E62 · Photoelectron spectroscopy

HAXPES: Depth & Buried Interfaces

Can higher-energy X-rays reveal a layer buried under the surface? They can. Higher-energy X-rays give the photoelectrons more energy, so electrons from a buried interface can still get out. Turning peak areas into layer thicknesses needs an assumed layer order and an escape model. With the wrong attenuation length, the result looks precise but is biased.

Compare soft and hard X-ray scans of the same layered silicon sample. Fit three chemical-state doublets to each scan's own counts, then use a stated escape model to work out the thickness of the cap and of the interface layer. Try grazing emission, weak counts and a wrong attenuation length.

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

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

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

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

More measurements and target

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

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

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

Supplied normal escape length $\text{—}$

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

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

Substrate analyzer kinetic energy $\text{—}$

Count-scaled fit residual $\text{—}$

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

Known cap thickness

t_c

6\,\mathrm{nm}

Emission angle from surface normal

\theta

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

Selected photon energy

h\nu

1500\,\mathrm{eV}

Processing multiplier for assumed escape length

q_\lambda

0.6

Advanced settings

Fit a curved continuumChanging the fitted background does not change either set of acquired counts.Known interfacial layer width

t_i

1.5\,\mathrm{nm}

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}

Learning goal and model limits

Estimating depth needs a model of electron escape

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

Model assumptions

The sample is synthetic: a cap, an intermediate-state layer and a semi-infinite silicon substrate, with the same silicon number density and the same escape length in every layer. The doublets are assumed to sit at 103.5, 101.4 and 99.4 eV, with a 0.63 eV splitting and a 2:1 area ratio. The intermediate state is a teaching template; it does not identify a unique phase. The escape length is 2 nm at the soft reference and grows with kinetic energy to the power 0.65. This escape model is invented and has no elastic scattering; it is not TPP-2M, NIST material data or a universal effective attenuation length. Equal line sensitivities and negligible photon attenuation make the layer integrals analytic. The count yield includes that escape integral and an assumed photon-energy power of minus 2.5, and the photon bandwidth grows linearly with photon energy. The two scans have separate noise but the same exposure and optics. Each exposure is the integration time for a whole scan, not for one energy bin. Nonnegative template areas and a smooth continuum are fitted only to the stored counts. The inversion assumes a known layer order, equal density, a common escape length and the supplied angle, so it cannot recover an arbitrary depth profile. Low component counts, broad resolution or a large residual stop the inversion. A wrong supplied length scales both inferred thicknesses. The 95% depth is a quantile of the ideal escape weight, not an information depth calibrated for a real material. Charging, recoil, real cross sections, satellites, damage and exact lens transport are left out. Noise is Poisson below 40 expected counts and rounded Gaussian above. The analyzer radius is 150 mm with a 2-degree entrance half-angle. Layer thicknesses in the scene are magnified, and the detector display gain is logarithmic.

Start with the essentials

Focus question: Can higher-energy X-rays reveal a layer buried under the surface?
One-sentence intuition: Fit each scan on its own; with a known three-layer order and a common escape model, the peak-area ratios give the cap and interface thicknesses, and a wrong model gives results that look precise but are biased.

Core mathematical model

Kinetic energy and escape length along the normal

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

The angle is measured from the outward surface normal. All three nearby states share one length, taken at the substrate kinetic energy. Elastic scattering and differences between materials are left out.

Assumed exponential depth weight

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

With equal silicon number density, a common sensitivity, a semi-infinite substrate and negligible photon attenuation, the normalized signal weight falls off exponentially with depth.

Signal share of each layer

\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 weights for the cap, the interface layer and the substrate add up to one. They are known references from the forward model and are kept apart from the 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}

The fitted areas come only from acquired counts. The inversion needs all three states to be clearly detected, and it needs the layer order and the escape model to be supplied.

Depth that holds 95% of the ideal signal

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

This depth encloses 95% of the ideal exponential weight for a semi-infinite sample. A real information depth or effective attenuation length needs a specified material, geometry and transport calculation.

Deeper weighting can come with fewer counts

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

The photon-energy exponent is an assumption here, and the photon flux is held fixed. It shows that harder photons can weight deeper layers more while giving fewer counts; it is not a universal cross-section law.

Common difficulties

Harder photons do not guarantee more counts

Typical misconceptionA photon that probes deeper must give a brighter buried-layer peak.

Better mental modelDeeper weighting and a falling cross section pull in opposite directions. Compare the normalized fractions and the absolute counts separately.

The depth inversion only works under its assumptions

Typical misconceptionThree peaks are enough to recover any unknown depth profile.

Better mental modelThis inversion assumes a known three-layer order, equal density and a common escape length. Many real depth profiles break those assumptions.

The escape length here is not calibrated data

Typical misconceptionThe power-law escape length comes from NIST material data.

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

Run the experiment

01
Predict the buried fraction
Keep the sample, exposure and analyzer settings fixed, and compare Soft X-rays with Infer buried layers.
What to observe: The buried layer's share grows with the assumed escape length, while the absolute counts can fall.
02
Fit the peaks and infer the thicknesses
Turn on Fit a curved continuum, choose a long enough exposure and supply the known teaching length model, then press Check target.
What to observe: The cap and interface estimates are calculated from the measured areas, not copied from the sample settings.
03
Change only the assumed length
After a successful acquisition, move the Processing multiplier for assumed escape length.
What to observe: The raw spectra and fitted areas stay the same, while both inferred thicknesses scale by the same factor.
04
Find where the method breaks down
Try Deep cap, harder photons, Grazing emission and Counting-starved, and look at each peak area.
What to observe: Grazing emission shortens the escape depth measured along the normal. When a component is missing, no inversion is reported.