---
title: "ARPES Band & Fermi-Surface Mapping Interactive Lab | Sandbox Physics"
description: "How does an emission angle reveal an electronic band? Each photoelectron's kinetic energy and emission angle give its binding energy and its momentum along the surface. Plot the counts against energy and angle, and the occupied band shows up directly. How strongly each part appears also depends on the photon energy, the temperature and the transition probability, known as the matrix element."
canonical_url: "https://sandboxphysics.com/labs/arpes-band-mapping/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/arpes-band-mapping/"
last_modified: "2026-10-25"
---

E53 · Electron imaging / spectroscopy

# ARPES Band & Fermi-Surface Mapping

**How does an emission angle reveal an electronic band?** Each photoelectron's kinetic energy and emission angle give its binding energy and its momentum along the surface. Plot the counts against energy and angle, and the occupied band shows up directly. How strongly each part appears also depends on the photon energy, the temperature and the transition probability, known as the matrix element.

Drag the gold crystal to choose an energy–momentum cut. Save cuts at different azimuths, then watch their near-Fermi intensity build a map with visible gaps where no samples were taken.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

The band is a synthetic tight-binding band on a square lattice, with Lorentzian spectral weight, Fermi occupation and Gaussian instrument blur. The polarization effect uses an assumed stand-in for the dipole weight. Perpendicular momentum uses a free-electron final state and an assumed 10 eV inner potential. The sample and the grounded analyzer both have an assumed 4.5 eV work function. The momentum image is resampled from the blurred angle data on the detector. The Fermi map averages saved, blurred detector samples within the selected energy window; gray cells have no samples. Each energy row uses its own kinetic energy. The entry example contains 48 separately simulated azimuths with co-rotated polarization. Manual cuts retain their chosen polarization, so brightness can differ between cuts. These are expected intensities in arbitrary units, with no counting noise or momentum-area density calibration. Changing the sample, photon energy or instrument response starts a new scan. With interlayer hopping, perpendicular momentum varies across this map. The hardware is a teaching schematic: an orbit at a fixed 20 eV pass energy shows how energies spread out, but the lens transfer and angular transport are not computed from real fields.

Physics tutorial

## How does an emission angle reveal an electronic band?

**Background** Monochromatic light strikes the crystal, and the emitted electrons, not the light, enter the analyzer. The lens keeps track of each electron's emission angle, the hemispherical electrodes spread the electrons out by energy, and a two-dimensional detector records one cut of energy against angle.

**Why it matters** Before drawing conclusions about the specimen, see how each instrument setting shapes the measured result.

Start with the essentials

- **Focus question**: How does an emission angle reveal an electronic band?
- **One-sentence intuition**: Measuring energy and emission angle together maps out the occupied part of the spectral function. A band can look dark because the matrix element suppresses it. Higher temperature smears the Fermi edge, and changing the photon energy changes which momenta can be reached.

### Core mathematical model

#### Azimuth and the Fermi energy window

$$
\begin{aligned}k_x&=k_\parallel\cos\varphi,\\k_y&=k_\parallel\sin\varphi,\\-\delta E_F&\leq E-E_F\leq\delta E_F\end{aligned}
$$

Every stored energy row uses its own kinetic energy. Grid cells show the sampled intensity averaged over the selected window, weighted by each row's overlap with that window. Unsampled cells stay gray; there is no interpolation across missing azimuths.

#### Energy and parallel momentum

$$
\begin{aligned}E_B&=h\nu-\phi_a-E_{\mathrm{kin}},\\k_\parallel&=\frac{\sqrt{2m_eE_{\mathrm{kin}}}}{\hbar}\sin\theta\end{aligned}
$$

The sample and the grounded analyzer are assumed to have the same 4.5 eV work function. The emission angle is measured from the sample surface normal. Each energy row of the image is converted with its own kinetic energy. The photon incidence angle is a separate geometric quantity.

#### Occupied spectral weight

$$
\begin{aligned}I&\propto [|M|^2Af(E,T)]\ast R,\\A&=\frac{\Gamma/\pi}{(E-\varepsilon_{\mathbf k})^2+\Gamma^2},\\R&=R_E\ast R_\theta\end{aligned}
$$

Instrument blur smears the occupied intensity; it does not shift the underlying band.

#### Synthetic band and final state

$$
\begin{aligned}\varepsilon_{\mathbf k}&=-2tC-\mu+2t_zD,\\C&=\cos(ak_x)+\cos(ak_y),\\D&=\cos(ck_z)\end{aligned}
$$

The hopping energy is 0.45 eV, the in-plane spacing 3.8 angstrom and the layer spacing 6 angstrom. Perpendicular momentum depends on the free-electron final-state assumption.

### Common difficulties

#### Why a band can look dark

**Typical misconception** An absent bright band proves there are no electronic states at that momentum.

**Better mental model** Photoemission intensity depends on spectral weight, occupation, dipole matrix elements and resolution. A gray cell has no samples; a dark colored cell has sampled but weak intensity. Neither proves a missing band. Each saved cut also retains its polarization, which can change brightness.

### Run the experiment

1. 01

   #### Build the map from saved cuts

   Choose New scan, then Save cut. Rotate Crystal azimuth and save another cut; Scan 48 azimuths supplies a complete acquisition example. Adjust Fermi window half-width to process those same saved images.

   **What to observe:** One slit leaves most momentum cells gray. More azimuths extend the coverage. Moving the crystal alone updates the current detector image; the stored map changes only after saving a cut. Changing sample or instrument conditions starts a new scan.

2. 02

   #### Locate the light and the electrons

   Follow the purple beam to the illuminated spot. Trace the emitted electrons through the lens and around the hemispheres. Tilt the sample: the white normal rotates while the lens axis stays fixed. Then compare Resolved cut and Suppressed weight.

   **What to observe:** Measuring energy and emission angle together maps out the occupied part of the spectral function. A band can look dark because the matrix element suppresses it. Higher temperature smears the Fermi edge, and changing the photon energy changes which momenta can be reached.

3. 03

   #### Capture a Fermi crossing

   Capture a Fermi crossing with momentum FWHM below 0.025 inverse angstrom, thermal edge width below 0.03 eV and matrix weight above 0.30. Then rotate the polarization to dim the band without moving it.

   **What to observe:** Check the numbers against the target, then compare the instrument view with the data it produces.

4. 04

   #### Change the cut and the photon energy

   Rotate the crystal azimuth to change the cut. Compare a hot, blurred acquisition with the cold target. Add interlayer hopping and vary the photon energy to see how it selects perpendicular momentum under the model assumption.

   **What to observe:** The band is a synthetic tight-binding band on a square lattice, with Lorentzian spectral weight, Fermi occupation and Gaussian instrument blur. The polarization effect uses an assumed stand-in for the dipole weight. Perpendicular momentum uses a free-electron final state and an assumed 10 eV inner potential. A slit analyzer records one cut at a time; the Fermi map combines the saved cuts and leaves unsampled cells gray. With interlayer hopping, its perpendicular momentum varies across the map.

Physical conclusion

Measuring energy and emission angle together maps out the occupied part of the spectral function. A band can look dark because the matrix element suppresses it. Higher temperature smears the Fermi edge, and changing the photon energy changes which momenta can be reached.

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)