In this section the mechanisms by which T1 relaxation occurs will be discussed.
According to the Boltzmann statistics discussed previously (section3.2.2) at finite temperatures there will be more electrons present at the lower energy level than the higher one. An incoming photon of microwave energy can induce an electron to move from the lower to the higher energy level or visa versa. Over time the two energy levels would become equally populated. However, this does not occur because relaxation causes re-population of the lower level. T1 relaxation involves
electrons spontaneously moving to the lower energy state by dissipating energy to the lattice [115, 123]. Solving the rate equation for the change in electron states gives equation 3.44, an exponential decay in magnetisation.
Despite the fact that photon-spin interactions are stronger than phonon-spin inter- actions at room temperature there are many more phonons present so the phonon relaxation mechanisms can compete with the photon excitation mechanisms, at low temperatures these phonons are less prevalent so the phonon-spin interaction reduces. This causes the relaxation mechanism to take longer [124].
The interaction between the lattice vibrations and the electrons can take different forms depending on the phonon energies and density. As T1 relaxation is deter-
mined by the time taken for the electron spin energy to dissipate into the lattice, experimental data can help understanding these interactions. Which relaxation mechanism dominates is dependent on temperature as this determines the number and energy of the phonons present. By understanding these different mechanisms experimental changes in T1 over temperature can be modelled to understand the
actual behaviours. Equation 3.45 describes the changes in T1 with temperature
[115].
1 T1
=aT +bTn+ c
exp(∆/kT)−1 (3.45)
In this equation the first term models the direct process, the second, the Raman process and the third, the Orbach process. T is the temperature. ∆ is the energy gap between the spin state at the level into which the electron is excited by the phonon during the Orbach process. a, b and c are coefficients which can be varied
it’s likelihood depends on the number of phonons at the correct energy. The effect is linear with temperature because a higher temperature environment has more phonons.
The direct process tends to dominate at low temperatures and at higher fields [123]. In equation 3.45 it is the first term.
Raman Process
The Raman process is a phonon process which occurs below the Debye temperature [123]. In diamond the Debye temperature is 2200 K [125]. A phonon is absorbed and subsequently emitted. The electron is excited to a virtual level and then relaxes back down. The change in energy of the phonon before and after the interaction is transferred to or from the electron and allows it to change energy state. Any phonon which has enough energy can be absorbed and hence the process is strongly temperature dependent. In equation3.45the term which relates to the Raman process is bTn. Where n depends on which type of spin state the electron is changing between.
Despite being a two phonon process, one phonon is absorbed and another of a different energy is emitted, the Raman process is much more likely to occur when compared to the direct process. This is because the direct process requires a phonon which has exactly the right energy, whereas the Raman process can absorb any phonon which has high enough energy.
Orbach
The Orbach process is independent of field [123]. It is a process which involves the electron absorbing a phonon with energy ∆. This causes the electron to be excited
(a) Electron moves to a lower energy level via the Orbach process.
(b) Electron moves to a higher energy level via the Orbach process.
Figure 3.6: The Orbach process showing a phonon, with energy∆, promoting an electron to a higher energy level and subsequently a phonon being emitted as the electron relaxes back down to a different energy level. Figure from [115].
up to a higher energy level. Another phonon, of a different energy, is emitted and the electron relaxes back down and thus has changed energy level. This process is shown in figure 3.6. This differs from the Raman process because the electron gets excited up to a real level.
The process depends on the number of phonons available which have energy ∆ and hence is temperature dependent. In equation 3.45 it is the final term.
Two additional process may also be relevant to the relaxation; local mode and thermally activated.
Local Mode
It may be difficult to distinguish between local mode interactions and thermally activated interactions when studying T1 behaviour [123]. The local mode mech-
anism involves a spin dissipating it’s energy into the heat bath. It occurs when a specific vibrational local mode pairs to an unpaired electron. As diamond has a symmetric, charge balanced, lattice, this effect only occurs where a defect is present. A defect, for example a single substitutional nitrogen atom, can vibrate with a characteristic frequency and the unpaired electron at the nitrogen atom will move. This causes a change in the environments of the surrounding electrons as a charge is moving and hence creating a field. This causes a change to the other electrons’ orbital angular momentum and hence via spin-orbit coupling dissipates
the form of rotation or vibration, either of which could change the symmetry of the crystal. If this motion occurs on the time scale of T1 it will affect EPR
spectra measurements. Energy is transferred to the lattice via molecular motion, for example methyl group rotation [126].