E52 · Photoelectron spectroscopy

UPS: Valence Bands & Work Function

Apply a negative bias, acquire a UV photoelectron spectrum and an independent metallic reference, then position the fitting windows. Recover the work function and occupied-band onset while testing a missing Fermi edge, an inaccessible cutoff and low counts.

Work function from measured edges $\text{—}$

Fitted secondary-electron cutoff $\text{—}$

Fitted Fermi reference $\text{—}$

Measured cutoff-to-Fermi energy span $\text{—}$

Measured occupied-edge binding offset $\text{—}$

Measured ionization energy $\text{—}$

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

Applied signed sample bias $\text{—}$

Recorded sample counts $\text{—}$

Independent reference counts $\text{—}$

Fermi-zero source $\text{—}$

Experiment target $\text{—}$

A band edge is not always a Fermi edge

Bias, select the edges, measure the span

Use the independent metal Fermi referenceNeeded when the sample has no occupied states at its Fermi level.Known sample work function

\phi_s

4.7\,\mathrm{eV}

Known occupied-edge offset below Fermi

E_V

0\,\mathrm{eV}

Negative sample bias · magnitude

|V_s|

0\,\mathrm{V}

Pass energy

E_p

30\,\mathrm{eV}

Equal entrance and exit slit width

s

1.2\,\mathrm{mm}

Scan-integrated exposure · sample and reference each

\tau

1\,\mathrm{s}

Temperature · edge broadening

T

300\,\mathrm{K}

Cutoff fitting window · raw kinetic center

K_c

0.2\,\mathrm{eV}

Fermi fitting window · raw kinetic center

K_F

16.7\,\mathrm{eV}

Occupied-edge window · binding offset

E_{V,\mathrm{seed}}

0\,\mathrm{eV}

Predict how negative bias moves both edges. Use the independent metal reference, at least 2 V negative bias and instrument width below 0.1 eV. Position all three fitting windows and recover work function and occupied-edge offset within 0.08 eV of their separate known model values, then check. Try the semiconductor preset without a metal reference.

Fixed He I photons at 21.218 eV and analyzer work function 4.5 eV. The sample and separate clean metallic reference receive the same ideal negative bias; their exposures and random counting streams are independent. The invented sample has adjustable work function and occupied-edge binding offset, a flat edge with two assumed valence peaks and an exponential secondary continuum. This is a synthetic spectrum, not a DOS or band-structure calculation. Gaussian broadening represents the analyzer and photon widths. Fermi broadening uses a Gaussian surrogate with thermal FWHM 3.53 times Boltzmann energy, not a full Fermi–Dirac convolution. An ideal detector rejects kinetic energies below 1 eV. A user-selected window is fitted to a rising cutoff or falling occupied edge with free amplitude and linear background, using only acquired counts and known broadening. Fits require support on both sides, sufficient counts, contrast and an interior optimum. Missing or incorrectly selected edges are invalid, not zero work function. A semiconductor-like occupied edge is below the Fermi level; the independent metal reference supplies that zero. Ionization energy adds the measured occupied-edge offset to the measured work function. The fitted cutoff edge model has a fixed assumed 2 eV secondary decay. Surface patches, extra false cutoffs, charging, space charge, contamination, angular acceptance distortion and matrix-element inversion are omitted. Exposures are assumed scan-integrated times; no new exposure is added by processing or animation. Counts use Poisson noise below 40 and a rounded Gaussian approximation above. Lens geometry and trajectory lengths are schematic; the quantitative analyzer radius is 150 mm.

Start with the essentials

Focus question: Which two edges measure a work function?
One-sentence intuition: Negative bias moves both edges upward by the same energy. Their difference cancels bias and analyzer work function. A separate metallic reference supplies the Fermi zero when the sample has no Fermi edge.

Core mathematical model

Analyzer-side energies under negative sample bias

\begin{gathered}K_F=h\nu-\phi_a+e|V_s|\\ K_c=\phi_s-\phi_a+e|V_s|\end{gathered}

Energies are measured in the analyzer vacuum. Sample and reference share the applied bias. The same bias offset enters the Fermi edge and cutoff.

Work function from a measured span

\phi_s=h\nu-(K_F-K_c)

The measured difference removes analyzer work function and bias. He I photon energy is fixed at 21.218 eV. Both edges must be inside the accepted energy range and correctly identified.

Occupied-edge binding offset and ionization energy

\begin{gathered}E_V=K_F-K_V\\ I=\phi_s+E_V\end{gathered}

The semiconductor-like occupied edge is below Fermi. Its offset comes from sample counts referenced to the separately measured metal edge. In a metal this offset approaches zero.

Acquisition and processing

D_i\sim\mathrm{Poisson}(\mu_i),\qquad E_{B,i}=K_{F,\mathrm{fit}}-K_i

The energy conversion reuses acquired counts. Expected rates and display gain are separate from measured spectra; changing fitting windows does not collect another scan.

Assumed Gaussian thermal surrogate

\Delta E_{F,\mathrm{thermal}}\simeq3.53k_BT

This matches the Fermi derivative FWHM with a Gaussian surrogate. It illustrates thermal broadening without claiming the exact Fermi–Dirac line shape.

Common difficulties

Absolute cutoff is not enough

Typical misconceptionThe cutoff energy alone equals sample work function.

Better mental modelAnalyzer vacuum offset and sample bias enter the absolute cutoff. Measure the Fermi-to-cutoff span.

The first occupied state is not Fermi

Typical misconceptionAny high-energy edge is a Fermi edge.

Better mental modelA semiconductor-like sample has an occupied-band offset. Using that edge as Fermi overestimates work function by the offset.

Valence spectra are weighted

Typical misconceptionThe displayed valence curve is the exact density of states.

Better mental modelPhotoemission weights and response influence the intensity. This Lab uses invented valence peaks and performs no DOS inversion.

Run the experiment

01
Recover detector access
Start without bias, then select Measure a metal.
What to observe: Negative bias lifts the sample cutoff above the 1 eV detector threshold; both edges move together.
02
Position the fitting windows
Move cutoff and Fermi windows across their visible measured edges; compare Wrong cutoff window.
What to observe: The fitted edge needs counts and two-sided support. A smooth continuum is not an accepted cutoff.
03
Separate Fermi and occupied-band edges
Use Occupied edge below Fermi, then switch off the metal reference.
What to observe: The sample lacks a Fermi edge. Restoring the independent reference recovers work function and ionization energy.
04
Complete and stress the target
Use sufficient counts and narrow instrument response, then check. Change bias and adjust the windows to follow the edges.
What to observe: The edge positions change while the recovered work function stays near the separate known value.