Electronic and vibronic transitions

In a solid the discrete available energy levels of neighbouring atoms combine result- ing in a continuum of states. This overlap leads to two or more continuous bands ink space. The two bands commonly discussed are referred to as the valence band and the conduction band with the gap between these two taking the name of the band gap. The introduction of impurities into the solid leads to additional states being added into the band gap. With this simple model of the electronic configu- ration of the energy states for a crystalline solid the possible interstate electronic absorption process can be divided into four categories:

• Intrinsic absorption: Whereby an electron is promoted from the valence band to the conduction band.

• Extrinsic absorption: In which an electron is transferred between either the valence or conduction band to a defect state within the band gap

Absorption Emission Conduction Band Valence Band ~ω Zero-phonon transitions E Q ~ω Vibronic transitions Absorption Emission

Zero phonon line Absorption

I

λ (b)

(a)

Figure 3.2: Figure (a) shows both zero-phonon transitions and vibronic tran-

sitions between two defect states in the band gap. Q is the configuration co- ordinate. At zero temperature both absorption and emission processes originate from the ground states but are not required to finish in a ground state. Figure (b) shows how the vibronic bands are observed symmetrically centred about the zero phonon line for an absorption and emission experiment, respectively.

Adapted from [165].

• Free carrier absorption: In this case the energy absorbed causes a transition within the continuum of states for a single band

• Inter defect state absorption: Whereby an electron is promoted from one defect state in the band gap to another.

The indirect band gap of diamond is approximately 5.45 eV which corresponds to 227 nm light [166]. Extrinsic absorption results in the observation of broad absorp- tion bands since states are available from anywhere within one of the continuous valence or conduction bands to a defect state. The minimum energy for transition of an extrinsic absorption is given by the smallest value of: ∆E =|Ev−Edef ect|or ∆E =|Ec−Edef ect|, where Ev andEc are the maximum valence band energy and the minimum conduction band energy, respectively. Such broad band absorption has been studied in diamond elsewhere with a comprehensive collection having been collated by Zaitsev [167].

Of most interest to this thesis are the inter defect state absorption processes. in order to discuss these properly it is necessary to realise that defect states in the band gap do not correspond to only a single energy level but as a set of energy levels as exist for a simple quantum well. The Born-Oppenheimer approximation approximation states that the heavy nature of the atoms in a material simply provide a periodic background potential to the electronic states therefore electronic and nuclear motions may be considered independently. In configuration coordinate diagrams this allows electronic transitions to be represented by vertical arrows and changes in vibrational states to be considered as a change in Q, an overall measure of the total motional atomic energy of the system. This also allows the wavefunction to be written as the product of the electronic, ψ(r), and vibrational,

χn(Q) (where Q is the configuration co-ordinate), parts wheren is the vibrational quantum number of the electronic state [165].

The shape of the vibrational potential is often considered to be harmonic where

V ∝Q2 . An anharmonic potential would manifest as a distortion to the quadratic approximation but this will not be considered in detail. An example of a configu- ration coordinate diagram can be seen in Figure3.2. The zero-phonon absorption energy corresponds to the energy difference between the lowest energy state of the ground state and the lowest energy state of the excited state. A transition for whichχf(Q) = χi(Q) is referred to as an electronic transition as there has been no change in Q. Transitions for whichχf(Q)6=χi(Q) are called vibronic transitions. Figure3.2 shows that for an absorption experiment, the vibronic transitions have a higher energy than the zero-phonon line (ZPL) whereas in luminescence experi- ments they have lower energy. Vibronic states are not observed as a series of sharp features but instead, since the thermally excited levels are separated by only ¯hω, they form a vibronic continuum in the observed optical absorption spectrum. The Huang-Rhys factor, S, is a measure of the probability of a transition occurring with no phonon interaction, i.e. a zero phonon transition. At low temperatures an electron is unlikely to have sufficient thermal energy to be excited out of its ground state. This means that the majority of electrons will undergo zero-phonon transitions at low temperatures instead of contributing to the vibronic side band.

In addition, and more importantly in diamond, the Huang-Rhys factor is tem- perature dependent. The Huang-Rhys factor is temperature-dependent because of the phonon population factor, and, for many optical centres, this leads to a modest increase (typically a factor of 2 to 3) in the zero-phonon transition proba- bility as the temperature is reduced from 300 K to 100 K or lower. The width of the ZPL also decreases as the temperature is reduced because of the reduction in Lorentzian broadening. Consequently, the ratio of the height of the ZPL to that of the vibronic band is increased by a factor of ≈ 10 to 20 on cooling from 300 K to 80 K. For many optical centres in natural diamonds little improvement is obtained by going to lower temperatures because other factors also lead to a broadening of the ZPLs. However, for some optical centres in diamonds with a very low defect concentration (such as those studied in this thesis) a further substantial narrowing of the ZPL can be obtained by cooling to 4K [168]. Consequently most experi- ments are performed at cryogenic temperatures, often around 80 K for diamond, in order to facilitate the study of the zero-phonon lines.