The electronic states of the impurity atoms doped into a crystal couple strongly to the vibrational modes of the host material through the electron-phonon interaction. This gives rise to continuous vibronic bands that are conceptually different from the electronic bands studied in the band theory of solids. The electronic states are localized near specific lattice sites in the crystal, and the continuous spectral bands arise by coupling the discrete electronic states to a continuous spectrum of vibrational (phonon) modes. This contrasts strongly with interband transitions which involve continuous bands of delocalized elec- tronic states.
The basic processes involved in the vibrational—electronic transitions in molecular materials were described in Sections 8.3.1-8.3.3. The principles developed there form a good starting point for the more general vibronic systems that we will be studying here. There are, however, two additional aspects of the physics that need to be discussed.
(1) We will usually be considering the optical transitions involving a low density of luminescent dopant ions or defects within an optically inert crystal. The interaction with the crystal host therefore has a strong effect on the spectra.
(2) We must consider the coupling of the electronic states to a continu- ous spectrum of vibrational modes, rather than the discrete modes of a
9.1
molecule. The density of states for the vibrational modes is determined by the phonon dispersion curves.
The formation of vibronic bands is depicted schematically in Fig. 9.1. Fig- ure 9.1(a) shows the optical transitions between the ground state of an isolated atom (e.g. a dopant ion) at energy E, and one of its excited states at energy E2.
If this atom is inserted into a crystalline host material, the electronic levels can couple to the vibrations of the lattice through the electron—phonon interaction.
At this stage, we do not wish to enter into the microscopic details of how such an interaction might occur, but merely consider the possibility that the coupling might be present. The presence of the coupling associates a continuous band of phonon modes with each electronic state, as shown in Fig. 9.1(b).
Optical transitions can occur between the vibronic bands if the selection rules permit them. We first consider an absorption transition. Before the photon is incident, the electron will be at the bottom of the ground state band. The absorption of a photon simultaneously puts the electron in an excited electronic state and creates a phonon, as shown in Fig. 9.1(b). Conservation of energy requires that the angular frequency {22 of the phonon involved must satisfy
hog = (F2 + hQ2) — Ey = (£2 -— Ey) + AQ, (9.1) where fiw, is the energy of the photon. Equation 9.1 shows that absorption is possible for a band of energies from (£2 — £1) up to the maximum energy of the phonon modes.
After the photon has been absorbed, the electron relaxes non-radiatively to the bottom of the upper band. The system then returns to the ground state band by a vibronic transition of energy:
howe = En — (EF, + hQ)) = (Eo — E1) — AQ, (9.2)
where Q, is the frequency of the phonon created in the ground state band.
Once the electron is in the ground state band, it relaxes to the bottom of the
band by non-radiative transitions, dissipating the excess vibrational energy as heat in the lattice.
excited state Ey + AQ,
Bygone E,
Ee] Ể Ig DP electron—phonon Ẽ : 5 TD
< ° coupling 6 ©
E,+hQ,
F\ ——— ---—- E,
ground state
(b) Atom doped into a
(a) Isolated atom vibronic solid
Vibronic absorption and emission 187
In the relaxation process, the vibrational en-
ergy of the localized phonon excited dur- ing the absorption transition rapidly spreads throughout the whole crystal and ultimately becomes heat.
Fig. 9.1 (a) Optical transitions between the ground state and an excited state of an isolated atom. (b) Absorption and emission transitions in a vibronic solid, in which the electron—phonon interaction couples each electronic state to a continuous band of phonons.
188 Luminescence centres
Fig. 9.2 Configuration diagram for the ground state and one of the excited electronic states of a vibronic solid. The optical transi- tions are indicated by the vertical arrows. The right hand side of the figure shows the general shape of the absorption and emission spectra that would be expected.
On first encountering configuration dia- grams, it is quite confusing to understand ex- actly what the configuration coordinate rep- resents physically. In the case of molecules
discussed in Section 8.3.2, it is easy to see
that Q corresponds to the amplitude of one of the normal modes of the vibrating molecule.
In a vibronic solid, Q might, for example, represent the average separation of the dopant ion from the cage of neighbouring ions in the
host lattice. In this case, the vibrations would
correspond to a breathing mode in which the environment pulsates radially about the optically active ion. This is equivalent to a localized phonon mode of the whole crystal.
In general there will a large number of vibra- tional modes in a solid, and the configuration coordinate can represent the amplitude of any one of these modes or perhaps a linear combination of several of them.
E ho ho
excited
State
| * zero-phonon
line absorption emission
ground |
state |
1 ]
On comparing eqns 9.1 and 9.2, we see that in a vibronic system the emis- sion generally occurs at a lower energy than the absorption. This red shift is called the Stokes shift. It is apparent from Fig. 9.1(b) that the Stokes shift arises from the vibrational relaxation that takes place within the vibronic bands. This contrasts with isolated atoms in which the absorption and emission lines occur at the same frequency.
The Stokes shift between absorption and emission can be understood in more detail by using configuration diagrams. The concept of configuration diagrams was introduced in Section 8.3.2 in the context of the vibrational- electronic spectra of molecules. This model carries over directly to the dis- cussion of the optical transitions in a vibronic solid. The electronic energy of the optically active species is a function of the vibrational configuration of the system as shown schematically in Fig. 9.2. This diagram shows the energy of two electronic states of a vibronic system as a function of Q, the
configuration coordinate. We have assumed that the electronic states are bound,
and they therefore have a minimum energy for some value of Q. In general, the equilibrium positions for the two states will occur at different values of the configuration coordinate. Therefore we label the position of the minima for the ground state and excited states as Qo, and Qp respectively.
The basic physical processes involved in the optical transitions of a vibronic solid are similar to those in a molecule, and we only give a brief summary here. More details can be found in Section 8.3.2. The energy of the electronic ground state can be expanded as a Taylor series about the minimum at Qo as follows:
dE 1 d*E
E(Ó) = E(Oo) + ——(O - Og)+>——<(O- Og)+---. dQ 2dQ? (93)
Since we are at a minimum, we know that ZE/đ(@ must be zero. Hence the E(Q) curve will be approximately parabolic for small displacements from Qo.
The same analysis can be applied to the excited state. This means that to first order we have harmonic oscillator potentials with a series of equally spaced energy levels as sketched in Fig. 9.2.
The Franck—Condon principle discussed in Section 8.3.3 tells us that optical transitions are represented by vertical arrows on the configuration diagram.
The absorption transition begins in the lowest vibrational level of the ground state, while the emission commences at the lowest vibrational level of the excited state following non-radiative relaxation. This gives rise to vibronic absorption and emission bands as shown in the right hand side of the figure.
In principle, the absorption and emission bands for a particular vibrational mode should consist of a series of discrete lines similar to those observed in molecules, each corresponding to the creation of a specific number of phonons.
However, in practice the electronic states can couple to many different phonon modes with a whole range of frequencies, and thus the spectra usually fill out to form continuous bands.
The transitions from the lowest vibrational level of the ground state to the lowest level of the excited state are called the zero-phonon lines. Since there are no vibrational quanta involved, the absorption and emission lines occur at the same frequency. In the absorption spectrum there will be a band of vibronic transitions to higher energy of the zero-phonon line, while in the emission spectra there will be a corresponding band to lower energy. The shape of the absorption and emission bands depends on the overlap of the vibrational wave functions as determined by the Franck—Condon factor given in eqn 8.11. In general, the peak occurs away from the zero-phonon line due to the difference between Qo and Oo: As with molecules, we would expect mirror symmetry between the emission and absorption about the zero-phonon line.
In the sections that follow, we will apply these general principles to the optical spectra of colour centres and luminescent impurities. In many cases, it will be sufficient to use simpler level diagrams of the type shown in Fig. 9.1(b) to explain the absorption and emission, without delving into the complications of the configuration coordinate model in any detail.