Model quantum occupancy across energy levels and temperatures. Analyze state probabilities with reliable physics-focused calculation outputs.
| Energy E (eV) | Occupation Probability f(E) | Occupied States N·f(E) |
|---|---|---|
| 0.1000 | 0.99956353 | 999,563.53 |
| 0.2000 | 0.95684522 | 956,845.22 |
| 0.2800 | 0.50000000 | 500,000.00 |
| 0.3600 | 0.04315478 | 43,154.78 |
| 0.5000 | 0.00043359 | 433.59 |
The graph tracks how occupation probability changes with energy, highlights the submitted point, and marks the selected Fermi level.
The Fermi-Dirac distribution gives the probability that a quantum state at energy E is occupied by a fermion in thermal equilibrium:
f(E) = 1 / [exp((E - Ef) / (kT)) + 1]
Here, Ef is the Fermi energy, k is the Boltzmann constant, and T is absolute temperature. The calculator also estimates occupied states using N · f(E) and electron count using g · N · f(E), where g is degeneracy and N is the available state count.
Fermi-Dirac statistics describe the chance that a fermionic state is occupied at thermal equilibrium. This calculator converts energy, Fermi energy, and temperature into a probability between zero and one. States below the Fermi level are usually highly occupied, states above it are weakly occupied, and E equal to Ef gives 0.5 exactly. That midpoint is a reliable validation checkpoint.
Temperature sets the steepness of the distribution curve. Low temperature keeps the transition sharp, while higher temperature broadens it and redistributes population around Ef. The main energy scale is kT, and 4kT is a practical broadening estimate. Near 300 K, kT is about 0.0259 eV, so small energy shifts can materially change occupancy in semiconductor calculations.
The core term is (E minus Ef) divided by kT. A strong negative value means the state is almost filled. A strong positive value means it is almost empty. A value near zero places the energy inside the transition zone. Because the ratio is dimensionless, it lets users compare different temperatures on a consistent physical basis for analysis and reporting.
The calculator also estimates occupied states with N multiplied by f(E). That translates probability into a useful count tied to available states. Including degeneracy extends the estimate to g times N times f(E). If N equals 1,000,000 and f(E) equals 0.62, the occupied-state estimate becomes 620,000 before degeneracy scaling.
The graph shows how occupancy changes across the selected energy interval and marks the submitted energy on the curve. A steep drop indicates limited thermal smearing, while a gentler drop indicates stronger temperature influence. When the marker lies near the midpoint, the operating condition is sensitive and small input changes can noticeably shift occupation probability during parametric studies.
This calculation supports semiconductor analysis, carrier statistics, solid-state teaching, and quick electronic screening. It helps identify whether a level is nearly full, half occupied, or nearly empty under specified conditions. Although it does not replace a full density-of-states integral, it provides fast, traceable insight for reports, design reviews, classroom demonstrations, preliminary modeling, educational lab exercises, and validation.
It means the evaluated energy equals the Fermi energy at the chosen temperature. That state has a fifty percent occupation probability under thermal equilibrium.
States above the Fermi reference require more thermal excitation to be occupied. As energy rises, the exponential term grows, so occupancy falls rapidly.
Yes. It is useful for quick semiconductor occupancy estimates, especially when comparing energy levels near conduction, valence, or defect-related references.
4kT is a practical estimate of the thermal broadening region around the Fermi level. It helps describe how wide the transition zone becomes.
They convert probability into estimated occupied counts. That makes the output easier to interpret for physical systems with multiple equivalent states.
No. It evaluates occupancy at selected energies and sample points. Full carrier modeling usually also integrates a density-of-states function across energy.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.