Defects in Functional Materials. Группа авторов

Defects in Functional Materials

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 (V_Zn) 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 V_Zn-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. where the summation includes all the electronic states i. ψ₊ is the positron wave function and ψi is the electron wave function with state i. The positron momentum is negligible as compared to that of its annihilating electron counterpart, and thus the total momentum of the positron-electron pair before the annihilation is effectively the electronic momentum (p in Fig. 4). Because of the linear momentum conservation, one of the annihilation gamma photons is Doppler blue shifted (p_γ,1 in Fig. 4) and the other one is red shifted (p_γ,2 in Fig. 4). The Doppler shifted energy is given by ΔE = pzc/2, where c is the speed of light and pz is the longitudinal momentum component of p in the direction of the 511 keV annihilation photon emission (i.e. z-direction in Fig. 4). For the Doppler broadening spectrum obtained from a single detector, the high momentum information is usually hidden behind the background noise. Introduction of a second detector for coincidence gating improves the resolution and the signal-to-noise ratio. Coincidence Doppler broadening spectroscopy (CDBS) can thus be used to explore the annihilation events originated from the high momentum electrons (i.e. the core electrons). As the core electron momentum distribution is the characteristic of a given element, CDBS are used to study the impurity decoration of vacancy type defects (for example Uedono et al. [40], Johansen et al. [41], and Rauch et al. [42]).

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 p_γ,1 and p_γ,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. where n₋ is the electron density. Since the electron density at the open volume defect is lower than the delocalized bulk state, the characteristic positron lifetime of the defect state is longer than that of the bulk state. The simple trapping model is normally used to analyse the positron lifetime spectrum; as such the positron lifetime spectrum modelled with the exponential components having different decaying time constants. Assuming for simplicity a system having only one positron defect trap, the spectrum is given by: where τ₁ and τ₂ are the constants. The long lifetime component with the decay time constant τ₂ (i.e. τ₂ > τ₁) is the positron defect trap component. Thus, τ₂ is the characteristic positron lifetime of the defect state and is the fingerprint of the defect. Application of PAS to study the vacancy type defects in SiC is discussed in Chapter 8.

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 (m_S) aligns itself either parallel (m_S = −1/2) or antiparallel (m_S = +1/2) to the external field, each alignment having a specific energy: E = m_Sg_eμ_BB₀, where B₀ is the external field, g_e is the g-factor for the free electron, μ_B is the Bohr magneton. For unpaired free electrons, the separation between the lower and the upper state is ΔE = g_eμ_BB₀. In classical theory, the g-factor is given by Landé formula in which the electron spin g_e factor equals to 2. Experimentally, it was found that the electron spin g_e factor for a free electron is ∼2.0023, indeed close to its theoretical value. As such, both μ_B and g_e 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 = g_eμ_BB₀. 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/g_eμ_B = 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 = μ_BB·g_e

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