E63 · Electron excitation and surface spectroscopy

Inverse Photoemission: Empty States

What can emitted photons tell us about empty electron states? Where they lie in energy. An incoming electron drops into an empty state and emits a photon, and the detector accepts only one fixed photon energy. Scanning the electron energy then locates the unoccupied states, broadened by the instrument resolution.

Step the energy of the incoming electrons while a detector counts photons in one fixed energy band. Find the two peaks where electrons drop into empty states, then see how photon bandwidth, electron energy spread, exposure and energy calibration change them.

First count-derived centroid $\text{—}$

Second count-derived centroid $\text{—}$

Weaker acquired peak SNR $\text{—}$

Combined instrument FWHM $\text{—}$

More measurements and target

Selected stored photon count $\text{—}$

Selected surface kinetic energy $\text{—}$

Larger centroid error against input $\text{—}$

Acquired electron-energy positions $\text{—}$

Summed incident dose $\text{—}$

Prescribed changed-response fraction $\text{—}$

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

Dwell per electron-energy position

\tau

0.1\,\mathrm{s}

Photon bandpass FWHM

w_\gamma

1\,\mathrm{eV}

Incident electron FWHM

w_e

0.5\,\mathrm{eV}

Assumed work function for display calibration

\widehat\Phi

3.5\,\mathrm{eV}

Advanced settings

Selected stored surface kinetic energy

E_{kin}

7\,\mathrm{eV}

Display smoothing radius in positions

r

0

Learning goal and model limits

Predict · acquire · process · explain

What can emitted photons tell us about empty electron states?

Target: both recorded centroids must lie within 0.12 eV of the undamaged input. The weaker peak needs an SNR of at least 10, the combined FWHM must be at most 0.8 eV, and the changed-response fraction must be at most 15 percent. The work-function calibration and the smoothing act on the stored scan and do not record a new one.

Model assumptions

This is isochromat inverse photoemission. The detector counts photons in a fixed band at 9.5 eV, modeled on published counters with an SrF2 window and approximated here by a Gaussian with adjustable FWHM. The electron current is fixed at 10 nA over one square millimetre. The scan has 220 electron-energy positions from 3 to 14 eV in 0.05 eV steps. Electron energy means the calibrated kinetic energy just outside the surface, not the gun voltage. The sample's true work function is 4.5 eV. The final-state energy equals the kinetic energy at the surface plus the work function minus the photon energy. The empty states are synthetic: two Gaussian peaks at 2 and 5 eV above the Fermi level. They are not gold (Au) data and not an absolute density of states. The numerical integral includes only empty final states. The Gaussian energy spread of the incoming electrons and the photon band both broaden the scan, and a wider photon band collects more counts. The assumed yield is two billionths per incident electron per unit spectral response, and the dark rate is five counts per second. Counts come from a fixed random seed and use a normal approximation above 40 expected counts. A set cumulative-dose rule shifts and weakens the empty-state response; it is not a calibrated damage model. Each centroid comes from one of two fixed kinetic-energy windows in the recorded scan, after the known dark counts are subtracted. Display calibration and smoothing reuse the same counts. The separate reference curves for filled and empty states are model inputs, not measured UPS spectra or reconstructed densities of states. The model leaves out momentum, spin, many-body matrix elements, charging and detector dead time.

Physics tutorial

Inverse Photoemission: Empty States

BackgroundAn electron that enters a surface can drop into an empty state and give off a photon. Here you step the energy of the incoming electrons while a detector counts photons in one fixed band. You find two empty-state peaks in the counts, then test how photon bandwidth, electron energy spread, exposure and energy calibration affect them.

Why it mattersThe goal is to place each peak within 0.12 eV of its undamaged position. Each peak needs a signal-to-noise ratio of at least 10, the combined instrument FWHM must be at most 0.8 eV, and the set damage rule may change at most 15 percent of the response.

Start with the essentials

Focus question: What can emitted photons tell us about empty electron states?
One-sentence intuition: The electron brings in energy and the photon carries a fixed energy away; the difference tells you where the empty state sits.

Core mathematical model

Energy of the empty final state

\begin{aligned}\varepsilon_f&=E_{kin}+\Phi-h\nu\\E_{kin}&=\varepsilon_f+h\nu-\Phi\end{aligned}

Kinetic energy is measured from the local vacuum level, and the final-state energy from the Fermi level. A calibration error shifts the whole energy axis.

Instrument resolution from two Gaussian widths

\begin{aligned}w_{inst}&=\sqrt{w_e^2+w_\gamma^2}\\\sigma&=\frac{w_{inst}}{2\sqrt{2\ln2}}\end{aligned}

The electron and photon widths are independent Gaussians, so they add in quadrature. The natural width of each state comes on top of this.

Expected photon counts at each electron energy

\begin{aligned}\varepsilon_0&=E+\Phi-h\nu\\R(E)&=\int_0^\infty\rho_u(u)\\&\quad\times G_\sigma(u-\varepsilon_0)\,du\\\mu(E)&=b\tau+\frac{I\tau}{e}\eta A_\gamma R(E)\end{aligned}

A wider photon band has a larger area, so it collects more counts. The recorded scan is the empty-state spectrum broadened and weighted by the instrument; it is not a reconstruction of the ideal input.

Common difficulties

Reference curves are not data

Typical misconceptionThe ideal input curve is the result inferred from the scan.

Better mental modelThe gold curves show the known input and serve only as an independent check. The teal result comes from the stored counts alone.

Longer exposure does not fix every error

Typical misconceptionIf I collect enough counts, the answer will be right.

Better mental modelMore counts reduce counting noise. Errors from the continuum background, the energy calibration, the finite instrument width and the set damage remain.

Run the experiment

01
Predict which energy is scanned
Pick a stored electron-energy position and compare its photon count with the energy bookkeeping card.
What to observe: The photon band the detector counts stays fixed; only the energy of the incoming electrons changes.
02
Recalibrate the same counts
Change the assumed work function from 3.5 to 4.5 eV. Then change the smoothing and check that the raw scan does not change.
What to observe: A wrong work function shifts both inferred peaks by the same energy.
03
Trade counts against resolution
Compare a wide photon band with a broad spread of incoming electron energies. Look at the count rate and the combined FWHM.
What to observe: More counts do not undo the broadening, and they cannot separate features that sit too close to resolve.
04
Recover the two peaks
Set narrow electron and photon widths, the calibrated work function and enough dwell time. Compare the centroid energies with the separate reference, then check the target.
What to observe: The target checks signal-to-noise, energy error, instrument width and the set amount of damage.

Core mathematical model

Energy of the empty final state

Instrument resolution from two Gaussian widths

Expected photon counts at each electron energy

Common difficulties

Reference curves are not data

Longer exposure does not fix every error

Run the experiment

Predict which energy is scanned

Recalibrate the same counts

Trade counts against resolution

Recover the two peaks