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2.1.1 Photoluminescence Properties Related to Points Defects: Electron–Phonon Coupling
ОглавлениеPoint 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:
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:
The vibration frequency in the lth electronic state is related to the coefficient of quadratic terms in expansion (2.6) by ω l = (a l /μ)1/2, where μ is the reduced mass of the system. The substitution of Eq. (2.6) into (2.5) transforms it into the Schrödinger equation of a harmonic oscillator. Its solution is well known: the energy levels are E l, n = W l (R l0) + ℏω l (n + 1/2), where n is the vibrational quantum number and φ l, n (R) are Hermite polynomials multiplied by Gaussian functions [3, 4].
Most generally, the defect‐matrix complex consists of N nuclei with f = 3N − 6 degrees of freedom and the function W l (R) can be expanded in a series analogous to Eq. (2.6). In this case, it is appropriate to introduce new variables (normal coordinates), q s (s = 1,2,…,f), so that the problem reduces to find the energy levels and wave functions for the stationary states of a set of f independent harmonic oscillators (normal modes). For each normal coordinate, the nuclear potential curve takes the form of a parabola centered at q l, s0:
(2.7)
The total energy is:
(2.8)
where the index s indicates the configurational coordinate with vibrational frequency ω l, s , and n denotes the set of vibrational quantum numbers n s . The total wave function is therefore the product of the individual normal oscillators:
The interplay between the normal modes of the defect and those of the whole solid is crucial to determine the optical lineshape. Indeed, it is known that the eigenfunction spectrum for the normal modes of a solid consists of alternating allowed and forbidden bands. When a defect, composed of a finite number of atoms undergoing vibrations, is introduced into the solid, we can distinguish two different cases [2].
Band vibrations: The vibrational frequency of the atoms included in the defect lies in one of the allowed bands of the solid matrix. The defect is in resonance with the eigenfrequencies of the host network and radiates elastic waves, thus losing energy. In this case, all N atoms, both belonging to the defect and to the host matrix, participate to the motion and share the finite energy of the normal mode. Then, each atom has energy depending on N −1 and its displacement is related to N −1/2. Vibrations of this kind are called band vibrations.
Localized vibrations: The vibrational frequency of the atoms included in the defect lies in one of the forbidden bands of the solid. In this case, since the defect is not in resonance with any eigenfrequency of the unperturbed host matrix, it does not radiate elastic waves. The vibration amplitudes of the atoms in the solid drop off rapidly with increasing the distance from the defect; only the atoms of the defect environment participate in the vibration and their displacements are independent of N. Such vibrations are called localized vibrations.
