Fermi Level Calculator — E_F, carrier densities and occupation

The Fermi level calculator above answers the question that sits underneath every semiconductor problem: given a material, a temperature and a doping level, where in the band structure does the Fermi level land? From that single position everything else follows — the electron density, the hole density, the barrier height at a junction, and the probability that any particular state is occupied.

Arb Digital builds free physics calculators that each finish one job properly rather than gesturing at several. This page computes the Fermi level and the carrier densities that go with it. The intrinsic carrier concentration calculator owns the undoped case in more depth, and the photon energy calculator converts a band gap into the wavelength that can bridge it. This page is the doped, finite-temperature statistics.

What This Fermi Level Calculator Does

The Fermi level, written E_F, is the energy at which the Fermi–Dirac occupation function equals exactly one half. It is the electrochemical potential of the electron population. In a metal it lies inside a band and marks the top of the filled states at absolute zero. In a semiconductor it usually lies inside the forbidden gap, where there are no states at all, and that is not a contradiction: the function still tells you how the states that do exist, above and below it, are filled.

The tool first works out the effective density of states in each band from the density-of-states effective masses and the temperature. It then computes the intrinsic carrier concentration, places the intrinsic level E_i near the middle of the gap, and shifts the Fermi level away from that midpoint by whatever the doping demands. The shift is upward for donors and downward for acceptors, and its size is logarithmic in the doping, which is why a thousand-fold increase in doping moves the level by only about a fifth of an electronvolt at room temperature.

How to Use It

1. Pick a material or enter your own parameters. The presets load a band gap and a pair of density-of-states effective masses. Everything remains editable, so a custom alloy or a value from your own reference works the same way.
2. Set the temperature. Every result on the page depends on it, and not weakly. The intrinsic concentration in particular changes by orders of magnitude across a modest temperature range.
3. Choose the doping type and the net concentration. Enter the net figure, donors minus acceptors or the reverse, because partial compensation is common and only the difference matters to the electrostatics.
4. Set the probe energy if you want an occupation number. It is measured upward from the valence band edge, so zero is the valence edge and the band gap value is the conduction edge.
5. Read the note under the grid. It reports the effective densities of states, the intrinsic concentration, and a warning when the Fermi level comes close enough to a band edge that the Boltzmann approximation is no longer trustworthy.

The Formula: How the Fermi Level Is Calculated

The occupation probability of a state at energy E is the Fermi–Dirac function, f(E) = 1 ÷ (1 + exp[(E − E_F) ÷ kT]). OpenStax University Physics Volume 3, section 9.4 on the free electron model of metals, introduces this factor and its zero-temperature limit, in which every state below E_F is filled and every state above it is empty. At any finite temperature the step softens over a width of a few kT, and that softening is the entire reason a semiconductor conducts at all.

The effective density of states in the conduction band is N_c = 2(2πm_n*kT ÷ h²)^3/2, with the same expression using the hole mass giving N_v. The Boltzmann constant used here is exactly 1.380649 × 10⁻²³ J/K, which NIST lists with no uncertainty because the kelvin has been defined in terms of it since 2019. With those two quantities the intrinsic concentration is n_i = √(N_cN_v) exp(−E_g ÷ 2kT), and the intrinsic level sits at E_i = E_g/2 + (3/4)kT ln(m_p*/m_n*) above the valence edge — slightly off centre whenever the two effective masses differ.

For an n-type sample, charge neutrality with full ionisation gives n = N_d/2 + √((N_d/2)² + n_i²), and then p = n_i²/n. The Fermi level follows from E_F − E_i = kT ln(n/n_i). For p-type the same algebra runs with holes as the majority carrier and the shift comes out negative. OpenStax section 9.6 on semiconductors and doping describes how donor and acceptor impurities create the shallow levels that supply these carriers.

Work the defaults by hand. At 300 K, kT is 0.025852 eV. Silicon with m_n* = 1.08 gives N_c = 2.82 × 10¹⁹ cm⁻³ and m_p* = 0.81 gives N_v = 1.83 × 10¹⁹ cm⁻³. Then n_i = √(2.82 × 1.83) × 10¹⁹ × exp(−1.12 ÷ 0.051704) = 2.27 × 10¹⁹ × 3.90 × 10⁻¹⁰ = 8.9 × 10⁹ cm⁻³, which is the familiar room-temperature silicon figure. With N_d = 10¹⁶ cm⁻³, n is 10¹⁶ to four figures, so E_F − E_i = 0.025852 × ln(1.13 × 10⁶) = 0.360 eV. Adding the intrinsic level at 0.554 eV puts E_F at 0.915 eV above the valence edge, which is 0.205 eV below the conduction edge — the number quoted in every device textbook for silicon doped at 10¹⁶.

Why the Fermi Level Moves So Little for So Much Doping

The relationship between doping and Fermi level position is logarithmic, and the practical consequences of that are easy to underestimate. Multiply the doping by ten and the Fermi level moves by kT ln(10), which at room temperature is about 60 millielectronvolts. Multiply it by a thousand and you have moved 0.18 eV. In silicon, whose gap is 1.12 eV, that means the entire useful doping range from 10¹⁴ to 10¹⁹ cm⁻³ shifts the level by less than a third of the gap.

This is why the built-in potential of a p-n junction saturates. The junction potential is the difference between the Fermi levels the two sides would have in isolation, so it grows logarithmically with the product of the doping levels and cannot be pushed much past the band gap in volts no matter how hard the material is doped. It is also why a diode's forward voltage is so stubbornly similar across an enormous range of devices.

The same 60 mV per decade appears in the subthreshold slope of a MOSFET, and for the same reason: the current is set by a Boltzmann factor whose exponent is a potential divided by kT. Our MOSFET calculator works the device side of that behaviour, while this page supplies the underlying statistics.

When the Boltzmann Approximation Stops Working

Almost every convenient semiconductor formula, including n = N_c exp[−(E_c − E_F)/kT], is a Boltzmann approximation to the true Fermi–Dirac integral. It is accurate as long as the Fermi level stays a few kT away from the band edge, which is the usual textbook condition of at least 3kT. At 300 K that is roughly 78 meV.

Push the doping toward 10¹⁹ cm⁻³ in silicon and the Fermi level crosses into the conduction band. The material is then degenerate: it behaves much more like a metal, the exponential relations overestimate the carrier density, and a proper treatment needs the Fermi–Dirac integral of order one half. The calculator flags this case explicitly in the note rather than reporting a number that looks fine and is not.

Fermi Level, Fermi Energy and Work Function

Three terms get used interchangeably and should not be. The Fermi energy, strictly, is the zero-temperature quantity: the highest occupied single-particle energy in a system of non-interacting fermions at absolute zero. The Fermi level is the finite-temperature chemical potential, the energy where occupation equals one half, and it drifts with temperature. In a metal the drift is tiny and the distinction rarely matters; in a semiconductor the Fermi level moves substantially with temperature and the distinction matters constantly.

The work function is a third quantity again: the energy needed to remove an electron from the Fermi level to the vacuum level. It is what governs photoemission thresholds and metal–semiconductor barrier heights, and it is a surface-sensitive property rather than a bulk one. Our photoelectric effect calculator works with the work function directly and shows how it sets the threshold frequency for electron emission.

One more property matters because it is why band diagrams are drawn as they are. At equilibrium the Fermi level is flat across an entire connected system, and everything else bends around it. Band bending at a junction is the bands moving, not the level.

Reading the Occupation Number Correctly

The occupation figure in the grid is f(E), the probability that a state at the probe energy holds an electron. It is not a carrier density and it is not a density of states. Multiply f(E) by the density of states at that energy and you get a carrier density per unit energy; that product, integrated over a band, is what the carrier densities in the grid actually represent.

Set the probe energy to the Fermi level itself and f returns exactly 0.5, whatever the temperature. Set it one kT above and f falls to 0.269; two kT above and it is 0.119. The function is symmetric about E_F in the sense that the probability of a state below being empty mirrors the probability of an equally distant state above being full — which is the formal statement of electron–hole symmetry in this picture.

A probe energy inside the forbidden gap still gives a well-defined f, which catches people out. The function has nothing to act on where there are no states — but a real crystal has defect and trap levels in the gap, and that same f decides whether they are filled.

Where This Sits Next to the Other Solid-State Tools

This page owns the equilibrium statistics of a doped semiconductor. Around it, the intrinsic carrier concentration calculator concentrates on the undoped baseline and its temperature dependence, and the Boltzmann factor calculator handles the classical exponential that the Fermi–Dirac function reduces to when occupation is sparse.

For transport rather than statistics, the electrical mobility calculator and the drift velocity calculator take a carrier density like the one produced here and turn it into a current. On the optical side, the photon energy calculator converts the band gap into the longest wavelength a material can absorb across it.

Common Mistakes to Avoid

• Treating the Fermi level as an occupied state — in a semiconductor it usually lies in the gap where no states exist. It is a reference energy for the statistics, not a place an electron sits.
• Setting n equal to N_d without checking against n_i — that shortcut only holds when doping comfortably exceeds the intrinsic concentration, and fails in narrow-gap material.
• Using conductivity effective masses in the density of states — the two are different averages over the band structure. Only the density-of-states mass belongs in N_c and N_v.
• Assuming full ionisation at low temperature — dopants freeze out below about 100 K, so the carrier density falls short of the doping level and this page overstates it.
• Holding the band gap fixed across temperature — real gaps shrink as temperature rises. Enter the gap for your working temperature, not the room-temperature value.

Related Free Tools From Arb Digital

For the undoped baseline use the intrinsic carrier concentration calculator, and for the classical limit of the same statistics use the Boltzmann factor calculator. Transport is covered by the electrical mobility calculator and the drift velocity calculator. Device behaviour built on these statistics is handled by the MOSFET calculator, emission thresholds by the photoelectric effect calculator, and band-gap optics by the photon energy calculator. Everything Arb Digital publishes is listed on the free online tools hub.

Frequently Asked Questions

This tool is provided for educational and study use. It assumes parabolic bands, complete dopant ionisation, Boltzmann statistics for the carrier densities and a temperature-independent band gap, so treat its output as a physics teaching result rather than a device-grade simulation.