula still hold for very heavy doping?
Not accurately once the semiconductor turns degenerate — roughly above 10^18 to 10^19 cm⁻³ in silicon, depending on the band. By that point the Fermi level has moved to within a few kT of the band edge or crossed into it, Boltzmann statistics no longer approximate the real Fermi-Dirac occupation well, and the logarithmic formula overstates the shift. Tabulated Fermi-Dirac integrals are needed instead.
Is Ef − Ei the same thing as a diode's built-in potential?
Not directly, but it is the building block. A p-n junction's built-in potential is the sum of the magnitudes of the shift on each side, V_bi = (E_f,n − E_i)/q + (E_i − E_f,p)/q, with both terms computed from this same formula using each side's own doping concentration. Run the calculator once per side and add the results to get the built-in voltage.