Angle-resolved photoemission spectroscopy

Angle-resolved photoemission spectroscopy measures the kinetic energy and emission angle of Electrons removed from a solid by light. With photon energy $h\nu$, sample work function $\Phi$, and binding energy $E_B$ measured downwards from the Fermi level, $E_{\mathrm{kin}}=h\nu -\Phi -E_B$. The measured parallel momentum is $k_{\parallel }=\sqrt{2m_e{E}_{\mathrm{kin}}}\sin \theta /\hslash$, modulo a surface reciprocal-lattice vector. Bright ridges reveal occupied electronic band structure; their slopes give the corresponding group velocity. Extracting a Fermi energy relative to a Dirac point requires a calibrated Fermi level, not merely an unlabelled image boundary.

For emission angle $\theta$ measured from the surface normal, a free outgoing photoelectron has parallel wavevector $k_{\parallel }=\sqrt{2m_e{E}_{\mathrm{kin}}}\sin \theta /\hslash$. Translation symmetry parallel to a clean periodic surface conserves this component modulo a surface reciprocal lattice vector, after accounting for photon momentum when needed. With initial binding energy $E_B$ below the Fermi level, its kinetic energy immediately outside the sample is $E_{\mathrm{kin}}=h\nu -{\varphi }_s-E_B$, where ${\varphi }_s$ is the sample work function. This sample-vacuum energy is distinct from the analyzer-referenced energy when contact potentials are present. Perpendicular crystal momentum generally needs a final-state model, whereas a strictly two-dimensional band has no perpendicular dispersion to reconstruct.

Within the dipole and sudden approximation, angle-resolved photoemission spectroscopy measures the electron removal spectral function multiplied by a transition matrix element. Finite instrumental energy and angular resolution convolve this product, and inelastic scattering adds a background. Peak positions reveal occupied electronic band structure, while intrinsic line widths contain lifetime information. Matrix elements depend on photon energy, polarization, orbital character, and geometry, so weak intensity does not by itself establish the absence of an electronic state. Equilibrium photoemission does not directly measure unoccupied bands far above the Fermi level.