Quantum Mechanics of XPS

 

Quantum mechanical treatment

When a photoemission event takes place, the following energy conservation rule holds:

{\displaystyle h\nu =|E_{b}^{v}|+E_{kin}}

where {\displaystyle h\nu } is the photon energy, {\displaystyle |E_{b}^{v}|} is the electron BE (with respect to the vacuum level) prior to ionization, and {\displaystyle E_{kin}} is the kinetic energy of the photoelectron. If reference is taken with respect to the Fermi level (as it is typically done in photoelectron spectroscopy{\displaystyle |E_{b}^{v}|} must be replaced by the sum of the binding energy (BE) relative to the Fermi level, {\displaystyle |E_{b}^{F}|}, and the sample work function, {\displaystyle \Phi _{0}} .

From the theoretical point of view, the photoemission process from a solid can be described with a semiclassical approach, where the electromagnetic field is still treated classically, while a quantum-mechanical description is used for matter. The one—particle Hamiltonian for an electron subjected to an electromagnetic field is given by:

{\displaystyle i\hbar {\frac {\partial \psi }{\partial t}}=\left[{\frac {1}{2m}}\left(\mathbf {\hat {p}} -{\frac {e}{c}}\mathbf {\hat {A}} \right)^{2}+{\hat {V}}\right]\psi ={\hat {H}}\psi },

where {\displaystyle \psi } is the electron wave function, {\displaystyle \mathbf {A} } is the vector potential of the electromagnetic field and {\displaystyle V} is the unperturbed potential of the solid. In the Coulomb gauge ({\displaystyle \nabla \cdot \mathbf {A} =0}), the vector potential commutes with the momentum operator ({\displaystyle [\mathbf {\hat {p}} ,\mathbf {\hat {A}} ]=0}), so that the expression in brackets in the Hamiltonian simplifies to:

{\displaystyle \left(\mathbf {\hat {p}} -{\frac {e}{c}}\mathbf {\hat {A}} \right)^{2}={\hat {p}}^{2}-2{\frac {e}{c}}\mathbf {\hat {A}} \cdot \mathbf {\hat {p}} +\left({\frac {e}{c}}\right)^{2}{\hat {A}}^{2}}

Actually, neglecting the {\displaystyle \nabla \cdot \mathbf {A} } term in the Hamiltonian, we are disregarding possible photocurrent contributions.[7] Such effects are generally negligible in the bulk, but may become important at the surface. The quadratic term in {\displaystyle \mathbf {A} } can be instead safely neglected, since its contribution in a typical photoemission experiment is about one order of magnitude smaller than that of the first term .

In first-order perturbation approach, the one-electron Hamiltonian can be split into two terms, an unperturbed Hamiltonian {\displaystyle {\hat {H}}_{0}}, plus an interaction Hamiltonian {\displaystyle {\hat {H}}’}, which describes the effects of the electromagnetic field:

{\displaystyle {\hat {H}}’=-{\frac {e}{mc}}\mathbf {\hat {A}} \cdot \mathbf {\hat {p}} }

In time-dependent perturbation theory,for an harmonic or constant perturbation, the transition rate between the initial state {\displaystyle \psi _{i}} and the final state {\displaystyle \psi _{f}} is expressed by Fermi’s Golden Rule:

{\displaystyle {\frac {d\omega }{dt}}\propto {\frac {2\pi }{\hbar }}|\langle \psi _{f}|{\hat {H}}’|\psi _{i}\rangle |^{2}\delta (E_{f}-E_{i}-h\nu )},

where {\displaystyle E_{i}} and {\displaystyle E_{f}} are the eigenvalues of the unperturbed Hamiltonian in the initial and final state, respectively, and {\displaystyle h\nu } is the photon energy. Fermi’s Golden Rule uses the approximation that the perturbation acts on the system for an infinite time. This approximation is valid when the time that the perturbation acts on the system is much larger than the time needed for the transition. It should be understood that this equation needs to be integrated with the density of states {\displaystyle \rho (E)} which gives:[8]

{\displaystyle {\frac {d\omega }{dt}}\propto {\frac {2\pi }{\hbar }}|\langle \psi _{f}|{\hat {H}}’|\psi _{i}\rangle |^{2}\rho (E_{f})=|M_{fi}|^{2}\rho (E_{f})}

In a real photoemission experiment the ground state core electron BE cannot be directly probed, because the measured BE incorporates both initial state and final state effects, and the spectral linewidth is broadened owing to the finite core-hole lifetime ({\displaystyle \tau }).

Assuming an exponential decay probability for the core hole in the time domain ({\displaystyle \propto \exp {-t/\tau }}), the spectral function will have a Lorentzian shape, with a FWHM (Full Width at Half Maximum) {\displaystyle \Gamma } given by:

{\displaystyle I_{L}(E)={\frac {I_{0}}{\pi }}{\frac {\Gamma /2}{(E-E_{b})^{2}+(\Gamma /2)^{2}}}}

From the theory of Fourier transforms, {\displaystyle \Gamma } and {\displaystyle \tau } are linked by the indeterminacy relation:

{\displaystyle \Gamma \tau \geq \hbar }

The photoemission event leaves the atom in a highly excited core ionized state, from which it can decay radiatively (fluorescence) or non-radiatively (typically by Augerdecay). Besides Lorentzian broadening, photoemission spectra are also affected by a Gaussian broadening, whose contribution can be expressed by

{\displaystyle I_{G}(E)={\frac {I_{0}}{\sigma {\sqrt {2}}}}\exp {\left(-{\frac {(E-E_{b})^{2}}{2\sigma ^{2}}}\right)}}

Three main factors enter the Gaussian broadening of the spectra: the experimental energy resolution, vibrational and inhomogeneous broadening. The first effect is caused by the non perfect monochromaticity of the photon beam -which results in a finite bandwidth- and by the limited resolving power of the analyzer. The vibrational component is produced by the excitation of low energy vibrational modes both in the initial and in the final state. Finally, inhomogeneous broadening can originate from the presence of unresolved core level components in the spectrum.

Theory of core level photoemission of electrons

In a solid, also inelastic scattering events contribute to the photoemission process, generating electron-hole pairs which show up as an inelastic tail on the high BE side of the main photoemission peak. In some cases, we observe also energy loss features due to plasmon excitations. This can either a final state effect caused by core hole decay, which generates quantized electron wave excitations in the solid (intrinsic plasmons), or it can be due to excitations induced by photoelectrons travelling from the emitter to the surface (extrinsic plasmons). Due to the reduced coordination number of first-layer atoms, the plasma frequency of bulk and surface atoms are related by the following equation: {\displaystyle \omega _{surf}={\frac {\omega _{bulk}}{\sqrt {2}}}}, so that surface and bulk plasmons can be easily distinguished from each other. Plasmon states in a solid are typically localized at the surface, and can strongly affect the electron Inelastic Mean Free Path (IMFP).

Vibrational effects

Temperature-dependent atomic lattice vibrations, or phonons, can broaden the core level components and attenuate the interference patterns in an XPD (X-Ray Photoelectron Diffraction) experiment. The simplest way to account for vibrational effects is by multiplying the scattered single-photoelectron wave function {\displaystyle \phi _{j}} by the Debye-Waller factor:

{\displaystyle W_{j}=\exp {(-\Delta k_{j}^{2}{\bar {U_{j}^{2}}})}},

where {\displaystyle \Delta k_{j}^{2}} is the squared magnitude of the wave vector variation caused by scattering, and {\displaystyle {\bar {U_{j}^{2}}}} is the temperature-dependent one-dimensional vibrational mean squared displacement of the {\displaystyle j^{th}} emitter. In the Debye model, the mean squared displacement is calculated in terms of the Debye temperature, {\displaystyle \Theta _{D}}, as:

{\displaystyle {\bar {U_{j}^{2}}}(T)=9\hbar ^{2}T^{2}/mk_{B}\Theta _{D}}