E59 · Surface spectromicroscopy

LEEM: Measure Island Growth

Can a changing LEEM image tell you a surface island is growing? Only with calibrated contrast. Selecting a different reflected beam can turn the same island from bright to dark without moving its boundary. Recorded electron numbers through a sequence let you estimate its changing area, provided the island shape and the signals from both regions are known.

Record a prescribed island expanding or retreating. Infer its boundary speed from image counts, reverse the selected reflection to test a contrast change, and compare with a stationary island or the retained static energy sweep.

Count-derived circular island radius $\text{—}$

Fitted boundary speed $\text{—}$

Speed difference from known input $\text{—}$

Recorded-region image counts $\text{—}$

More measurements and target

Conditional counting uncertainty of speed $\text{—}$

Fitted initial radius $\text{—}$

Selected exposure midpoint $\text{—}$

Resolved circular-radius frames $\text{—}$

Imaged order $\text{—}$

Selected nominal landing energy $\text{—}$

Recorded region contrast $\text{—}$

Separate micro-LEED counts $\text{—}$

First-order propagation threshold: phase A $\text{—}$

Assumed image FWHM $\text{—}$

Acquisition elapsed time $\text{—}$

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

Record: 0 energy sweep · 1 growth

m_r

1

Initial circular island radius

R_0

0.7\,\mathrm{\mu m}

Prescribed signed boundary speed

v_b

0.05\,\mathrm{\frac{\mu m}{s}}

Stored nominal landing energy

E_0

25\,\mathrm{eV}

Contrast-aperture reciprocal radius

q_a

1.2\,\mathrm{\mathring A^{-1}}

Imaged order: 0 BF, 1 A, 2 B

0

Advanced settings

Dead time after each exposure

t_g

0.6\,\mathrm{s}

Selected stored frame · zero-based

n

23

Counting noise: 1 on

n_P

1

Source-energy standard deviation

\sigma_E

0.15\,\mathrm{eV}

Exposure per image

t

0.4\,\mathrm{s}

Field-aperture center: horizontal

x_s

-1.5\,\mathrm{\mu m}

Field-aperture center: vertical

y_s

0\,\mathrm{\mu m}

Selected-area radius

r_s

0.6\,\mathrm{\mu m}

Learning goal and model limits

Predict, then pick the signal that answers it

Can a changing LEEM image tell you a surface island is growing?

Drag the gold island-edge handle to change the initial radius and acquire a new sequence. Target: speed error at most 10 percent or 0.003 micrometre per second, whichever is larger; conditional counting uncertainty at most 0.002 micrometre per second, at least 20 resolved frames and full island coverage. The static energy-sweep preset keeps its separate dark-field selection target below.

Model assumptions

The growth branch prescribes a circular 1 nm phase-B island, rotated by 30 degrees with a 0.4 eV potential offset, in a 0.6 nm phase-A background. Its radius advances at a supplied constant signed speed and stays zero after dissolution. This is boundary kinematics, not a thermodynamic, nucleation or deposition-rate model for a real material. Each run stores 24 exposures at one fixed landing energy; the user sets exposure duration and the dead time after it, so exposures do not overlap. Every expected image uses the evolving material geometry, four by four spatial samples per pixel and 16 temporal samples per exposure. A padded point-spread convolution is cropped to the 8 micrometre field; the 40 by 40 recorded pixels are 0.2 micrometre apart. Counts use the same source and elastic channel calibrations as the static branch. The inverse reads whole-image counts, timing and independently given signals for the two material states. It fits initial radius and boundary speed to the analytically integrated area of a circle, without accessing the actual radius, growth speed, material mask or micro-LEED counts. It does not discover an arbitrary island shape or identify a new phase. Negative noisy area estimates are retained; a displayed radius requires positive area above a heuristic three-standard-deviation threshold. The local Poisson information estimate is conditional on the supplied optical and shape calibrations, excludes systematic and numerical model error, and is not a confidence interval. Near the field boundary the complete-island calibration fails and the target is rejected. Display colors alone clip, with one fixed scale within each record. The physical surface shows the prescribed instantaneous geometry at the selected exposure midpoint; the count image averages that exposure. Micro-LEED is an independent repeated run of exactly the same prescribed evolution, using a field aperture and a separate diffraction configuration. It is not simultaneous image and diffraction acquisition on one collector. Presentation motion does not advance the recorded growth. The retained static-energy branch is a teaching model of a lossless attractive slab. It is not a dynamical LEED calculation for a real material, and it does not reconstruct thickness. Both domains have an assumed 3 angstrom square lattice. Their thicknesses are 0.6 and 1 nanometre, their rotations 0 and 30 degrees and their relative surface-potential offsets 0 and 0.4 eV. The source energy spread is averaged with a Gaussian before recording. The total elastic reflectivity comes from the exact solution for a symmetric slab. When the first orders can propagate, an invented split sends 40 percent of it into those four orders. The reciprocal-space spots are normalized on the grid, so they conserve that elastic probability. The LEEM image counts are recorded through a contrast aperture. The micro-LEED pattern is a separate exposure, taken with that aperture retracted and a real-space field aperture inserted. Selected-area spectra are sums of stored LEEM pixels. The image stack has 0.5 eV steps and 0.2 micrometre pixels. The 18 keV relay, the spatial-width law and the drawn rays are schematic. Inelastic absorption, multiple-scattering crystal intensities, growth kinetics and real lens fields are left out. Below the local landing threshold, the electrons turn back as mirror electrons before they reach the surface.

Physics tutorial

LEEM: Measure Island Growth

BackgroundA low-energy electron image changes when a surface island grows, but it also changes when the microscope selects a different reflected beam. Record a circular island at one fixed energy, infer its area from electron counts, then reverse the selected reflection. The two movies can look very different while giving the same boundary speed.

Why it mattersInfer the signed boundary speed within 10 percent or 0.003 micrometre per second, whichever is larger, with conditional counting uncertainty at most 0.002 micrometre per second and at least 20 resolved radius frames. Keep the complete island and its image blur inside the field. The static energy-sweep preset retains a different target: isolate phase A above 22 eV with aperture radius 0.2–0.5 inverse angstrom, spread at most 0.3 eV, more than 500 region counts and contrast above 0.8.

Start with the essentials

Focus question: Can a changing LEEM image tell you a surface island is growing?
One-sentence intuition: Reflection selection can reverse the brightness of an unchanged boundary. Growth requires a size history with a known contrast calibration.

Core mathematical model

Prescribed circular boundary motion

\begin{aligned}R(t)&=[R_0+v_bt]_+\\A(t)&=\pi R(t)^2\end{aligned}

The positive-part operation holds the radius at zero after dissolution. The two material states and the boundary speed are assigned inputs; this is a kinematic example, not a thermodynamic growth law.

Area from calibrated image counts

\begin{aligned}S&=\sum_iN_i-nN_{\rm inc}T_A\\\widehat A&=\frac{p^2S}{N_{\rm inc}(T_B-T_A)}\\\widehat R&=\sqrt{\widehat A/\pi}\end{aligned}

The field contains two known material signals. Reversing their contrast changes the sign of the denominator as well as the image. Negative noisy area estimates stay in the fit; a radius is displayed only when the positive area is resolved. The complete island and its blur must remain inside the recorded field.

Average the area over each exposure

\begin{aligned}\overline R&=R_0+v_b(t_0+\Delta t/2)\\\overline A&=\pi\left(\overline R^2+\frac{v_b^2\Delta t^2}{12}\right)\end{aligned}

This expression applies when the radius stays positive throughout the exposure. At dissolution the integration is split at zero radius. The count fit uses actual exposure times, with the chosen dead time between frames; it never receives the input speed or geometry mask.

When a diffracted order can propagate

E_\perp=E_0-\frac{\hbar^2|\mathbf G|^2}{2m_e}

At normal incidence, an elastic order can leave the surface only if its perpendicular kinetic energy is zero or positive. A relative surface potential changes the local landing energy.

Reflection from a symmetric attractive slab

R=\frac{\eta\sin^2(qd)}{1+\eta\sin^2(qd)}

This lossless slab has an attractive 10 eV well and can be solved exactly. The phase the wave gathers inside the film makes the reflectivity oscillate.

Mismatch factor and wave number inside the film

\begin{aligned}\eta&=\frac{V_0^2}{4E(E+V_0)}\\q&=\frac{\sqrt{2m_e(E+V_0)}}{\hbar}\end{aligned}

Thickness, local energy and source-energy averaging each change the intensity curve. One curve therefore does not give a unique thickness for a real material.

Common difficulties

An image change does not identify a growth mechanism

Typical misconceptionIf an island gets brighter, I have measured its growth rate or identified a phase transition.

Better mental modelThe reflection setting, source spread and counting noise also change the image. This example assumes a circular island and known signals for two specified material states. Its fit estimates a boundary speed under those assumptions; it does not identify an unknown phase, recover an arbitrary shape or explain why the material grows.

Run the experiment

01
Measure the recorded growth
Start with the expanding island. Move the selected stored frame in advanced settings, then drag the gold edge handle in the surface detail view to change the initial radius.
What to observe: Frame selection reuses the recorded counts. Changing the initial condition acquires a new sequence. The circular radius in the image comes from the measured area.
02
Reverse contrast without changing the growth law
Choose Reverse contrast, same growth. Compare the image with the fitted speed and its counting uncertainty.
What to observe: The island changes from bright to dark, yet a known two-region calibration can still recover its expanding boundary.
03
Test noise and a retreating boundary
Compare the stationary and retreating islands, then reduce exposure time or choose the closed order.
What to observe: A stationary island can give a small signed apparent speed. A retreating island gives a negative speed. Too little admitted contrast or too few counts prevents a reliable estimate.
04
Inspect diffraction as a separate repeat
Open the micro-LEED data and move its field aperture. Use the static energy-sweep preset to inspect the original fixed domains and energy curves.
What to observe: The evolving micro-LEED pattern is an independent repeated run of the same prescribed growth, not a transformation of the LEEM image counts or simultaneous acquisition on one camera.