E13 · Field emission and beam deceleration

Low-energy SEM: Slow at the Surface

Keep the electron column energetic and retard the beam at a negatively biased specimen. Recover a thin-film pattern while balancing depth sensitivity, probe width, counting noise and prescribed charging.

Selected stored detector counts $\text{—}$

Actual landing energy $\text{—}$

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

Prescribed film depth weight $\text{—}$

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

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

Prescribed negative charging magnitude $\text{—}$

Prescribed frame displacement span $\text{—}$

Combined Gaussian response FWHM $\text{—}$

Incident dose per pixel $\text{—}$

Experiment target $\text{—}$

Predict · acquire · compare · explain

Can electrons travel fast and land gently?

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

Known surface film thickness

t

30\,\mathrm{nm}

Selected stored pixel index

k

528

Derived-image display gain only

G

1

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

Seeded synthetic counts on a 32 by 32 raster, 20 nm pitch, 640 nm field and fixed 20 pA current. Dwell is per pixel; collection efficiency is fixed at 15 percent. Poisson counts below expectation 40 use a normal approximation above. Hardware and rays are schematic; the source, exposed face and same-face collectors are connected. Display gain and selected pixel reuse stored counts. No Monte Carlo transport, real detector calibration, contamination, damage kinetics or expert certification. Negative bias magnitude is subtracted from column energy; an additional prescribed negative charging potential reduces actual landing energy. Nonpositive landing energy records zero counts. A declared exponential depth kernel has length 12 nm times landing energy in keV to power 1.5; this is not a Kanaya–Okayama range or calibrated information depth. A sinusoidal film yield of mean 0.22 and amplitude 0.08 on an 80 nm period overlies uniform substrate yield 0.30. Gaussian probe and lateral spread attenuate the harmonic before counts are acquired. The lens-width and charge/displacement laws are prescribed, not an electrostatic field solution or material charging calculation. Count-only harmonic inference never receives the input amplitude, depth kernel or charging law. BSE collection remains fixed even with bias; no realistic bias-dependent detector acceptance or SE transport. Depth and lateral specimen scale are 8 nm per scene unit; hardware is separately schematic.

Start with the essentials

Focus question: Can electrons travel fast in the column and land gently?
One-sentence intuition: Landing energy sets depth sensitivity; column energy remains part of the probe budget.

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}

Negative stage bias and negative charging reduce landing energy; nonpositive energy blocks the scan.

Declared depth and transfer

\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 weighting kernel and Gaussian transfer attenuate the prescribed film pattern.

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}

Normalized counts determine the fit; known film and instrument parameters stay outside inference.

Common difficulties

Separate appearance from information

Typical misconceptionA clearer-looking image guarantees the right interpretation.

Better mental modelDisplay gain does not change counts or undo physical mixing. The input reference is separate from measurement.

Run the experiment

01
Predict surface sensitivity
Compare the deep preset with the known film pattern and depth kernel.
What to observe: A narrow beam can still collect mostly substrate response.
02
Separate column from landing
Compare a 0.5 keV column with a 5 keV column and 4.5 kV negative stage bias.
What to observe: Equal landing energies have equal depth weights but different prescribed probe budgets.
03
Expose the charging limit
Use the charging preset, then increase charge drainage. Inspect displacement and the stored row fit.
What to observe: Longer dwell raises counts while charge can change where the beam lands.
04
Recover the surface pattern
Retard the beam, retain column energy, drain charge and balance dwell. Check the target.
What to observe: Display gain changes appearance without repairing a lost harmonic.

Core mathematical model

Energy and dose

Declared depth and transfer

Harmonic from recorded counts

Common difficulties

Separate appearance from information

Run the experiment

Predict surface sensitivity

Separate column from landing

Expose the charging limit

Recover the surface pattern