E11 · Electron imaging / spectroscopy

SEM Secondary-Electron Topography

Does a brighter surface mean a taller surface? Not necessarily. In a secondary-electron image, brightness depends on how each part of the surface faces the detector: edges glow, slopes turned away fall into shadow, and counting noise adds speckle. Move the detector and the same terrain can look like a completely different object.

Move the side detector around a fixed terrace with three particles on it. Tilt the specimen and change the working distance and landing energy, then compare a saved frame of electron counts with the known height map. Inspecting pixels or changing display gain reuses that same frame. The same terrain can look like a different object.

Saved flank mean counts $0$

Saved flank contrast $0\,\mathrm{\%}$

Flank count CNR $0$

Probe blur proxy $0\,\mathrm{nm}$

More measurements and target

Assumed escape depth $0\,\mathrm{nm}$

Fixed grain height $0\,\mathrm{nm}$

Selected pixel counts $0$

Selected face $\text{—}$

Experiment task $\text{—}$

Simulated examples; each plot identifies its data source.

In-lens collectionUse an axial collection proxy instead of the side detector.Working distance

w

10\,\mathrm{mm}

Specimen tilt

\beta

0\,\mathrm{{}^\circ}

Detector azimuth

\varphi

30\,\mathrm{{}^\circ}

Advanced settings

Landing energy

E_0

10\,\mathrm{keV}

Detector elevation

\theta_d

35\,\mathrm{{}^\circ}

Beam current

I

80\,\mathrm{pA}

Pixel dwell

\tau

10\,\mathrm{\mu s}

Inspect column

j

113

Inspect row

i

104

Display gain

g

1

Learning goal and model limits

Instrument settings

SEM Secondary-Electron Topography

In the saved frame, reach at least 15 percent contrast between the two flanks, a count CNR of at least 5 and a probe-blur estimate below 8 nm. Then reverse the detector and explain why the height map does not change.

Model assumptions

The image is a teaching raster laid out in specimen coordinates: a 192 by 192 grid of sites spanning 1 micrometre, with the same dwell time at each site. It leaves out the foreshortening of a projected image and does not redistribute the probe where terrain blocks it. Yield, isotropic blur and collection are all assumed. Opaque terrain blocks both direct excitation and side collection; axial collection uses a field-guided approximation. Saved counts use a fixed random seed, drawn from a Poisson distribution below a mean of 40 and from a Gaussian approximation above it. The detector field, charging, calibrated efficiency and recovering height from the image are not modeled.

Start with the essentials

Focus question: Does a brighter surface mean a taller surface?
One-sentence intuition: Moving the detector changes the image even though the specimen stays the same. Edge enhancement, line of sight to the detector and counting noise all affect brightness, so brightness alone is an ambiguous measure of height.

Core mathematical model

Electron counting

\begin{gathered}N_0=I\tau/e\\\bar N_{ij}=N_0(G_\sigma*S)_{ij}\end{gathered}

The ideal signal is yield times collection. An assumed isotropic Gaussian blurs it in specimen coordinates, and then counts are sampled with a fixed seed. The 192 by 192 pixel centers span 1 micrometre, and the dwell time applies to each site. This teaching raster stays registered to the specimen, so it leaves out projected foreshortening and the redistribution of a blocked probe.

Side-detector collection estimate

\begin{gathered}C_{\mathrm{side}}=(0.08+0.55c_d)V\\c_d=\max(0,\mathbf n\cdot\mathbf d)\end{gathered}

The direction points from the surface point to the receiving window drawn on the detector. Visibility is zero when that ray points into the surface or the terrain blocks it. This straight-line estimate leaves out field-guided collection around obstacles; in-lens mode uses a separate prescribed estimate. There is no solid-angle or distance calibration.

Contrast and shot-noise test

\begin{gathered}c=\frac{|N_1-N_2|}{N_1+N_2}\\\mathrm{CNR}=\frac{|N_1-N_2|}{\sqrt{N_1+N_2}}\end{gathered}

Both numbers use the two saved pixels closest to the two chosen flanks of the grain, after blur and counting. The CNR divides the count difference by the independent Poisson noise of the two pixels; it is not the uncertainty of a region average. If the total is zero, the flanks remain unresolved. Display gain and pixel inspection cannot change this test.

Common difficulties

Reading brightness as height

Typical misconceptionA bright rim is a taller rim, and an SEM image is already a height map.

Better mental modelThe fixed height map and the profile seen by the opposite detector are counterexamples. Both the generation and the collection of secondary electrons depend on geometry, so a quantitative height needs more information than brightness.

Run the experiment

01
Predict the bright flank
Predict which flank of the central grain will brighten when the detector turns by half a revolution. Compare Surface target and Opposite detector before changing the specimen tilt.
What to observe: The image changes even though the specimen stays the same. Edge enhancement, line of sight to the detector and counting noise all affect brightness, so brightness is an ambiguous measure of height.
02
Meet the imaging target
Reach at least 15 percent contrast between opposite flanks, a CNR of at least 5 and a probe-blur estimate below 8 nm. Then reverse the detector and explain why the height map stays the same.
What to observe: Press Check target; the test uses the saved counts. Then inspect pixels with the row and column controls. Changing the gain or camera view reuses the same saved frame. The known height map is a separate input, not something recovered from brightness.
03
Separate the causes of contrast
Switch to In-lens collection, then raise the working distance and lower the current. Change the gain and confirm that the count-based task result does not change. Tell apart directional shading, blur and counting noise.
What to observe: The image is a teaching raster laid out in specimen coordinates: a 192 by 192 grid of sites spanning 1 micrometre, with the same dwell time at each site. It leaves out the foreshortening of a projected image and does not redistribute the probe where terrain blocks it. Yield, isotropic blur and collection are all assumed. Opaque terrain blocks both direct excitation and side collection; axial collection uses a field-guided approximation. Saved counts use a fixed random seed, drawn from a Poisson distribution below a mean of 40 and from a Gaussian approximation above it. The detector field, charging, calibrated efficiency and recovering height from the image are not modeled.