Kramers-Kronig analysis

Dielectric constants of metals, semiconductors and insulators can be detennined from ellipsometry measurements [38, 39]. Since the dielectric constant can vary depending on the way in which a fihn is grown, the measurement of accurate film thicknesses relies on having accurate values of the dielectric constant. One connnon procedure for detennining dielectric constants is by using a Kramers-Kronig analysis of spectroscopic reflectance data [39]. This method suffers from the series-tennination error as well as the difficulty of making corrections for the presence of overlayer contaminants. The ellipsometry method is for the most part free of both these sources of error and thus yields the most accurate values to date [39]. 

The 1/to4 high frequency limit for R can be useful in determining optical constants from Kramers-Kronig analysis of reflectance data (see Section 2.7). Reflectances at frequencies higher than the greatest far-ultraviolet frequency for which measurements are made can be calculated from (9.15) and used to complete the Kramers-Kronig integral to infinite frequency. 

Rasigni, M., and G. Rasigni, 1977. Optical constants of lithium deposits as determined from Kramers-Kronig analysis, J. Opt. Soc. Am., 67, 54-59. 

The Pt + Pt intervalence transitions of such chain complexes occur in the regions 25,000-18,200 cm 1, 23,600-14,300 cm 1 and 20,600-7,500 cm 1 for chloro-, bromo-, and iodo-bridged complexes, respectively, the trend Cl > Br > I being the reverse of that of the conductivity of the complexes. The transition wavenumbers may be determined either by Kramers-Kronig analysis of specular reflectance measurements or from plots of the excitation profiles of Raman bands enhanced at or near resonance with the Pt I-PtIV intervalence band. The maxima have been found to be related to the Pt —PtIV chain distance, the smaller the latter the less being the intervalence transition energy (3). 

Fourier transform infrared microscopes are equipped with a reflection capability that can be used under these circumstances. External reflection spectroscopy (ERS) requires a flat, reflective surface, and the results are sensitive to the polarization of the incident beam as well as the angle of incidence. Additionally, the orientations of the electric dipoles in the films are important to the selection rules and the intensities of the reflected beam. In reflectance measurements, the spectra are a function of the dispersion in the refractive index and the spectra obtained are completely different from that obtained through a transmission measurement that is strongly influenced by the absorption index, k. However, a complex refractive index, n + ik can be determined through a well-known mathematical route, namely, the Kramers-Kronig analysis. 

E. Shiles, T. Sasaki, M. Inokuti, and D. Y. Smith, "Self-consistency and sum-mle tests in the Kramers-Kronig analysis of optical data Applications to aluminum," Phys. Rev. B, 22, 1612-28 (1980). 

In Section I, the spectra of e"(ai) consist of Dirac 5 peaks (1.79). In a real crystal these peaks are broadened by static disorder, thermal fluctuations, and excitation-relaxation processes. Discarding for the moment the static disorder, we focus our attention on broadening processes due to lattice phonons, which may be described alternatively in terms of fluctuations of the local energies of the sites, or in terms of exciton relaxation by emission and absorption of phonons. These two complementary aspects of the fluctuation-dissipation theorem64 will allow us to treat the exciton-phonon coupling in the so-called strong and weak cases. The extraordinary (polariton) 0-0 transition of the anthracene crystal will be analyzed on the basis of these theoretical considerations and the semiexperimental data of the Kramers-Kronig analysis. 

Figure 2.12. Real and imaginary parts of the dielectric permittivity e(co) around the fe-polarized 0-0 transition, obtained from Kramers-Kronig analysis of rellectivity spectra at temperature ranging from 7 to 77 K. The arrow on the r. curves indicates the point where e = 0. We note the stepwise threshold of the 46-cm-1 phonon sideband (a). At higher temperatures (b-h) it broadens to give the smooth asymmetrical absorption curve at SO K (g).

Figure 2.14. Evolution with temperature of the full width y0 at half maximum of the 0-0 absorption peak. Hollow circles represent our results from Kramers-Kronig analysis, (a) Evolution between 0 and 77 K. The solid line was drawn using equation (2.126) and adjusted parameters y, =72cm 1, hfi = 27cm" . The dashed line connects the results of our model (2.127)—(2.130) for six different temperatures, (b) Evolution between 0 and 300 K. The full circles are taken from ref. 62. This summary of the experimental results shows the linear behavior between 30 and 50 K., and the sublinear curvature at temperatures above 200 K. 

Figure 3.12. Simulation of the b-polarized (0-0) reflectivity of the anthracene crystal using the bulk reflectivity amplitude derived from a Kramers-Kronig analysis (Section II.C). The total reflectivity is calculated from the scheme of Fig. 3.11 and (3.24)-(3.25) for various values of the nonradiative broadening parameter 7% Comparison with spectra of our best crystals gives the value / ° = 3cm 1 for T = 1.7 K.

Usually, the absorption coefficient of the conducting crystals is so high that producing a crystal sufficiently thin and suitable for absorption measurements presents a great difficulty. If this is so, the bulk optical constants of a solid may be computed from the normal-incidence reflectivity of that material over an extended range of frequencies, followed by a Kramers-Kronig analysis of the measurements [12,14]. In this method the real, n, and imaginary, k, parts of the complex index of refraction... 

Figure 3. Polyethylene measured energy loss function, Im(—l/e) ( ). Results of a Kramers-Kronig analysis e,(--------------) (---) (i).

The role of anisotropy and interchain interaction has been identified through dc and ac conductivity of stretch films, Kramers-Kronig analysis of reflectance on stretched films, anisotropic microwave frequency conductivity [44,52] and muon spin relaxation [62]. The role of interaction of spin of interstitial dopants with the conduction electrons at low temperatures has recently been explored by Y.W. Park and coworkers [63]. 

Far-IR reflectivity spectra of the (Pbo 5Cao 5)(Eeo 5Tao 5)03 specimens sintered at 1250°C for 30 min were taken to calculate the intrinsic dielectric loss at microwave frequencies. The spectra of the specimens were fitted by 10 resonant modes. The calculated reflectivity spectra are well fitted with the measured ones, as shown in Figure 22.4 and Table 22.2. The dispersion parameters of the specimens in Table 22.2 were determined by the Kramers-Kronig analysis and the classical oscillator model. The calculated values were higher than the measured ones by Hakki and Coleman s method, which is due to extrinsic effects such as grain size and porosity. Assuming the mixture of dielectrics and spherical pore with 3-0 connectivity, the measured loss quality also depends on the porosity as well as the intrinsic loss of materials, and Equation 22.24 may be modified for the loss quality, as in Equation 22.25 ... 

Mathematical analysis Heaviside theory Capacitance vs. frequency Vf Fitting Kramers-Kronig analysis, assumed error structure Measurement model, measured error structure... 

A Kramers-Kronig analysis of the data (63) yields the optical constants n,k. As can be concluded from the reflectivity curves recalculated from n and k, the accuracy is rather low, mainly because of the limitation of data at low energy and beginning interference of the quartz at highest energies. The preliminary character of these reflectivity measurements is evident. 

Experimentally, crystals of polyacenes have been studied by absorption and reflection spectroscopy. The pioneering work by Clark and his students, using a microtome to deliberately cut specific crystal faces of organic crystals which were then studied by reflection spectroscopy was very important. The corresponding absorption spectra have been derived by Kramers-Kronig analysis (63) (equa-... 

Searching for possible, less pronounced differences, we have performed a Kramers-Kronig analysis of the r(e 1 a) and R(E c)specrfea. Due to uncertainties in the high frequency extrapolations, the resulting oscillator strengths in the... 

To gain more insight, we carried out a Kramers-Kronig analysis to extract the excitation spectrum, or 2( 5 ( ), from the EELS data. We used E q, 0) — Q) K, 0 = e q = 0, m = 0) where 0 = 4.83 was obtained from previous optical measurements on this material [8.40, 8.41]. Figure 8.12 shows the resultant dispersion and the peak intensity of the optically allowed excitation, or the peak position and the weight of 2, respectively, as functions of the scattering angle. The transition peak position at 2.8 eV for small q in the... 

Fig. 8.12. The dispersion (a) and the intensity (b) of the optically allowed excitation along the [100] and [110] directions in 2 obtained by Kramers-Kronig analysis.

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