Calculation results
Calculation details
Experimental fitting result
Sweep data
| Input | Transmission | Current density |
|---|
Formula used
How to use this calculator
Select a calculation mode and a suitable barrier model. Enter electron, barrier, voltage, area, and material properties. Press Calculate for results, warnings, charts, and calculation steps.
Use parameter sweep mode to study sensitivity across a chosen range. Use fitting mode for measured voltage-current pairs. Export results after reviewing every assumption and model limitation.
Example data
| Scenario | Energy | Barrier | Width | Suggested model |
|---|---|---|---|---|
| Vacuum nanogap | 1.0 eV | 4.5 eV | 1.0 nm | Exact or WKB |
| Oxide tunnel junction | 0.5 eV | 3.1 eV | 1.5 nm | Simmons |
| Strong field emission | 0.0 eV | 4.5 eV | Field controlled | Fowler–Nordheim |
| Asymmetric dielectric | 0.8 eV | 3.2 to 2.1 eV | 2.0 nm | Trapezoidal WKB |
| Resonant double barrier | 0.3 eV | 1.0 eV | 0.7 nm each | Double barrier numerical |
Model notes and assumptions
The exact rectangular model assumes a one-dimensional, constant barrier. WKB works best when the barrier changes gradually. Accuracy can decrease near sharp classical turning point regions.
The current estimate uses independent tunneling attempts. Simmons assumes a metal-insulator-metal junction and an idealized barrier. Fowler–Nordheim applies mainly to strong-field emission through triangular barriers.
Temperature is reported and included in sweep controls. The present probability equations remain primarily zero-temperature approximations. Devices may require scattering, band, contact, and image-force corrections.
Frequently asked questions
What is electron tunneling?
Electron tunneling permits quantum transmission through a classically forbidden barrier. The probability falls rapidly with increasing barrier width. Barrier height and effective mass also strongly influence transmission.
When should I use the exact rectangular model?
Use it when the potential barrier is approximately constant and one-dimensional. It handles energies below and above the barrier. It offers a useful benchmark for approximate WKB calculations.
When is the WKB approximation reliable?
WKB is strongest for smooth and relatively opaque barriers. It is weaker near abrupt changes and turning points. Compare WKB against exact results whenever practical and available.
Why is the probability extremely small?
Tunneling probability depends exponentially on width and energy difference. Small parameter changes can create enormous result changes. Scientific notation keeps extremely small computed values clearly visible.
What does penetration depth mean?
Penetration depth is the inverse barrier decay constant. It indicates how quickly the wavefunction decreases inside the barrier. Barrier width and penetration depth represent different physical quantities.
How is current estimated?
The simple current mode multiplies charge, attempt frequency, electron count, and transmission. Junction models calculate current density directly. Device geometry converts current density into total electrical current.
What is Fowler–Nordheim tunneling?
It describes field-assisted electron emission through a triangular barrier. Strong electric fields narrow the effective barrier. The model is widely used for vacuum field emission.
What is the Simmons model?
It approximates tunneling current through thin insulating films between metals. The model uses barrier height, thickness, effective mass, and voltage. Real junction asymmetry often requires further model corrections afterward.
Can this calculator fit experimental data?
Yes, fitting mode estimates barrier height and width from voltage-current pairs. It uses a bounded numerical grid search. Treat fitted parameters as preliminary estimates requiring experimental validation.
Can electron energy exceed barrier height?
Yes, but the process is then partly above-barrier transmission. The exact model switches to an oscillatory solution. WKB tunneling formulas are not appropriate in that region.