Matrix Elements PhotoemissionEdit
Matrix Elements Photoemission
Matrix element photoemission concerns how the intrinsic electronic structure of a material is projected into the signals detected by photoemission experiments. In practice, the measured intensity in angle-resolved photoemission spectroscopy (ARPES) is not a direct map of the spectral function of the solid alone; it is modulated by quantum-mechanical transition amplitudes, or matrix elements, that depend on the geometry of the experiment, the photon energy and polarization, and the character of the initial and final electronic states. This interplay between the intrinsic electronic structure and the matrix elements is what scientists must untangle to draw reliable conclusions about band structure, orbital makeup, and many-body effects from photoemission data.
Historically, matrix-element effects have moved ARPES from a simple “band-mapping” tool toward a more nuanced spectroscopy in which careful control of experimental conditions reveals or hides specific orbital contributions. The field relies on a mix of approximate models and increasingly sophisticated calculations to predict how matrix elements transform A(k, ω) into I(k, ω) in a variety of materials. As a result, researchers often use multiple photon energies, different light polarizations, and varied geometries to build a complete picture of a material’s electronic structure.
Fundamental framework
The ARPES intensity and the role of matrix elements
In the dipole regime, the photoemission intensity at momentum k and energy ω is often expressed, schematically, as I(k, ω) ∝ |Mfi(k, ω)|^2 × spectral function × f(ω), where f(ω) is the Fermi-Dirac distribution. Here, Mfi(k, ω) is the transition matrix element between an initial Bloch state and a final state of the photoelectron. The spectral function A(k, ω) encodes the intrinsic electronic structure and many-body effects, while |Mfi|^2 encapsulates how the experimental setup couples to those states. Practical interpretations must recognize that I(k, ω) is not a direct one-to-one image of A(k, ω) unless Mfi is understood and accounted for. See also angle-resolved photoemission spectroscopy for a broader context.
Initial and final states
The initial state is typically a Bloch-like state in a crystal, carrying information about orbital character, lattice symmetry, and correlations. The final state describes the escaping photoelectron and is often approximated as a free-electron-like state, though more realistic descriptions may use time-reversed LEED-type states or other final-state constructs to capture surface and boundary effects. The choice of final-state model strongly influences the calculated matrix element and thus the predicted angular and energy dependence of I(k, ω). For a modern treatment, see discussions of final state in photoemission and the contrast between plane-wave and more elaborate final-state descriptions.
Polarization, geometry, and selection rules
Matrix elements are highly sensitive to experimental geometry: the polarization vector of the incident light, the incidence angle, the detection angle, and the alignment of the sample surface all enter the calculation of Mfi. Selection rules, derived from symmetry and orbital character, determine which transitions are allowed or suppressed. As a result, certain bands may appear with strong intensity under one polarization and be nearly invisible under another, even if they share the same intrinsic spectral weight. See also dipole approximation and selection rules for foundational concepts.
K-space and conservation laws
Under typical ARPES conditions, the in-plane momentum component k|| is conserved across the photoemission process up to a reciprocal lattice vector, while the out-of-plane component kz remains more model-dependent due to the surface potential. The matrix element thus also carries a kz dependence through the initial-state character and the chosen final-state model. See k-space and Brillouin zone for related concepts.
Theoretical models of matrix elements
The three-step model
In the traditional three-step picture, photoemission is treated as a sequence: (1) optical transition from the initial bound state to a final intermediate state within the crystal, (2) propagation of the excited electron to the surface, and (3) transmission into the vacuum as a free electron. The resulting matrix elements incorporate aspects of both initial-state symmetry and the boundary conditions at the surface. This model has been instrumental for intuition and semi-quantitative analysis, and it is often used as a practical framework to predict how I(k, ω) should change with experimental geometry. See three-step model.
The one-step model
A more unified approach treats photoemission as a single, coherent process governed by the same Hamiltonian that defines the solid and the vacuum, including surface effects and final-state scattering in one framework. The one-step model provides a more rigorous description of matrix elements, especially in regimes where surface sensitivity and final-state interactions are strong. It is computationally more demanding but is increasingly used for quantitative comparisons across materials. See one-step model of photoemission.
Final-state effects and surface sensitivity
Final-state effects—how the outgoing electron interacts with the surface and how the surface potential shapes the detected signal—can significantly modulate matrix elements. These effects are essential for accurate interpretations in materials with strong surface reconstruction, layered structures, or anisotropic orbital makeup. See final-state effects in photoemission for a broader treatment.
Practical interpretation and debates
Extracting the spectral function
A central practical question is when I(k, ω) can be treated as a proxy for spectral function. In many materials this is an approximation; a well-behaved A(k, ω) can be teased out only after careful accounting of Mfi. Debates in the field focus on how best to deconvolve or model the matrix element to reveal intrinsic electronic properties, particularly in strongly correlated systems where A(k, ω) contains nontrivial structure such as incoherent continua or momentum-dependent self-energies.
The role of polarization and photon energy
Because |Mfi|^2 can vary strongly with photon energy and polarization, researchers routinely exploit this to highlight or suppress specific orbital characters. For example, certain d-orbitals may couple more strongly to a particular polarization geometry, illuminating orbital textures that would be hidden otherwise. This technique is a practical strength of ARPES, but it also requires careful interpretation to avoid misattributing spectral weight to intrinsic effects when it is in fact a matrix-element artifact. See orbital symmetry and spin-orbit coupling for related ideas.
Sudden approximation and beyond
The traditional interpretation often relies on the sudden approximation, which assumes the photoelectron escapes without affecting the residual system. In some materials, especially those with strong correlations or near surfaces with unusual screening, this approximation can be questionable. Contemporary work explores beyond-sudden approaches to capture more accurately the interplay between the photoelectron and the many-body system. See sudden approximation for background and current discussions in the literature.
Controversies and ongoing debates
- To what extent can different final-state models be reconciled across diverse materials? The consensus is that no single model perfectly captures every material, but cross-checks among three-step and one-step predictions can provide robust insights when used judiciously.
- How much spectral weight in I(k, ω) is intrinsic versus a matrix-element artifact? The answer depends on material class, photon energy, and geometry; the best practice is often a multi-geometry, multi-energy experimental program complemented by theory.
- Do matrix elements obscure or reveal many-body effects? The consensus is nuanced: matrix elements can both mask and reveal features, depending on how well they are modeled and how carefully the data are analyzed.
Applications and case studies
Cuprate superconductors and orbital character
In cuprate superconductors, ARPES has been crucial for mapping the Fermi surface and the d-orbital character of bands near the Fermi level. Matrix elements help identify whether observed features arise from particular copper-oxide plane orbitals or from more three-dimensional components, and they have informed interpretations of pseudogap phenomena and superconducting gaps. See high-temperature superconductivity and orbital symmetry for context.
Graphene and other carbon-based materials
Graphene and related carbon systems present a relatively clean testing ground for matrix-element effects because of their simple band structure near K points and strong sensitivity to polarization. Matrix elements aid in distinguishing between σ- and π-bands and in revealing subtle changes due to substrate or strain. See graphene and carbon for broader coverage.
Topological materials and spin textures
In topological insulators and Dirac/Weyl semimetals, matrix elements can be leveraged to highlight spin textures and orbital-manifold contributions to surface states, while also posing challenges in disentangling surface vs. bulk signals. See topological insulators and Dirac semimetals for related topics.