Atomic corral / Electron standing waves

Build an Atomic Corral

Why does a ring of atoms on a metal surface make the surface electrons form rings of standing waves? Because electron waves reflected from all around the ring interfere. Each atom in the fence scatters the surface waves, and at a given energy the reflections settle into standing waves: rings where electrons of that energy are likely to be found, separated by rings where they are not. The fence leaks through its gaps, so the pattern depends on the energy, the wall strength and the shape.

Reshape a fence of atoms, open a gap in it and move the probe. The standing-wave pattern of the surface electrons and the local spectrum under the probe both come from one scattering calculation.

Atomic landscapes

Build a fence of atoms and let the electrons draw the pattern.

Change the shape or open a gap. Watch the standing-wave pattern reorganize, then move the probe to read a different local spectrum.

Set up the corral

Probe energy

E/J

-3.5

Major radius

a_{\rm c}/a

16

Minor / major radius

b_{\rm c}/a_{\rm c}

1

Wall scattering strength

U_{\rm wall}/J

8

Opening toward the right

\theta_{\rm gap}/{}^\circ

0

Spectral half-width

\eta/J

0.04

Positions snap to a square lattice. Changing the radius also changes the number of wall sites. Spectral broadening averages resonances; it is not a measured decoherence rate.

Move the probe

x/a

y/a

Surface electrons are spinless and noninteracting and hop on a square lattice. Each fence atom is a finite on-site scatterer. The images show the broadened local density of states, not electron paths or surface height. The model is dimensionless and inspired by real atomic corrals; it is not a calibrated STM simulation of Fe/Cu(111).

Atoms set the boundary; the waves make the pattern

SPECTRAL DENSITY AT THE PROBE

0

SELECTED ENERGY

E/J=-3.5

Compare the prediction with your own scan

Read the pattern and test what causes it

The predicted surface

(x,y)/a=(0,0)

Drag the white cross to move the probe. Arrow keys move one site; Shift moves five. Display scale is fixed across presets.

The spectrum under the probe

DRAG ENERGY

Cyan: with the wall. Dashed gray: bare surface. Gold: acquired counts divided by exposure.

LOCAL DENSITY

0

REFERENCE

0

LOCAL CHANGE

0

Density units: states per site per hopping energy. A local spectrum is not normalized as a position probability.

Simulated acquisition

Make your own noisy scan

Each surface pixel and spectral bin is an independent Poisson count. The scan reveals recorded data row by row; the prediction above stays separate.

Exposure per pixel / spectral bin

\mu=\mathcal E\,\rho J

Counts use a fixed-height, constant tip-DOS, low-temperature narrow-bin idealization. Exposure is a dimensionless count scale, not calibrated seconds. Changing the model or probe clears records; pin first to preserve them.

What is solved · what is schematic

The retarded resolvent is evaluated on a periodic 256 by 256 square lattice. Nearest-neighbor hopping is negative; energy is in hopping units and distance in lattice spacings. The displayed window is 45 by 45 sites. Finite real on-site potentials form the wall. The model keeps multiple scattering to all orders.

H=-J\sum_{\langle r,s\rangle}(|r\rangle\langle s|+|s\rangle\langle r|)+\sum_j U_j|j\rangle\langle j|

G(z)=(z-H)^{-1},\qquad \rho_r(E)=-\operatorname{Im}G_{rr}(E+i\eta)/\pi

All maps and spectra use the same Green function. Positive spectral half-width smooths the spectrum with a Lorentzian; it is not a separately solved bath or a direct coherence-time measurement. The periodic host is numerical, not a claim that the physical surface is a torus. The published control range is checked against a doubled host.

The tip is schematic. Its height, atom size and colors are enlarged for legibility. No tunneling barrier, height feedback, finite-temperature convolution, realistic crystal band structure, spin, Coulomb interaction or Kondo screening is solved. Difference maps subtract two predictions at identical settings; they are not reconstructed from the acquired counts.

Start with the essentials

Focus question: Why does a ring of atoms on a metal surface make the surface electrons form rings of standing waves?
One-sentence intuition: A wall of atoms is a set of separate scatterers with gaps between them, not a perfectly sealed box.

Core mathematical model

The spectrum at one point

\rho(\mathbf r,E)=-\frac{1}{\pi}\operatorname{Im}G_{\mathbf r\mathbf r}(E+i\eta)

A positive broadening gives each level a Lorentzian line shape, which sets the spectral resolution. The result is a density of states per lattice site per unit of hopping energy.

How one impurity changes the spectrum far away

\delta G_{rr}=\frac{U_{\rm imp}G^{\rm c}_{ri}G^{\rm c}_{ir}}{1-U_{\rm imp}G^{\rm c}_{ii}}

The propagator of the empty corral carries the impurity's effect to the probe. Because the propagator is complex, its phase can turn a dip into a peak.

Common difficulties

A spectrum is not topography

Typical misconceptionThe bright ridges are bumps in the metal, or paths the electrons follow.

Better mental modelThe surface is flat. The color shows the local spectral density at one energy. The tip is drawn only as a sketch, and this model has no feedback loop.

What the model leaves out

Typical misconceptionThe image reproduces the original Kondo mirage experiment quantitatively.

Better mental modelSurface electrons are spinless and noninteracting and hop on a square lattice. Each fence atom is a finite on-site scatterer. The images show the broadened local density of states, not electron paths or surface height. The model is dimensionless and inspired by real atomic corrals; it is not a calibrated STM simulation of Fe/Cu(111).

Run the experiment

01
Read the surface
Drag the probe across the map; then drag the energy cursor across the spectrum.
What to observe: Dark nodes and bright regions have different local spectra.
02
Change the setup and compare
Open the wall, stretch the ellipse, then remove the scattering.
What to observe: How strongly the standing waves stand out depends on the corral's shape, the energy and the finite strength of each scatterer.
03
Acquire a scan
Scan the surface and the spectrum, increase the dwell time, pin the result and export the raw counts.
What to observe: The scanned image has Poisson shot noise. The estimate divides the counts by the set exposure; it is never scaled to the brightest pixel.