Calculator
Example Data Table
| E (eV) | V₀ (eV) | a (nm) | Regime | Exact T | WKB-like T |
|---|---|---|---|---|---|
| 1 | 5 | 1 | Tunneling (E < V₀) | 0 | 0 |
| 0.5 | 3 | 0.5 | Tunneling (E < V₀) | 0 | 0 |
| 8 | 5 | 1.2 | Tunneling (E < V₀) | 0 | 0 |
Formula Used
This tool models a one-dimensional particle scattering from a finite rectangular barrier of height V₀ and width a. The wave numbers are:
- k = √(2mE) / ħ outside the barrier.
- If E < V₀: κ = √(2m(V₀ − E)) / ħ inside the barrier.
- If E ≥ V₀: q = √(2m(E − V₀)) / ħ inside the barrier.
The exact transmission probability for a rectangular barrier is:
- For E < V₀: T = 1 / [1 + (V₀² sinh²(κa)) / (4E(V₀ − E))].
- For E ≥ V₀: T = 1 / [1 + (V₀² sin²(qa)) / (4E(E − V₀))].
For thick barriers, a quick estimate often used is T ≈ exp(−2κa).
How to Use This Calculator
- Enter the particle energy E and choose its unit.
- Enter the barrier height V₀ in the same energy scale.
- Set the barrier width a and select a length unit.
- Pick a particle mass preset or choose “Custom” with your mass unit.
- Press Compute Tunneling to view results above the form.
- Use the download buttons to export the latest result as CSV or PDF.
Professional Notes on Quantum Tunneling
1) Why tunneling matters in real systems
Quantum tunneling enables particles to cross classically forbidden regions. It explains scanning tunneling microscopy, alpha decay in nuclei, and current flow in tunnel diodes. In nanodevices, it also sets practical leakage limits when barriers become only a few nanometers thick.
2) What the calculator models
This calculator uses a one-dimensional finite rectangular barrier. Inputs are particle energy E, barrier height V₀, width a, and mass m. It returns an exact transmission probability for the barrier and, for E < V₀, a quick WKB-like estimate.
3) Typical numbers engineers and physicists see
In solid-state contexts, barriers of 1–3 eV and widths of 0.5–3 nm are common for thin oxides and interfaces. For an electron, changing width from 1 nm to 2 nm can reduce transmission by orders of magnitude, even when E is unchanged.
4) Sensitivity to width: the exponential lever
When E < V₀, the decay constant κ controls the “opacity” of the barrier. Many cases follow the intuition of T ≈ exp(−2κa): doubling a roughly squares the exponential suppression. This is why nanoscale fabrication tolerances matter.
5) Sensitivity to mass: heavier particles tunnel far less
Mass enters κ and the wave numbers through √m. A proton is about 1836× heavier than an electron, so for the same barrier geometry and energy scale, electron tunneling can be observable while proton tunneling may be negligible. Use the mass selector to compare scenarios quickly.
6) Above-barrier transmission and resonances
If E ≥ V₀, the barrier no longer forbids passage, but partial reflection still occurs because the wave number changes inside the barrier. The exact formula contains a sin²(qa) term that can create resonance-like peaks where transmission approaches one for particular widths and energies.
7) Interpreting results for devices and experiments
For tunnel junctions, a transmission around 10⁻³ to 10⁻⁶ can already produce measurable currents depending on area and bias. In STM, the tip–sample gap acts like a barrier; a sub-nanometer change in separation can strongly change current, matching experimental observations.
8) Practical tips and limitations
The model is one-dimensional and assumes a clean, rectangular barrier. Real materials can have effective masses, band bending, and non-rectangular potentials. Still, this calculator is excellent for first-pass estimates, sensitivity checks, and teaching reliably how E, V₀, a, and m shape transmission.
FAQs
1) What does “transmission probability” mean?
It is the fraction of an incoming quantum wave that passes through the barrier rather than reflecting. A value of 0.01 means about one percent transmission under the model’s assumptions.
2) Should E and V₀ use the same unit?
Yes. The calculator converts both to joules internally, but physical meaning is clearest when you use the same energy scale (for example, both in eV) so comparisons stay intuitive.
3) Why are results so sensitive to width a?
For E < V₀, the wave decays inside the barrier. Transmission often behaves like exp(−2κa), so even small increases in width can reduce transmission by large factors.
4) What is the WKB-like estimate used here?
It is a common approximation for “thick” barriers in the tunneling regime: T ≈ exp(−2κa). The calculator also provides an exact rectangular-barrier expression for more reliable values.
5) Can I use effective mass instead of free electron mass?
Yes. Select “Custom” mass and enter an effective mass in units of mₑ or in kilograms. This is useful for semiconductor band-structure approximations and material comparisons.
6) Why can T be near one even with a barrier?
When E ≥ V₀, the barrier is not classically forbidden. The exact solution can show resonance-like conditions where reflections cancel and transmission approaches one for specific widths and energies.
7) What are common pitfalls when choosing inputs?
Mixing units, using unrealistic widths, or selecting the wrong mass are typical issues. Start with eV and nm for nanoscale problems, then adjust one parameter at a time to understand sensitivity and trends.