Quantum mechanics: step, barrier and well
An electron can appear on the other side of a wall that, according to all of classical physics, it should not be able to cross. That result is not a laboratory curiosity: it is the reason a chip cannot be thinned beyond a certain limit.
01The equation that replaces Newton
If a particle is also a wave, there must be an equation that governs that wave, just as Newton's second law governs a trajectory. That equation was written by Schrödinger in 1926, building directly on the de Broglie hypothesis from the previous topic. In its time-independent form:
None of the four cases has an exact solution in classical mechanics. All four do have one with this equation, and all four end up being the model of a real device.
02The step: reflection where there should be none
A classical particle with energy \( E \) greater than a potential step \( V_0 \) always gets past it, only slower. The Schrödinger equation gives a different answer: part of the wave is reflected, exactly as light is partially reflected on changing medium.
It is the same mechanism, in miniature, that makes an electron slow down and partly bounce back on entering a region with a different potential inside a crystal: every interface in a semiconductor device is, in this sense, a step.
03The barrier: tunneling
If the step has a finite width \( L \) and then drops again, the most radical of the four results appears: with \( E < V_0 \) —which classical physics forbids without exception— the particle has a nonzero probability of appearing on the other side. The wave function does not cut off abruptly inside the barrier: it decays exponentially, and if the barrier is narrow, some wave is still left when it ends.
Three real devices, three different barrier heights, the same formula. You vary the width and watch how many orders of magnitude the tunneling probability changes.
04The infinite well and the harmonic oscillator: quantized energies
Confining a particle between two infinite walls —a well of width \( L \)— requires the wave function to vanish at the edges, just like a guitar string fixed at both ends. That allows only certain wavelengths, and therefore only certain energies.
The harmonic oscillator —a parabola-shaped potential, the model of any small vibration, from a spring to an atom in a crystal lattice— gives a different but equally quantized result:
05Where it appears in electronics
| Device | What it uses | Consequence |
|---|---|---|
| Scanning tunneling microscope (STM) | Tunneling through vacuum | Atomic resolution: the current changes by an order of magnitude per angstrom |
| Advanced-node MOSFET | Tunneling through the gate oxide | Set a physical limit —not a manufacturing one— on thinning the SiO₂ oxide below 1.2 nm |
| Tunnel diode (Esaki) | Tunneling through a heavily doped p-n junction | Negative resistance: the current decreases as the voltage rises, over one stretch |
| Flash memory | Fowler-Nordheim tunneling (triangular barrier, with field) | Writes and erases with no moving parts, through the same oxide that limits the MOSFET |
| SIS junction (superconductor-insulator-superconductor) | Tunneling through a 1-2 nm oxide | Superconducting gap spectroscopy (Giaever, 1960) and the basis of SQUIDs |
For twenty years, shrinking a transistor was mainly a lithography problem. Around 2005 it stopped being one: with the gate oxide at 1.2 nm —about four atoms thick— the tunneling leakage current became comparable to the useful current of the transistor. A better factory was not needed; a different physics was. The solution was to replace SiO₂ with dielectrics of higher dielectric constant (high-k), which allow the same capacitance with a thicker layer and less tunneling.
06In the lab
Calculate the reflection coefficient \( R \) of an electron with \( E = 3 \) eV facing steps of 0.5, 1 and 2 eV. Verify that \( R \to 0 \) when \( V_0 \to 0 \) and that it grows with the height of the step.
With the lab on this page, calculate by how many decades \( T \) changes when the barrier width is moved by 2 Å, for the three devices. Relate the result to why an STM needs to isolate picometer vibrations.
Calculate the first three allowed energies of an electron confined in an infinite well of 2, 5 and 10 nm. Plot \( E_1 \) against the width and explain why the smallest quantum dots emit higher-energy (bluer) light.
07Common mistakes
- Believing that with \( E > V_0 \) there is never any reflection. Quantum mechanically there is always some, even if small.
- Thinking that tunneling violates conservation of energy. The particle that crosses keeps its energy \( E \); what changes is that the probability of finding it on the other side is no longer zero.
- Using the formula \( T \approx e^{-2\kappa L} \) with very narrow or very low barriers. It only holds if \( \kappa L \gg 1 \); otherwise the full formula is needed.
- Forgetting the zero-point energy of the well and of the oscillator. Neither of the two allows zero energy, not even in the ground state.
- Confusing Fowler-Nordheim tunneling (with field, triangular barrier) with simple tunneling through a rectangular barrier. They are relatives, not the same calculation.
- Thinking that the gate-oxide limit was a manufacturing problem. It was a physical limit: tunneling exists even if the oxide is perfect.
08Self-assessment
What does quantum mechanics predict at a potential step with \( E > V_0 \)?
That part of the wave is reflected, something classical mechanics does not allow if the particle has energy to spare.
What does the tunneling probability depend on most strongly?
On the width of the barrier: \( T \) falls exponentially with \( L \), so one angstrom of difference can change it by an order of magnitude.
Why did the chip industry stop being able to thin the SiO₂ gate oxide?
Because below 1.2 nm the tunneling leakage current becomes comparable to the useful current of the transistor: it is a physical limit, not a manufacturing one.
What do the energies of an infinite well and of a harmonic oscillator have in common?
Both are quantized and neither allows zero energy: both have a zero-point energy in their ground state.
How do the energy spacings of the well and the oscillator differ?
In the well the levels grow as \( n^2 \), ever more widely spaced; in the oscillator they are equally spaced, each one \( \hbar\omega \) from the previous one.
What does a scanning tunneling microscope measure?
The current that tunnels through the vacuum between a tip and a surface; since that current depends exponentially on the distance, it resolves relief of atomic size.
09Further reading
- Robert Eisberg and Robert Resnick. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles. 2nd ed., John Wiley & Sons, 1985 (Spanish edition, “Física cuántica: átomos, moléculas, sólidos, núcleos y partículas”, Limusa). The four potentials of this topic, solved step by step with the Schrödinger equation.
- C. Julian Chen. Introduction to Scanning Tunneling Microscopy. 2nd ed., Oxford University Press, 2008. The formula \( \kappa \approx 0.512\sqrt{\varphi} \) and its role in the atomic resolution of the STM, from the specialist who wrote the reference book on the subject.
- Simon M. Sze and Kwok K. Ng. Physics of Semiconductor Devices. 3rd ed., Wiley, 2007. The MOSFET chapter develops the tunneling leakage through the gate oxide and the physical limit that forced the switch to high-k dielectrics.
- Ivar Giaever. Energy Gap in Superconductors Measured by Electron Tunneling. Physical Review Letters, vol. 5, 1960. The experiment that opened tunneling spectroscopy and earned him the 1973 Nobel Prize.