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Build an Atomic Corral

Reshape an atomic fence, open a gap and move the probe. Read the standing-wave pattern and its local spectrum from the same scattering calculation.

ATOMIC LANDSCAPES

Build the fence. Let electrons draw.

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

SHAPE THE SURFACE

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

A spinless, noninteracting square-lattice surface with finite on-site scatterers. Images show broadened local density of states, not electron trajectories or surface height. This dimensionless model is inspired by atomic corrals; it is not a calibrated Fe/Cu(111) STM simulation.

ATOMS SET THE BOUNDARY

SPECTRAL DENSITY AT THE PROBE

0

SELECTED ENERGY

E/J=-3.5

MOVE / COMPARE / ACQUIRE

Read the pattern. Test the signature.

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: A ring of atoms makes electrons draw rings of their own.
One-sentence intuition: An atomic wall is a collection of scatterers with gaps, not an ideal 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)

Positive broadening produces Lorentzian spectral resolution. This is density per site per hopping-energy unit.

One impurity, a remote change

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

The empty-corral propagator carries the perturbation to the probe; its complex phase can turn a dip into a peak.

Common difficulties

A spectrum is not topography

Typical misconceptionBright ridges are raised metal or electron trajectories.

Better mental modelThe surface is flat. Color encodes local spectral density at one energy. The tip is schematic and has no feedback loop here.

What the model omits

Typical misconceptionThe image quantitatively reproduces the original Kondo mirage experiment.

Better mental modelA spinless, noninteracting square-lattice surface with finite on-site scatterers. Images show broadened local density of states, not electron trajectories or surface height. This dimensionless model is inspired by atomic corrals; it is not a calibrated Fe/Cu(111) STM simulation.

Run the experiment

01
Read the surface
Drag the probe across the map; then drag the energy cursor across the spectrum.
What to observe: Nodes and bright regions have different local spectra.
02
Make the comparison
Open the wall, stretch the ellipse and remove the scattering.
What to observe: Standing-wave contrast depends on geometry, energy and finite scattering strength.
03
Acquire a scan
Scan the surface and spectrum, increase dwell, pin the result and export raw counts.
What to observe: The acquired image contains Poisson shot noise. Its estimate divides the counts by the declared exposure, never by the image maximum.