= −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 ·S+S·D·S+S·A·I (Loubser and van Wyk [44]). The first term represents the electronic Zeemann interaction, the second is the interaction of the electron spin with the crystal fields produced by the surroundings, and the last is the magnetic hyperfine interaction due to the nuclear spins. S and I represent operators related with electron and nuclear spins, respectively. ge, D and A are tensors, which are to be determined from experiment and carry the information about the defects (impurities). These interactions will add pronounced features and great complexity in the ESR spectrum of a real system. The unpaired electron can gain or lose angular momentum and therefore change the value of the g-factor to be different from that for free electrons. For a spin system with S > 1/2, the spin-spin interaction among the unpaired electrons arises, splitting the energy level in absence of the external magnetic field. The magnetic moment of nucleus with a non-zero nuclear spin (I ≠ 0) will affect any unpaired electrons which have wavefunctions overlapping with that atom. This leads to the hyperfine interaction, splitting the ESR resonance signal into doublets, triplets and so forth. Note that the split of the energy levels caused by nuclear spins is often smaller than that of the electronic Zeeman interaction and spin-spin interaction. Moreover, the g-factor and hyperfine interaction are generally not the same for all orientations of an unpaired electron in an external magnetic field. This anisotropy depends on the electronic structure of the environment of the unpaired electron, thus containing information about the atomic or molecular orbital with which the unpaired electron is associated.
For over six decades, ESR has played a key role in the study of point defects in semiconductors and many other materials even in liquids. Under optimized conditions with narrow resonance line, the sensitivity can be very high, i.e. as low as the defect concentration of ∼1010 cm−3 at room temperature. The sensitivity can be increased at lower temperatures. The ESR information, which can be obtained, about the defects and impurities in semiconductors was nicely summarized by Loubser and van Wyk [44]:
(1) In the same crystal, different types of paramagnetic centers result in well distinguishable ESR spectra and the intensity of a particular center is proportional to the concentration of this kind of centers.
(2) The shift of the g-factor is a measure of the orbital contribution to the magnetic moment: is generally negative for electrons and positive for holes.
(3) From the resolved fine-structure lines, one can obtain the number of unpaired electrons associated with the defect.
(4) The hyperfine interaction due to the nucleus of spin I associated with the defects leads each fine-structure line split into 2I + 1 hyperfine lines. Therefore, impurities with non-zero nuclear spin can be identified.
(5) The anisotropy of the spectra, e.g., the angular dependence of g as the crystal is rotated in the external field, gives information about the symmetry of the defects.
(6) The widths of the ESR resonance lines contain information about the magnetic and exchange interaction between defects and about spin-lattice relaxation.
The above-mentioned information is essential for the defect identification and understanding its electronic structure. In many cases, the identification of point defects in semiconductors needs additional support from theoretical modelling. A careful correlation between ESR experiments and theoretical calculations allows an unambiguous identification of the defect. Along with the new development of optical and electrical detection methods, the sensitivity of ESR can be further increased (Kennedy and Glaser [45] and Spaeth [46]). For example, single isolated nitrogen-vacancy pair in diamond can be resolved and optically detected (Abe and Sasaki [47]) using the optical confocal microscopy.
Figure 6 shows another example, which is the ESR spectra of Se+ in isotopically pure Si (Nardo [48]). The electronic g factor is g = 2.0057. All stable Se isotopes X Se+ (X = 74, 76, 78, 80) have zero nuclear spin and the 77Se isotope has nuclear spin I = 1/2 and the isotropic hyperfine coupling of A = 1.6604 GHz with the donor electron spin. The central line (around g = 2) is corresponding to Se isotopes with zero nuclear spin and the hyperfine-split lines are corresponding to 77Se. The remaining features in the 77Se+ ESR spectrum are due to Se-H pairs. Those resonance peaks have a very small linewidth (<5 μT) due to Si isotopic purification.
Figure 6. ESR spectra of Se+ in 28Si for 28Si:Se (upper, nature Se) and 28Si:77Se (lower). The natural abundance of 77Se is 7.5%, and the remaining isotopes (92.5% abundance) possess zero nuclear spin. In the lower panel, the 77Se-doped
