E62 · Photoelectron spectroscopy

HAXPES: Depth & Buried Interfaces

Compare soft and hard X-ray scans of the same layered silicon specimen. Fit three chemical-state doublets from independent acquired counts, then infer cap and interface thicknesses with an explicit escape model. Test grazing emission, weak counts and a wrong attenuation length.

Cap thickness inferred from counts $\text{—}$

Interface width inferred from counts $\text{—}$

Fitted cap / interface / substrate signal $\text{—}$

Fitted three-state areas · counts $\text{—}$

Instrument Gaussian-equivalent FWHM $\text{—}$

Recorded selected-energy counts $\text{—}$

Supplied normal escape length $\text{—}$

Known teaching-model normal length $\text{—}$

Known normal depth · 95% escape weight $\text{—}$

Known cap / interface thicknesses $\text{—}$

Substrate analyzer kinetic energy $\text{—}$

Count-scaled fit residual $\text{—}$

Experiment target $\text{—}$

Depth needs an explicit escape model

Fit a curved continuumChanging the fitted background keeps both acquired count buffers fixed.Known cap thickness

t_c

6\,\mathrm{nm}

Known interfacial layer width

t_i

1.5\,\mathrm{nm}

Emission angle from surface normal

\theta

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

Selected photon energy

h\nu

1500\,\mathrm{eV}

Pass energy

E_p

100\,\mathrm{eV}

Entrance and exit slit width

s

1.2\,\mathrm{mm}

Scan-integrated exposure per scan

\tau

1\,\mathrm{s}

Processing multiplier for assumed escape length

q_\lambda

0.6

Predict which scan sees the buried state. Use at least 3 keV photons, a cap at least 3 nm thick and enough counts in every component. Fit the curved background, achieve resolution below 0.8 eV and total fitted area above 8000 counts. Supply the correct assumed escape length to recover cap thickness within 0.4 nm and interface width within 0.3 nm. Then change only the length multiplier.

A synthetic cap / intermediate-state layer / semi-infinite silicon substrate has equal Si number density and a common escape length in all layers. Assumed doublet anchors are 103.5, 101.4 and 99.4 eV, with 0.63 eV splitting and 2:1 area ratios; the intermediate state is a teaching template, not a unique phase identification. A 2 nm length at the soft reference grows with kinetic energy to the power 0.65. This is an invented escape model without elastic scattering, not TPP-2M, NIST material data or a universal effective attenuation length. Equal line sensitivities and negligible photon attenuation give analytic layer integrals. Count yield includes that escape integral and an assumed photon-energy power of minus 2.5; photon bandwidth grows linearly with photon energy. Both scans have separate noise and identical exposure and optics; each exposure is scan-integrated time, not per energy bin. Nonnegative template areas and smooth continuum are fitted only from stored counts. The inverse assumes a known layer order, equal density, common length and supplied angle; it cannot recover an arbitrary depth profile. Low component counts, broad resolution or large residuals suppress the inversion. A wrong supplied length scales both inferred thicknesses. The 95% depth is an ideal escape-weight quantile, not a material-calibrated information depth. Charging, recoil, real cross sections, satellites, damage and exact lens transport are omitted. Noise is Poisson below 40 expected counts and rounded Gaussian above. Analyzer radius is 150 mm with a 2-degree entrance half-angle. Scene thicknesses are magnified; detector display gain is logarithmic.

Start with the essentials

Focus question: Can a harder photon reveal a buried layer?
One-sentence intuition: Fit each acquired scan independently. With a stated three-layer order and common escape model, area ratios yield cap and interface thicknesses. A wrong model gives precise-looking biased results.

Core mathematical model

Energy and normal escape length

K_a=h\nu-\phi_a-E_B,\qquad L=\lambda(K_a)\cos\theta

The angle is measured from the outward normal. The common length uses substrate kinetic energy for all three nearby states, omitting elastic scattering and material differences.

Assumed normalized depth weight

w(z)=\frac{1}{L}\exp\!\left(-\frac{z}{L}\right)

Equal Si number density, common sensitivity, semi-infinite substrate and negligible photon attenuation yield a normalized exponential depth model.

Three layer weights

\begin{aligned}f_c&=1-e^{-t_c/L}\\f_i&=e^{-t_c/L}(1-e^{-t_i/L})\\f_s&=e^{-(t_c+t_i)/L}\end{aligned}

The cap, interfacial layer and substrate weights sum to one. These are known forward-model references, kept separate from fitted fractions.

Thicknesses from fitted areas

\begin{aligned}\widehat t_c&=-L_{\mathrm{assumed}}\ln\frac{A_i+A_s}{A_c+A_i+A_s}\\\widehat t_i&=L_{\mathrm{assumed}}\ln\frac{A_i+A_s}{A_s}\end{aligned}

Only acquired counts determine fitted areas. The inversion assumes all three states are sufficiently detected and the layer order and escape model are supplied.

Ideal 95% escape-weight depth

d_{95}=-L\ln(0.05)

This quantile encloses 95% of the ideal semi-infinite exponential weight. Real information depth and effective attenuation length require a specified material, geometry and transport calculation.

Depth and count yield need separate axes

N\propto\tau\,\lambda\cos\theta\,(h\nu)^{-2.5}

The photon-energy exponent is assumed here, with fixed photon flux. It illustrates the possibility of deeper weighting with lower counts, not a universal cross-section law.

Common difficulties

Harder does not guarantee more counts

Typical misconceptionA deeper probing photon must give a brighter buried-state peak.

Better mental modelDepth weighting and cross section compete. Compare normalized fractions separately from absolute acquired counts.

Depth inversion is conditional

Typical misconceptionThree peaks recover any unknown depth profile.

Better mental modelThis inverse uses a known three-layer order, equal density and a common escape length. Many real profiles violate those assumptions.

A length is not universally calibrated

Typical misconceptionThe power-law escape length is NIST material data.

Better mental modelIt is an invented teaching model. An inelastic mean free path, effective attenuation length and information depth are different quantities.

Run the experiment

01
Predict the buried fraction
Compare Soft X-rays and Infer buried layers at fixed specimen, exposure and analyzer settings.
What to observe: The buried fraction increases with the assumed escape length; absolute counts can fall.
02
Fit and invert
Fit the curved continuum, choose sufficient exposure and supply the known teaching length model, then check the target.
What to observe: The cap and interface estimates come from measured areas, not copied specimen dimensions.
03
Change only the model
Move the processing length multiplier after a successful acquisition.
What to observe: Raw spectra and fitted areas stay fixed while both inferred thicknesses scale.
04
Find the practical limit
Use Deep cap, harder photons, Grazing emission and Counting-starved. Inspect individual peak areas.
What to observe: Grazing emission reduces normal escape depth; missing components suppress the inversion.