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Why does the sample work function drop out of the energy accounting for photo-electrons in XPS?
An XPS spectrum reports photoelectron intensity, typically expressed as counts or counts per second, as a function of electron binding energy. The analyzer measures the kinetic energy of the emitted electrons. Their binding energies are then determined using conservation of energy, accounting for the incident photon energy and the work function. The work function that explicitly enters this calculation is the analyzer work function and not the sample work function. In this post, we will attempt to understand why the sample work function does not enter this calculation.
What is the origin of the work function?
Before following the energy balance, it helps to understand the origin of the work function. Within the Born-Oppenheimer approximation, the ions inside a solid are effectively held fixed. Their charge density abruptly terminates at the surface. The electron wavefunctions, however, do not end abruptly at that boundary. They extend a short distance into the vacuum. This extension is of the electron wavefunction into the vacuum outside of the surface of a solid is a manifestation of quantum mechanical tunneling.
Figure 1. Electron wavefunctions extend beyond the solid’s surface into vacuum. The resulting electron spill-out creates a separation of charge, indicated by δ⁺ and δ⁻, forming a surface dipole that constitutes an energy barrier for the electrons.
The resulting electron density outside the surface creates a separation of charge at the surface, forming a surface dipole as is illustrated in fig 1. The surface dipole lowers the electrostatic potential outside of the surface, creating an energy barrier that an electron must overcome to leave the solid. This is essentially the origin of the work function.
Accounting for the energy at emission
Returning to XPS, we can now account for this energy barrier as we follow a photoelectron from its initial bound state to the analyzer.
Consider an electron emitted from a particular core level. Its binding energy, Eb, is referenced to the fermi level. To escape into vacuum, the electron must also overcome the sample work function, φs, which is the energy difference between the Fermi level and the vacuum level just outside the surface.
For an incident photon of energy hν, conservation of energy gives:
Here, KEs is the electron’s kinetic energy just outside the sample, assuming it escapes without an inelastic energy loss. Rearranging,
The sample work function appears explicitly in this expression. It would therefore seem that we need to know the sample work function to determine core-level binding energies of its constituents. However, the kinetic energy is measured at the analyzer, and we still need to account for the energy change an electron undergoes through its passage from the sample surface to the analyzer.
Following the electron to the analyzer
When a conducting sample is in good electrical contact with the analyzer through the grounded sample stage, their Fermi levels align. Their work functions can still differ, so their local vacuum levels need not align. This difference produces a contact potential that changes the electron’s kinetic energy as it travels to the analyzer as illustrated in fig 2.
Figure 2. Electrical contact aligns the Fermi levels of the conducting sample and analyzer. Their work-function difference produces a contact potential that changes the photoelectron’s kinetic energy by φₛ − φₐ during transit. This compensates for the sample work function in the energy balance, leaving only the analyzer work function in the binding-energy calculation. The illustrated case shows φₛ > φₐ, so the electron gains kinetic energy.
Writing the analyzer work function as φA, the kinetic energy at the analyzer is:
If φs > φA, the electron gains kinetic energy on its way to the analyzer. If φs < φA, it loses kinetic energy. Substituting our expression for KEs,
The sample work function cancels, leaving:
