E52 · Photoelectron spectroscopy

UPS: Valence Bands & Work Function

Which two edges of a UPS spectrum give the work function? The secondary-electron cutoff at the low-energy end and the Fermi edge at the high-energy end. A negative sample bias moves both edges by the same amount, so their spacing stays fixed. Subtract that spacing from the photon energy and you have the work function. If the sample has no Fermi edge, take it from a metal reference in electrical contact with the sample.

Apply a negative bias, record a UV photoelectron spectrum and a separate metal reference, then place the fitting windows on the edges. Find the work function and the onset of the occupied band, and see what goes wrong with a missing Fermi edge, a cutoff the detector cannot reach, or too few counts.

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

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

Fitted Fermi reference $\text{—}$

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

More measurements and target

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 $\text{—}$

Simulated examples; each plot identifies its data source.

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}

Negative sample bias · magnitude

|V_s|

0\,\mathrm{V}

Cutoff fitting window · raw kinetic center

K_c

0.2\,\mathrm{eV}

Advanced settings

Known occupied-edge offset below Fermi

E_V

0\,\mathrm{eV}

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}

Fermi fitting window · raw kinetic center

K_F

16.7\,\mathrm{eV}

Occupied-edge window · binding offset

E_{V,\mathrm{seed}}

0\,\mathrm{eV}

Learning goal and model limits

A band edge is not always a Fermi edge

Apply a bias, pick the edges, then measure the span

First predict how a negative bias moves both edges. Use the independent metal reference, a negative bias of at least 2 V and an instrument width below 0.1 eV. Place all three fitting windows so that the work function and the occupied-edge offset each come within 0.08 eV of their separate known model values, then check the target. Afterward, try the semiconductor preset without the metal reference.

Model assumptions

The He I photons are fixed at 21.218 eV, and the analyzer work function is 4.5 eV. The sample and a separate clean metal reference get the same ideal negative bias, with independent exposures and independent random counts. The invented sample has an adjustable work function and an adjustable binding offset of its occupied edge. Its spectrum has a flat edge, two assumed valence peaks and an exponential secondary continuum. This spectrum is synthetic; it is not a DOS or band-structure calculation. Gaussian broadening stands for the analyzer and photon widths. Fermi broadening uses a Gaussian stand-in with a thermal FWHM of 3.53 times the Boltzmann energy, not a full Fermi–Dirac convolution. An ideal detector rejects kinetic energies below 1 eV. Each window you choose is fitted with a rising cutoff or a falling occupied edge, with a free amplitude and a linear background, using only the acquired counts and the known broadening. A fit needs data on both sides of the edge, enough counts, enough contrast and a best fit inside the window. A missing or wrongly chosen edge is marked invalid; it does not give a zero work function. A semiconductor-like occupied edge lies below the Fermi level, so the separate metal reference supplies that zero. The ionization energy is the measured work function plus the measured occupied-edge offset. The cutoff fit assumes a fixed 2 eV secondary decay. Surface patches, extra false cutoffs, charging, space charge, contamination, distortion from angular acceptance and matrix-element inversion are left out. Exposures are taken as scan-integrated times; processing and animation add no exposure. Counts use Poisson noise below 40 and a rounded Gaussian approximation above. The apparatus shows a fixed window of raw energies; fitting windows and reference corrections affect only the analysis. Lens geometry and trajectory lengths are schematic; the analyzer radius used in the calculation is 150 mm.

Start with the essentials

Focus question: Which two edges of a UPS spectrum give the work function?
One-sentence intuition: A negative bias moves both edges up by the same energy, so their difference cancels the bias and the analyzer work function. When the sample has no Fermi edge, a separate metal reference supplies the Fermi zero.

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. The sample and the reference share the applied bias, and the same bias offset appears in both the Fermi edge and the cutoff.

Work function from a measured span

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

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

Occupied-edge binding offset and ionization energy

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

The occupied edge of a semiconductor-like sample lies below the Fermi level. Its offset comes from the sample counts, measured from the separately recorded metal edge. In a metal this offset goes to zero.

Counts and the binding-energy scale

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

The energy conversion reuses the acquired counts. Expected rates and display gain are kept apart from the measured spectra, and moving a fitting window does not record a new scan.

Gaussian stand-in for thermal broadening

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

A Gaussian with the same FWHM as the derivative of the Fermi function shows thermal broadening. It does not reproduce the exact Fermi–Dirac line shape.

Common difficulties

The cutoff position alone is not enough

Typical misconceptionThe cutoff energy by itself is the sample's work function.

Better mental modelThe analyzer vacuum offset and the sample bias both shift the absolute cutoff. Measure the span from the Fermi edge to the cutoff instead.

The highest occupied edge is not always the Fermi level

Typical misconceptionAny edge at the high-energy end is the Fermi edge.

Better mental modelA semiconductor-like sample has an offset between its occupied band and the Fermi level. Treating that band edge as the Fermi edge overestimates the work function by the offset.

A valence spectrum is not the density of states

Typical misconceptionThe valence curve on screen is the exact density of states.

Better mental modelPhotoemission weighting and the instrument response both shape the intensity. This Lab uses invented valence peaks and does not invert for the density of states.

Run the experiment

01
Bring the cutoff into view
Start with no bias, then choose Measure a metal.
What to observe: Negative bias lifts the sample cutoff above the 1 eV detector threshold, and both edges move together.
02
Place the fitting windows
Move the cutoff and Fermi fitting windows over the measured edges you can see, then compare with Wrong cutoff window.
What to observe: A fitted edge needs enough counts and data on both sides. A smooth continuum is not accepted as a cutoff.
03
Tell the Fermi edge from the band edge
Choose Occupied edge below Fermi, then turn off Use the independent metal Fermi reference.
What to observe: The sample has no Fermi edge of its own. Turning the separate reference back on recovers the work function and the ionization energy.
04
Meet the target, then push it
Use enough counts and a narrow instrument response, then press Check target. Change the bias and move the windows to follow the edges.
What to observe: The edges move, but the recovered work function stays close to the separately known value.