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Research
Angle-Resolved Photoelectron Spectroscopy
Photoelectron spectroscopy is a general term which refers to all techniques
based on the photoelectric effect originally observed by Hertz. This
was later explained as a manifestation of the quantum nature
of light by Einstein, who recognized that when light is incident
on a sample, an electron can absorb a photon and escape from the
material with a maximum kinetic energy, hυ-φ, where hυ is
the light energy and φ is the work-function of the material. The
energy of an electron inside a solid can be obtained using photoelectron
spectroscopy—the core electrons will have a lower kinetic energy
than the valence electrons when absorbing the same photon energy. In
principle, the momentum of the electrons can also be obtained—different
momentum electrons will escape at different angles from the surface
of a material. However, since the electrons are being projected
through the surface, the momentum perpendicular to the surface is
not conserved. Therefore, angle-resolved photoemission is ideal
for 2D materials where the principle momentum directions of interest
are parallel to the surface. One of the primary materials of
interest are the 2D copper-oxide perovskite family in which the undoped
parent compounds are anti-ferromagnetic insulators and the doped
compounds have the highest known superconducting transition temperatures,
yet to be described by theory.
In practice, the electrons ejected from the material are collected using a hemispherical detector in which lens voltages direct the electron onto a two-dimensional (energy, momentum) multi-channel plate. The sample and the detector are kept in an ultra high vacuum (UHV) chamber in order to minimize surface contamination. Light sources are either synchrotron radiation at ~20-100eV, plasma Helium discharge at ~20eV, or more recently, modern-day lasers ~ 6 eV.
Since ARPES measures the energy and momentum of electrons inside a solid directly, it has a very natural theoretical description. The initial state of the electrons evolves, upon interaction with light, to a final state with a transition probability given by Fermi’s golden rule:
where the interaction Hamiltonian is given by:
The initial state of this dipole-interaction Hamiltonian is a neutral solid, while the final state is a dipole—the hole in the material left by the ejected electron and the photoelectron itself, which is a combination of the light and the ejected electron. If one assumes that the solid does not relax in the time it takes to measure the photoelectron (the so-called “sudden approximation”), then the intensity (I) of the spectral weight can be thought of as:
where I0 describes a matrix element dependent on incident photon energy, and f(w) is the Fermi function, signifying that ARPES measures the occupied density of states.
A (k, ω) is the single-particle spectral function that describes, theoretically, the energy and momentum of an electron in a solid. The “bare-band” electron dispersion is denoted by εk, while the “self-energy”, Σ, describes all the complex interactions that an electron has with other electrons and the lattice in a solid. This description is based on Landau’s Fermi liquid theory in which electrons, though part of a solid, remain independent “quasi-particles” near the Fermi level. They can therefore be understood as relatively free electrons, but with a renormalized dispersion due to interactions. Differences in how electrons interact lead solids to form metals, insulators, magnets, or superconductors. ARPES allows direct access to the electron spectral function through which these interactions take place.
