Spin Hamiltonian

2.3.1 Electron Zeeman interaction

In section 2.1, we derived the Zeeman interaction term, 𝓗_EZ= g_eμ_BST_B

0, for the case of the free electron, where g_eis the isotropic g-value of the free electron. In general, how- ever, the interaction of a paramagnetic centers with an external magnetic can dependent on the relative orientation of S and B₀due to asymmetries in the wavefunction or the sur- rounding crystal field. Such anisotropies are taken into account by replacing the scalar g-value with a 3 × 3 interaction matrix g. The general expression for the electron Zeeman term then becomes

𝓗_EZ= μ_BSTgB₀. 2.26

The interaction matrix g is commonly referred to as the g-tensor, although, strictly speak- ing, g itself is not a tensor in the physical sense. However, the experimentally observed g-value for a given orientation n = B₀/B₀is [41]

g (n) = √nT_(gT_{g)n, 2.27}

and this measurable quantity √gT_{g is a symmetric second-rank tensor. In the following,} we will refer to this quantity as the g-tensor. The symmetric g-tensor, while not being diagonal in general, can be diagonalized via a rotational transformation R(α, β, γ) to its principal axis system:

g_diag= RT_{(α, β, γ) gR(α, β, γ) =} ⎛⎜ ⎜ ⎝ g_x g_y g_z ⎞ ⎟ ⎟ ⎠ 2.28

The g-tensor can thus be fully characterized be its three real principal values g_{x, y, z}and the three so-called Euler angles α, β, γ that define the orientation of its principal axis system. In EPR, the latter is usually regarded as the molecular frame and all interaction tensors are referred to this frame.

The orientation of B0with respect to the molecular frame can be expressed with two polar angles θ and ϕ, such that B₀ = B₀(sin θ cos ϕ, sin θ sin ϕ, cos θ). The observed ef- fective g-valueis then given by

g(θ, ϕ) = √(gxsin θ cos ϕ)2_{+ (g}

ysin θ sin ϕ)2+ (gzcos θ)2. 2.29

Based on the symmetry of the paramagnetic center, three types of g-tensor anisotropies can be distinguished: (i) For cubic symmetry, g is isotropic with g = g_x= g_y = g_z and the same EPR spectrum is observed for all orientations. (ii) For axial symmetry, two principal g-values g_⟂= g_x= g_y and g_∥ = g_z are observed for perpendicular and parallel orientation of B₀ with respect to the symmetry axis. The experimentally observed ef- fective g-value in this case is determined by the angle θ between B₀ and the symmetry axis: g(θ) = (g2⟂sin2θ + g2∥cos2θ)

1/2

. (iii) A more complex structures of the paramag- netic center, where the axial symmetry is lifted, is referred to as rhombic symmetry. In this case, the three principal values g_{x, y, z} are non-degenerate and the observed g-value depends on both polar angles θ and ϕ as described by eq. 2.29.

For a crystalline sample, where all paramagnetic sites exhibits the same orientation, the g-tensor anisotropy can be directly measured by rotating the sample and recording

the effective g-value as a function of θ and ϕ. If a sample, however, is present in the form of a powder or when dealing with disordered materials such as a-Si, the orientation of an individual paramagnetic center is random. For truely disordered samples, all orientations occur with equal probability and a distribution of all possible orientations is observed at once. Examples of the resulting characteristic EPR spectra, referred to as powder patterns, are shown in fig. 2.3 for the three cases of an isotropic g-value (fig. 2.3a) and a g-tensor with axial (fig. 2.3b) and rhombic (fig. 2.3c) anisotropy.

The principal g-values of real paramagnetic centers, for example, defect states in silicon, deviate from the free-electron value g_e. This is due to an interaction of the electron-spin magnetic moment with the orbital angular momentum of the charge carrier by means of spin-orbit coupling. Although, for a non-degenerate electronic ground state, the or- bital angular momentum is quenched (L = 0), a perturbation arises from the ground state being mixed with excited states, such that an orbital momentum L is admixed into the ground state. The Hamiltonian given in eq. 2.26 can then also be written as

𝓗_EZ = μ_B(LT+ g_eST_{) B0}+ λLTS. 2.30

The first part is the electron Zeeman term including the admixed orbital angular mo- mentum, and 𝓗_SO= λLT_{S is the spin-orbit interaction, where λ is the spin-orbit coupling} constant. By means of second-order perturbation theory and comparison with eq. 2.26, an expression for gij, the elements of g, can be derived [42]:

g_ij= g_eδ_ij+ 2λ_∑ n≠0

⟨0|L_i|n⟩ ⟨n|L_j|0⟩

E₀− E_n , 2.31

where |0⟩ and |n⟩ denote the wavefunctions of the ground state and the nth excited state with energies E₀ and E_n, respectively. Thus, the smaller the energy gap between the ground state and the excited states, and the larger the spin-orbit coupling, the larger the deviation of the principal values of g from the free-electron value g_e. For all paramagnetic centers discussed within this work, only small g-value are observed (Δg < 0.01). However,

giso 341.5 342 342.5 Magnetic field (mT) EPR sig nal (ar b .units) (a) g⟂ g∥ 341.5 342 342.5 Magnetic field (mT) (b) gx gy gz 341.5 342 342.5 Magnetic field (mT) (c)

FIGURE 2.3 Simulated X-band (ν =9.6 GHz) EPR powder spectra for different g-tensor symmetries. Absorption spec- tra are shown for(a)an isotropicg-tensor with g=2.0055,(b)an axially symmetricg-tensor with g⟂=2.0065 and

g∥=2.0042, and(c)a rhombicg-tensor with principal values gx=2.0079, gy=2.0061 and gz=2.0034. These g-values correspond to the principal values found for the dangling-bond defect in a-Si:H [148]. Note thatg-strain has been ne- glected for the simulations, such that the simulated spectra do not resemble the actual EPR signature of DB defects. Instead, the spectra have been convolved with a field-independent Gaussian line width of0.05 mT. Simulations have been carried out using the EasySpin [43] MATLAB toolbox.

2.3 Spin Hamiltonian

for sufficiently strong doping and large λ, it is possible to obtain strong g-value shifts for paramagnetic states in amorphous silicon [44].

For disordered materials such as amorphous silicon (see chapter 3 for details), g-values are not sharply defined, as indicated by fig. 2.3. Owing to the disorder, variations of the local structural and electronic environment of paramagnetic states result in a distribution of g-tensors, commonly referred to as g-strain (Δg). Due to the magnetic-field dependence of the Zeeman interaction, g-strain leads to a broadening of the EPR spectrum that is pro- portional to B₀. Thereby, field-dependent g-strain broadening counteracts the resolution enhancement obtained by carrying out EPR experiments at high MW frequencies and magnetic fields. On the other hand, multifrequency/-field experiments can be used to distinguish between g-strain and other field-independent broadening mechanisms (see also section 2.4). We will employ this strategy in chapter 5, where multifrequency EDMR is used to separated signals stemming from different paramagnetic states by the field de- pendencies of the respective spectral line widths.

2.3.2 Hyperfine interaction

The hyperfine interaction (HFI) between an electron spin S and a nuclear spin I is de- scribed by the Hamiltonian

𝓗_HFI/h = STAI, 2.32

where A denotes the hyperfine-coupling tensor. Note that, as for the g-tensor discussed in the previous section, A itself is only a 3 × 3 matrix, but the observable √AT_{A is a sym-} metric second-rank tensor, which we will refer to as the tensor A (see also ref. [41]).

The HFI can be split apart into the isotropic Fermi contact interaction and the electron- nuclear dipolar interaction, such that

A = a_isoI₃+ T, 2.33

where I₃ denotes the 3 × 3 identity matrix, a_iso is the isotropic hyperfine-coupling con-