E13 · Field emission and beam deceleration

Low-energy SEM: Slow at the Surface

Can electrons travel fast in the column and land gently? Yes. A negative voltage on the specimen slows the electrons just before they land. They travel down the column at high energy and land with low energy, so the interaction stays shallow, close to the surface.

Run the electron column at high energy, then slow the beam down with a negative bias on the specimen. Try to recover a thin-film pattern by trading off depth sensitivity, probe width, counting noise and the assumed charging.

Actual landing energy $\text{—}$

Prescribed film depth weight $\text{—}$

Modulation to representative pixel noise $\text{—}$

Prescribed negative charging magnitude $\text{—}$

More measurements and target

Selected stored detector counts $\text{—}$

Count-derived film harmonic amplitude $\text{—}$

Amplitude error against separate film $\text{—}$

Prescribed frame displacement span $\text{—}$

Combined Gaussian response FWHM $\text{—}$

Incident dose per pixel $\text{—}$

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

Electron column energy

E_c

1\,\mathrm{keV}

Negative stage bias magnitude

|V_b|

0\,\mathrm{kV}

Dwell per pixel

\tau

0.05\,\mathrm{ms}

Prescribed charge drainage: 1 complete

c

0.2

Advanced settings

Known surface film thickness

t

30\,\mathrm{nm}

Selected stored pixel index

k

528

Derived-image display gain only

G

1

Learning goal and model limits

Predict, then test it on the microscope

Can electrons travel fast and land gently?

Target: recover the film harmonic amplitude within 0.02 of the separate input, with a representative pixel CNR of at least 8 and a response FWHM of at most 8 nm. The film depth weight must be at least 90 percent, and the prescribed charging at most 20 V.

Model assumptions

The counts are synthetic and drawn from a fixed random seed. The raster is 32 by 32 pixels with a 20 nm pitch and a 640 nm field, and the beam current is fixed at 20 pA. Dwell time is set per pixel, and collection efficiency is fixed at 15 percent. Counts with an expected value below 40 are Poisson samples; above 40 they use a normal approximation. The hardware and rays are schematic, but the rays do run from the source to the exposed specimen face and on to collectors above that same face. Display gain and the selected pixel reuse the stored counts. Monte Carlo electron transport, real detector calibration, contamination and damage kinetics are not modeled, and the Lab has no expert certification. The landing energy is the column energy minus the size of the negative bias, lowered further by an assumed negative charging potential. If the landing energy is zero or below, no counts are recorded. An assumed exponential depth kernel has a length of 12 nm times the landing energy in keV raised to the power 1.5; this is not the Kanaya–Okayama range or a calibrated information depth. A thin film whose yield varies sinusoidally, with mean 0.22, amplitude 0.08 and an 80 nm period, sits on a substrate with a uniform yield of 0.30. A Gaussian probe and the lateral spread weaken the film pattern before the counts are recorded. The laws for lens-limited width, charging and beam displacement are assumed; they are not an electrostatic field solution or a material charging calculation. The harmonic fit uses only the counts and never sees the input amplitude, depth kernel or charging law. Backscatter collection stays fixed when the bias changes; realistic bias-dependent detector acceptance and secondary-electron transport are not modeled. Specimen depth and lateral distances are 8 nm per scene unit; the hardware is drawn to a separate, schematic scale.

Physics tutorial

Low-energy SEM: Slow at the Surface

BackgroundAn SEM can run its column at high energy and still land electrons gently by putting a negative bias on the specimen. Here you try to recover a thin-film pattern while trading off depth sensitivity, probe width, counting noise and the assumed charging.

Why it mattersThe task is to recover the film's harmonic amplitude to within 0.02 of the separate input. Along the way, keep the contrast-to-noise ratio at a representative pixel at 8 or more, the response FWHM at 8 nm or less, the film depth weight at 90 percent or more, and the assumed charging at 20 V or less.

Start with the essentials

Focus question: Can electrons travel fast in the column and land gently?
One-sentence intuition: The landing energy decides how deep the signal comes from, but the column energy still helps set the probe width.

Core mathematical model

Energy and dose

\begin{aligned}E_l&=E_c-e|V_b|-e|V_q|\\N_{inc}&=I\tau/e\\D&=N_{inc}/p^2\end{aligned}

A negative stage bias and negative charging both lower the landing energy. If it falls to zero or below, the electrons never land and the scan records nothing.

Assumed depth weighting and blur

\begin{aligned}L&=12\,\mathrm{nm}(E_l/\mathrm{keV})^{1.5}\\w_t&=1-e^{-t/L}\\H(k)&=e^{-k^2\sigma^2/2}\end{aligned}

An exponential depth weighting and a Gaussian blur both weaken the film pattern in the signal.

Harmonic from recorded counts

\begin{aligned}\widehat C&=\frac{2}{n}\sum_jy_j\cos(kx_j)\\\widehat S&=\frac{2}{n}\sum_jy_j\sin(kx_j)\\\widehat A&=\sqrt{\widehat C^2+\widehat S^2}\end{aligned}

Only the normalized counts enter the fit; the known film and instrument parameters are not used.

Common difficulties

A clearer image does not carry more information

Typical misconceptionIf the image looks clearer, my interpretation must be right.

Better mental modelDisplay gain does not change the counts, and it cannot unmix the physical effects. The input reference is shown separately and is not a measurement.

Run the experiment

01
Predict surface sensitivity
Load the Buried substrate dominates preset and compare it with the known film pattern and depth kernel.
What to observe: Even a narrow beam can pick up mostly substrate signal.
02
Separate column from landing
Compare a 0.5 keV column with a 5 keV column plus a 4.5 kV negative stage bias.
What to observe: Both beams land at the same energy and weight depth the same way, but their assumed probe widths differ.
03
See how charging limits the scan
Load the Charging and frame shear preset, then raise the charge drainage. Watch the beam displacement and the fit to the stored rows.
What to observe: A longer dwell gives more counts, but the charge that builds up can also shift where the beam lands.
04
Recover the surface pattern
Slow the beam with the bias, keep the column energy up, drain the charge and pick a balanced dwell. Then press Check target.
What to observe: Display gain changes how the image looks, but it cannot bring back a harmonic that was lost.

Core mathematical model

Energy and dose

Assumed depth weighting and blur

Harmonic from recorded counts

Common difficulties

A clearer image does not carry more information

Run the experiment

Predict surface sensitivity

Separate column from landing

See how charging limits the scan

Recover the surface pattern