Angle-resolved photoemission and spectral functions

Let binding energy be positive below the sample chemical potential. Energy conservation relates it to photon energy, measured kinetic energy, and the calibrated work function. For a clean crystalline surface, parallel momentum conservation includes the incident photon momentum, modulo a surface reciprocal vector. The displayed mapping neglects that photon contribution and assumes nonrelativistic vacuum-electron kinematics, appropriate approximations for the low-energy example below. More generally, initial parallel crystal momentum is the outgoing parallel electron momentum minus incident parallel photon momentum, modulo a surface reciprocal vector. Analyzer energy calibration and the reference Fermi edge should be established before comparing small shifts or gaps across temperature and photon-energy changes.

Momentum normal to the surface is less direct because translation symmetry is broken and the electron has a finite escape depth. Estimating it often introduces a final-state dispersion and inner potential; varying photon energy can help test a three-dimensional assignment. Surface reconstruction, charging, and cleavage-dependent termination may change the electronic structure itself. A surface-sensitive spectrum and a bulk oscillation experiment can therefore disagree for physical reasons even if both instruments and analyses are functioning correctly.

For a resolved band in equilibrium, the sudden approximation and a suitable photoemission model motivate intensity proportional to a matrix-element factor times the Fermi occupation times the single-particle spectral function. Use energy epsilon relative to the chemical potential, so occupied states lie mostly at negative epsilon. This epsilon is an energy, not an angular frequency. The spectral function includes both electron addition and removal, while ordinary direct photoemission predominantly accesses the occupied removal part.

For a canonical fermionic orbital, the complete spectral function integrates to one over all energies. Raw ARPES intensity generally does not: its matrix element varies with momentum and photon configuration, the positive-energy part is suppressed by occupation, and the experiment has a finite window. A polarization selection rule can suppress a band that still exists. Dividing by a small Fermi factor near or above the chemical potential can amplify noise dramatically and does not reconstruct missing information without assumptions.

A retarded self-energy changes both peak position and width. For an isolated simple pole with nearly constant negative imaginary self-energy, the spectral peak is Lorentzian; its half width is the positive broadening parameter. A lifetime interpretation also requires the appropriate quasiparticle residue and a clear definition of amplitude versus population decay. Momentum-distribution widths can be converted into energy scales only when dispersion, self-energy dependence, and the measurement cut justify that conversion.

Measured spectra convolve the occupied, matrix-weighted signal with energy and momentum resolution and add background. Gaussian widths add in quadrature only for genuinely Gaussian convolutions; a Lorentzian intrinsic peak convolved with a Gaussian becomes a Voigt profile. An unresolved pair can mimic one broad band, and a leading-edge shift can reflect thermal or resolution changes as well as a gap. Varying polarization, photon energy, temperature, and resolution tests these alternatives before a detailed many-body interpretation is assigned.