varying positron incidence energies. PAS has been used to study the correlations between the materials properties and the vacancy type defects in different materials. For example, Krause et al. [35] studied the EL2 and its metastable state in GaAs; Lawther et al. [36] studied the compensating defect complexes in Group-V heavily doped Si; Tuomisto et al. [37] identified zinc vacancy (VZn) as the dominant acceptor in undoped ZnO; Ling et al. [38] identified the shallow acceptors in undoped GaSb; Khalid et al. [39] correlated magnetic data in undoped ZnO with VZn-related defects. Kilpeläinen et al. [49] studied the thermal evolution of the defect complexes in P-doped SiGe.
Two techniques are typically used in PAS, namely the Doppler broadening spectroscopy (DBS) and the positron lifetime spectroscopy (PLS). DBS measures the Doppler broadening of the line shape of the annihilation photopeak of the annihilation gamma ray energy spectrum. Doppler broadening spectrum reveals the electronic momentum distribution seen by the positrons, i.e.
Figure 4. Schematics of the DBS momentum conservation, accounting for the momentum before the annihilation p (i.e. effectively the electron momentum), and the momenta of the two annihilation gamma photons pγ,1 and pγ,2.
PLS measures the positron lifetime distribution in the sample. The positron lifetime is inversely proportional to the overlapping of the electron and positron density, i.e.
5. Magnetic Characterization
The resonant absorption of electromagnetic radiation by unpaired electrons is known as electron spin resonance (ESR) [43]. An electron has a spin S of 1/2 and an associated magnetic moment. In an external magnetic field, two spin states have different energies and this is called Zeeman effect. The electron’s magnetic moment (mS) aligns itself either parallel (mS = −1/2) or antiparallel (mS = +1/2) to the external field, each alignment having a specific energy: E = mSgeμBB0, where B0 is the external field, ge is the g-factor for the free electron, μB is the Bohr magneton. For unpaired free electrons, the separation between the lower and the upper state is ΔE = geμBB0. In classical theory, the g-factor is given by Landé formula in which the electron spin ge factor equals to 2. Experimentally, it was found that the electron spin ge factor for a free electron is ∼2.0023, indeed close to its theoretical value. As such, both μB and ge may be seen as constants. Therefore, the splitting of the energy levels is directly proportional to the magnetic field’s strength (see Fig. 5). An unpaired electron can transfer between two energy levels by either absorbing or emitting a photon of energy hν setting the resonance condition at hν = ΔE. This result is the fundamental equation for the ESR spectroscopy technology: hν = geμBB0. In principle, this equation holds for a large combinations of frequency (ν) and magnetic field (B0) values. Practically, most of the ESR measurements are performed with microwaves in the 9–10 GHz region. The ESR spectrum is usually taken by fixing the microwave frequency and varying the magnetic field. At the condition of the gap between two energy states matching the energy of the microwaves, the unpaired electrons can jump between their two spin states. Following Maxwell–Boltzmann distribution, there are typically more electrons in the lower state, leading to a net absorption of energy. This absorption is monitored and converted into a spectrum. For the microwave frequency of 9.388 GHz, the resonance should occur at the magnetic field of about B0 = hν/geμB = 0.3350 T.
Figure 5. Schematics of the energy splitting of an unpaired electron under external magnetic field.
In real systems, electrons are generally associated with one or more atoms of their surroundings. The spin Hamiltonian can be written as H = μBB·ge