If, for the moment, we neglect HFIs and assume only isotropic g-values, the fit param- eters required to model the EDMR spectra comprise the g-value, field-independent and -dependent line widths, as well as a weighting factor for each spectral component. Assum- ing three line contributions for each spectrum (CBT/VBT and DB resonances at room temperature; CBT/VBT and TE lines at T = 10 K), one obtains 12 fitting parameters for each spectrum (three g-values, six line-width parameters, and three weighting factors). If applied to a single spectrum, such a fit is clearly over-parametrized and will most cer- tainly result in a best-fit spectrum that reproduces the experimental data. However, the present case offers means to substantially reduce the degrees of freedom of the fit.
The availability of EDMR spectra recorded at different MW frequencies and tempera- tures allows to follow a global fitting strategy by fitting all datasets with the same shared parameter set, instead of fitting each spectrum one at a time. Since spin-Hamiltonian parameters and line-broadening parameters do not depend on the MW frequency, the fit result is constrained to those sets of parameters that reproduce both the X-band and the 263 GHz spectra recorded at the same temperature. Global fitting of multifrequency spectra thereby allows to distinguish field-dependent and independent line-broadening mechanisms, which cannot be achieved bv fitting a single EDMR spectrum. Moreover, our fitting model assumes the presence of the CBT and VBT resonance at all temperatures (altough as the result of different spin-dependent transport processes). To our knowledge, no temperature dependencies of the CBT and VBT line shapes have been reported in previous EDMR or LEPR studies. Therefor, the fit parameters should not only be shared
amongst spectra at different MW frequencies, but also amongst spectra at different tem- peratures. As a result, the spin-Hamiltonian and line-broadening parameters of all four spectral components will be fit globally, leaving solely the weighting factors of the indi- vidual components to vary between the EDMR spectra at different temperatures and MW frequencies,1The number of fitting parameters is thereby reduced from 12 parameters per spectrum to 12 global parameters (g-values, g-strain broadenings and field-independent line widths) and 12 weighting factors for the three components per spectrum.
The degrees of freedom of the fit are further reduced by prior knowledge of a large part of the fit parameters. Especially, the EPR resonances of the DB defect and CBT/VBT states have been studied before using both EPR and/or EDMR/ODMR techniques (see chapter 3). From this earlier research, g-values, line broadenings and HFI parameters are in part well known and can then be either kept fixed during the fit, or at least serve as a valid starting point. These known parameters are summarized below for all four spectral components that we assume for our fitting model:
Dangling bond The DB resonance has been extensively studied in the past by both EPR and EDMR (see sections 3.2.1 and 3.4.1). Fehr et al. [148] determined the spin-Hamiltonian parameters in great detail using a combined approach of multifrequency EPR and DFT calculations. We thus use these parameters, comprising a rhombic g-tensor, an ax- ially symmetric A-tensor for the HFI with 29Si nuclei, g- and A-strain and a field- independent Voigtian line width to account for lifetime broadening and unresolved HFI. These parameters will be kept fixed and only the weighting factors will be fitted. The DB resonance will only be included for the room-temperature spectra.
CBT/VBT states The g-values of CBT and VBT states are known from earlier LEPR and EDMR/ODMR studies (see sections 3.2.3 and 3.4.1). For CBT electrons, an isotropic g-value of 2.0044 was found, with only little deviation between different studies. This value will thus also be excluded from the fit. The g-values for VBT holes reported by different groups, however, spread over a much wider range at about 2.009 to 2.015, such that we will leave this value as a fit parameter.2In addition, g-strain and line-width pa- rameters will be included in the fit for both the CBT and the VBT line, since a systematic line-shape study that distinguishes between field-dependent and -independent broad- ening for these resonance is not available in the literature. As for the DB defect, Umeda et al. [150] found HFI with29Si nuclei for the CBT resonance as the result of a mul- tifrequency LEPR study. This result is consistent with theoretical calculations by Ishii and Shimizu [151] (see section 3.2.3 for details). We will include their findings when simulating the CBT resonance, though without varying the HFI parameters during the fit. In order to fit the simulations to the experimental data, also A-strain parameters are required. Since no values are known from literature, we adopt the A-strain param- eters from the DB defect, noting that the HFI parameters for the CBT state are very similar to the A-tensor reported for DBs. For the VBT resonance, HFIs were found to be negligible [150, 151] and will thus be disregarded for the simulations.
TE line While the TE resonance has been observed before in both EDMR and ODMR
1Note that it is tempting to assume also the same spectral weighting factors for X-band and 263 GHz spectra recorded at the same temperature.
However, as described in chapter 4, different illumination intensities were used in the different experimental setups. As it is known from photoluminescence studies that transport channels in a-Si:H can strongly dependent an the charge-carrier generation rate (see section 3.3), different spectral weights at X-band and 263 GHz thus cannot be excluded.
2It is to be noted that several studies reported axial or rhombic g-tensor symmetries [26, 150]. While we will explore this possibility later on,
5.4 Spectral fitting
spectra of a-Si:H (see section 3.4.3), both its origin and the spin-Hamiltonian param- eters are to date unknown. Previous studies reported g-values around 2.008, which we will use as an initial value. Stutzmann and Brandt [244] reported a Gaussian line shape, which is consistent with our findings. Besides an isotropic g-value, we thus also include g-strain, as well as a field-independent Gaussian line width into the fit. Note that this line width is purely phenomenological, since the EDMR spectra alone do not allow a clear assignment to, e. g., dipolar coupled triplet excitons and a related broadening mechanism such as D-strain.
Besides the spin-Hamiltonian parameters, two additional field-offset parameters at X- band and 263 GHz, respectively, were included into the fit, as, unfortunately, the magnetic field was not calibrated properly at the time of the experiments. In particular, at 263 GHz, RF coils were attached to the sample holder (see section 4.2.2), which later turned out to be inadvertently made of ferromagnetic material and thus exposed the EDMR samples to a constant offset field of about 8 mT. At X-band, an NMR teslameter was used to mea- sure the magnetic field. Nevertheless, the teslameter was positioned outside of the EPR resonator, such that a constant offset with respect to the measured field remains at the sample position. In later experiments (see, e. g., chapter 6), this offset was calibrated by attaching an EPR field standard close to the EDMR sample. However, during the present experiments, such a field calibration was not carried out, such that the field-offset has to remain as a fitting parameter. This, in turn, requires at least one g-value parameter to be kept fixed during the fit (here, the g-values of the CBT and DB components).
Altogether, we obtain a total of 22 parameters that need to be adjusted to globally fit the four EDMR spectra:
Two g-values (VBT and TE).
Three g-strain broadening parameters (CBT, VBT and TE). Three field-independent line widths (CBT, VBT and TE).
Three weighting factors per spectrum (CBT/VBT/TE at T = 10 K; CBT/VBT/DB at room temperature).
Two field-offset parameters (X-band and 263 GHz).
The fitting routine was implemented in MATLAB. Simulated solid-state EPR spectra were calculated based on eq. 5.1 using thepepperfunction for powder-averaged EPR spectra from the EasySpin [43] toolbox. Global fitting was achieved by scaling all four spectra to equal amplitude and concatenating them into a single dataset. The simulated spectra were combined in the same way to achieve simultaneous fitting of all experimental spec- tra. The customly written fitting function is based on theesfitfunction for nonlinear least-squares fitting provided by the EasySpin library and uses a Nelder-Mead downhill- simplex algorithm.