Physics · Atomic and quantum physics · Year 3

Schrödinger and quantum tunnelling

How does an electron pass through a barrier higher than its energy, and how does an STM “see” atoms?

The situation

Your smartphone stores photographs in flash memory. Every time it writes, electrons “leak” through an insulating layer about 10 nm thick — impossible in classical physics, because the electron's energy is below the barrier. The same phenomenon lets the scanning tunnelling microscope “see” individual atoms: the current flowing between the tip of the needle and the surface is extremely sensitive to distance. How does a particle get through a barrier higher than its own energy, and why does a change of 0.1 nm change the current 10 times?

Topic

Schrödinger equation, tunnelling. Scanning tunnelling microscope: “seeing” atoms

Key ideas

Schrödinger equation
Tunnelling
Scanning tunnelling microscope

Formulas

i ℏ \frac{\partial ψ}{\partial t} = - \frac{ℏ^{2}}{2 m} \frac{\partial^{2} ψ}{\partial x^{2}} + V (x) ψ

The time-dependent Schrödinger equation; $| ψ |^{2}$ is the probability density of finding the electron.

κ = \frac{\sqrt{2 m (V_{0} - E)}}{ℏ}, ψ \propto e^{- κ x}

Inside a barrier the wave function does not oscillate; it decays exponentially.

T = {[1 + \frac{V_{0}^{2} \sinh^{2} (κ a)}{4 E (V_{0} - E)}]}^{- 1} \approx \frac{16 E (V_{0} - E)}{V_{0}^{2}} e^{- 2 κ a}

The probability of passing a rectangular barrier ( $E < V_{0}$ , $a$ is the barrier width).

I \propto e^{- 2 κ d}, κ = \frac{\sqrt{2 m φ}}{ℏ} \approx 11 {nm}^{- 1}

The tunnelling microscope current: $d$ is the distance between tip and surface, $φ$ is the work function (≈ 4–5 eV).

Δ d = \frac{\ln 10}{2 κ}

The change in distance that reduces the current 10 times.

Theory

Wave packet

In quantum mechanics an electron is described by a wave function $ψ (x, t)$ . It is a complex number: the height of the edge of the twisted ribbon in the scene is $Re ψ$ , and its depth is $Im ψ$ . The orange region on the back screen is $| ψ |^{2}$ — the probability of finding the electron at that place. The pitch of the twist of the ribbon is the de Broglie wavelength $λ = h / p$ : if you increase the energy, the twist gets tighter.

What happens inside the barrier?

A classical particle with $E < V_{0}$ is certain to bounce back from a barrier. A wave, however, enters the barrier and decays there as $e^{- κ x}$ . If the barrier is thin, a small but non-zero wave remains on the far side — this is the tunnel effect. The packet splits in two: part is reflected ( $R$ ) and part is transmitted ( $T$ ), with $T + R = 1$ . $T$ depends exponentially on the barrier width: if you double the width, $T$ falls roughly to its own square (1 % → 0.01 %). Reflection also occurs when $E > V_{0}$ — impossible in classical physics too.

Stationary states, transmission and reflection coefficients

If the potential does not depend on time, the equation has solutions of the form $ψ (x, t) = φ (x) e^{- i E t / ℏ}$ — stationary states: in them $| ψ |^{2}$ does not change with time, and $φ$ satisfies the stationary Schrödinger equation: $- \frac{ℏ^{2}}{2 m} φ^{''} + V φ = E φ$ . To the left of the barrier $φ = e^{i k x} + r e^{- i k x}$ (incident and reflected waves), inside it $e^{\pm κ x}$ , and on the right only $t e^{i k x}$ . The condition that $φ$ and $φ^{'}$ are continuous at the boundaries gives $r$ and $t$ , and hence the formula for T(E). Through the probability current $j = \frac{ℏ}{m} Im (φ^{*} φ^{'})$ the transmission coefficient is $T = j_{t r} / j_{i n c} = | t |^{2}$ and the reflection coefficient is $R = | r |^{2}$ , with $T + R = 1$ . The wave packet in the scene is a sum of stationary states with different $E$ ; that is why it moves, and the measured $T$ is the average of $T (E)$ over the packet's spectrum.

Scanning tunnelling microscope

The single atom at the tip of the needle sits 0.3–1 nm from the surface. The vacuum between them is a barrier of height $φ$ (the work function). At a small voltage the current is $I \propto e^{- 2 κ d}$ : for $φ \approx 4.6$ eV, $κ \approx 11 {nm}^{- 1}$ , so moving 0.1 nm closer raises the current ≈ 9 times. A feedback system holds the current constant and moves the needle up and down: the tip “glides” over the atoms, and an image of the surface is drawn from its height (constant-current mode).

Alpha decay and field emission

G. Gamow (1928) explained α decay as the tunnel effect. The α particle inside the nucleus ( $r < R \approx 9$ fm) is held by nuclear forces; outside it is surrounded by a Coulomb barrier ≈ 25–28 MeV high, while its own energy is only 4–9 MeV. The particle strikes the barrier ≈ 10²¹ times a second and each time gets through with probability $e^{- G}$ ; $G$ is very sensitive to energy, so if the energy differs by a factor of ≈ 2 the half-life changes from microseconds to billions of years — the Geiger–Nuttall law ( $\lg T_{1 / 2}$ is almost linear in $Q^{- 1 / 2}$ ). In the “Alpha decay (Gamow theory)” section, calculate it for real nuclei. Field (cold) emission: if a field of $10^{9}$ – $10^{10}$ V/m is applied to a metal surface, the work-function barrier becomes a thin “triangle” and electrons tunnel out of an unheated metal. The current density is given by the Fowler–Nordheim formula: $j \propto E^{2} e^{- b φ^{3 / 2} / E}$ — exponentially sensitive to the field. This is how the field-emission cathodes of electron microscopes and the write–erase process of flash memory work.

About the model

The one-dimensional Schrödinger equation for an electron is solved on the interval $x \in [- 40; 40]$ nm, on a grid with $Δ x = 0.01$ nm, by the Crank–Nicolson method ( $Δ t = 0.04$ fs); at the edges there is a layer that absorbs the wave. The measured $T$ is the probability to the right of the barrier together with that absorbed at the right edge. The packet's energy is slightly “spread” ( $σ_{E} \propto 1 / σ$ ), so $T$ may differ from $T (E_{0})$ in the formula — widen the packet and the difference shrinks. Time is slowed down $10^{14}$ times (1 s = 10 fs). STM current: $I = A e^{- 2 κ d}$ , with $A$ chosen so that graphite gives 1 nA at 0.5 nm; the surface corrugation is approximate.

Did you know?

The scanning tunnelling microscope was created by Gerd Binnig and Heinrich Rohrer in an IBM laboratory; in 1986 they received the Nobel Prize in Physics.
In an experiment published in 1990, IBM scientists used an STM needle to place 35 xenon atoms on a nickel surface and spell out “IBM”.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Detecting particles — Grade 11
Elementary particles — Grade 11
Spin, the Zeeman effect and MRI — Year 1
Many-electron atoms — Year 1
Bands in solids — Year 1
Quantum dots and the potential well — Year 3
The molecule: bonding and spectra — Year 3
Superconductivity — Year 3