Spectroscopy for Materials Characterization. Группа авторов

2 Time‐Resolved Photoluminescence

       Marco Cannas and Lavinia Vaccaro

       Department of Physics and Chemistry – Emilio Segrè, University of Palermo, Palermo, Italy

2.1 Introduction to Photoluminescence Spectroscopy

      Luminescence is a paramount property for several optical applications including lighting, laser sources, detectors, and, recently, modern nanotechnologies (bioimaging, optoelectronics). To this aim, the research is currently active toward the development of production methods successful to finely control the physical and chemical characteristics of materials, thus tailoring their emission. This challenge strongly stimulates the luminescence spectroscopy to be more and more performing by new solutions that improve the efficiency of excitation sources and detectors used in the experimental setup. A crucial part for the study of luminescence properties is covered by the time‐resolved technique that allows to characterize in detail the properties of excited state from which the photon emission originates. Time‐resolved luminescence provides indeed the measure of spectroscopic parameters (lifetime, quantum yield, oscillator strength, and electron–phonon coupling) useful to fully describe the optical cycle excitation/emission.

      The purpose of this chapter is to provide a theoretical background of the luminescence properties related to color centers in wide band‐gap insulators. This is a very important issue that has been dealt with in several textbooks (see, for example, Refs. [1–7]). In particular, we will focus on specific features arising from the electron–phonon coupling (Zero‐Phonon Line and vibrational sidebands), which are fundamental for the interpretation of luminescence experiments in solids.

      2.1.1 Photoluminescence Properties Related to Points Defects: Electron–Phonon Coupling

Point defects are usually defined in the context of a crystalline network: if the regular array of atoms is interrupted, the lattice site is occupied differently than in the ideal crystal and it is called point defect. Defects include unoccupied sites (vacancies); occupied sites that in the perfect crystal are unoccupied (interstitial); impurities at sites that in the crystal lattice either are occupied by atoms of the pure material (substitutional impurities) or are unoccupied (interstitial impurities). The concept of defect may be also extended to amorphous materials even if the lack of regularity (long‐range order) introduces differences respect to the crystal, where a defect has fixed orientation and symmetry. The presence of defects in a crystalline or amorphous matrix may drastically modify the optical properties of the host material. In fact, they exist in different localized electronic states that cause optical transitions as absorption and luminescence with lower energies than the fundamental absorption edge of the material, from valence to conduction band. For these reasons, point defects are also called color‐centers or chromophores. Even if these transitions are localized at the defect site, the optical spectra will be influenced, to a greater or a lesser extent, by the fact that the color‐center is embedded in a solid matrix, either crystalline or amorphous. Indeed, it is closely surrounded by neighboring atoms with which it interacts; then, the description of its optical properties requires the defect‐matrix complex to be considered.

      To compute the electronic states involved in the optical transitions, such a complex is treated as a system of n electrons (mass m ≅ 9.109 × 10⁻³¹ kg, coordinate r) and N nuclei (mass M _α , coordinate R) which interact by Coulomb forces. In some ways, the defect‐matrix complex could be considered as an oversimplification of a molecule, whose energy levels and optical transitions are treated in the previous chapter. For the case considered here, the Hamiltonian is given by:

      (2.1)

      where the first and the second terms are the kinetic energy of electrons and nuclei, respectively, and V(r, R) is the interaction potential energy given by:

      (2.2)

      where e is the electron charge (e≅1.602 × 10⁻¹⁹ C) and Z is the atomic number.

      The Schrödinger's equation Hψ(r, R) = Eψ(r, R), where ψ(r, R) and E are the wave function and the energy eigenvalue of the defect‐matrix complex, is a many‐body problem that cannot be exactly solved. To this purpose, it is usual to adopt the Adiabatic Approximation based on the substantial difference between the electron mass m and the nuclear mass M _α (M _α /m ≥ 10³) so that electrons move much faster than nuclei, namely, the nuclei are almost fixed. According to this approximation, the wave function ψ _{l, n} (r, R) which describes the stationary state of the system is given by the product:

      (2.3)

      where ϕ _l (r, R) and φ _{l, n} (R) are the electronic and nuclear wave functions and are solutions of the two equations:

(2.4)

(2.5)

where ℏ=h/2π is the reduced Planck's constant (h≅6.626 × 10⁻³⁴ J⋅s), and the indices l and n represent the electronic and nuclear states, respectively. Equation (2.4) describes the stationary states of the electrons moving in the field of fixed nuclei and experiencing a potential energy V(r, R). For different nuclear positions, V(r, R) changes and both ϕ _l (r, R) and W _l (R) depend parametrically on R. The motion of the nuclei is governed by the second equation (Eq. 2.5), where W _l (R) plays the role of the potential energy and E _{l, n} represents the eigenvalue of the total energy of the defect‐matrix complex.

If the system is in a stable state, the nuclear motions reduce to small vibrations about the equilibrium positions R _l0. In the simplest case, when R is the distance between two nuclei, W _l (R) can be expanded in a Taylor series up to quadratic terms: