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Dispersion analysis with inverse dielectric function modelling
Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 168 (2016) 212–217
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Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy journal homepage: www.elsevier.com/locate/saa
Dispersion analysis with inverse dielectric function modelling Thomas G. Mayerhöfer a,c,⁎, Vladimir Ivanovski b, Jürgen Popp a,c a b c
Leibniz Institute of Photonic Technology (IPHT), Albert-Einstein-Str. 9, D-07745 Jena, Germany Institute of Chemistry, Faculty of Natural Sciences and Mathematics, Sts. Cyril and Methodius University, Arhimedova 5, 1000, Skopje, Macedonia Institute of Physical Chemistry, and Abbe Center of Photonics, Friedrich Schiller University, Jena D-07743, Helmholtzweg 4, Germany
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 5 February 2016 Received in revised form 24 May 2016 Accepted 31 May 2016 Available online 02 June 2016
We investigate how dispersion analysis can profit from the use of a Lorentz-type description of the inverse dielectric function. In particular at higher angles of incidence, reflectance spectra using p-polarized light are dominated by bands from modes that have their transition moments perpendicular to the surface. Accordingly, the spectra increasingly resemble inverse dielectric functions. A corresponding description can therefore eliminate the complex dependencies of the dispersion parameters, allow their determination and facilitate a more accurate description of the optical properties of single crystals. © 2016 Elsevier B.V. All rights reserved.
Keywords: Dispersion analysis Reflectance spectroscopy Infrared spectroscopy Dielectric function modelling Perpendicular modes
1. Introduction In his often-cited work “Infrared Absorption at Longitudinal Optic (LO) Frequency in Cubic Crystal Films” Berreman assumed “that the equivalent of longitudinal optic modes in an infinite medium actually are stimulated in films” [1]. Since then this assumption is the object of highly controversial discussions in literature and a number of theories on the microscopic level were provided to explain a potential stimulation of longitudinal optic modes. Among the manifold explanations are surface charges, [2] interference optical effects, [3] intensified-absorption-loss (an extension of the prior explanation) [4], surface polariton modes [5] and surface longitudinal optical mode absorption [6]. All these explanations are closely connected to films and their properties. It is less well known that also bulk samples of cubic materials in polycrystalline form and glasses show peculiarities which are closely related to the Berreman-effect in their reflectance spectra. [7–9] Indeed, it is easy to demonstrate that all polycrystalline materials, as long as they consist of crystallites which are randomly oriented and small compared to the wavelength as well as glasses show peaks close to LO frequencies when they are excited with parallel-polarized light at higher angles of incidence. According to the textbooks of optics, Rp (reflectance with ppolarized light) is given by [10] 2 2 ε cosα− ε− sin2 α ε cos2 α − 1− 1ε sin2 α Rp ¼ ¼ ; 2 2 1 2 ε cosα þ ε− sin α ðε cos α Þ þ 1− ε sin α 1 2
1 2
1 2
1 2
⁎ Corresponding author. E-mail address: [email protected] (T.G. Mayerhöfer).
http://dx.doi.org/10.1016/j.saa.2016.05.055 1386-1425/© 2016 Elsevier B.V. All rights reserved.
1 2
1 2
ð1Þ
wherein ε is the complex dielectric function of the glass or the polycrystalline material and α is the angle of incidence, when we assume incidence from vacuum. From the right form of Rp in Eq. (1) it is obvious that for higher angles of incidence the term εcos2α loses importance while at the same time the terms 1ε sin2 α gains. Therefore, for higher angles of incidence, Rp is increased if 1ε becomes large, which is near the LO frequencies, where the real part of ε tends to become zero. It seems therefore plausible that p-polarized reflectance spectra of materials which can be characterized by a scalar dielectric function can be better modelled by using 1ε instead of ε at higher angles of incidence. The basis
of this modelling is that 1ε can, like ε, be represented by a constant and a sum of Lorentz-type oscillators as was first noted by Humlíĉek [11]. Humlíĉek investigated the transverse and longitudinal vibration modes of α-quartz by employing “the most symmetric orientations” by infrared spectroscopic ellipsometry. By using this technique, he could directly gain the real- and the imaginary part of the two principal dielectric functions εa and εc of α-quartz. To obtain oscillator parameters, he then fitted not only these dielectric functions (a very common procedure), but also their negative inverse with help of Lorentz-type oscillators. The latter procedure was taken over in a number of other investigations [12–15]. As already stated we will employ this idea for dispersion analysis of reflectance spectra. In contrast to spectroscopic ellipsometry, it is not possible to gain the dielectric function directly from reflectance spectra. Instead, spectra are usually fitted by repeatedly generating an improved dielectric function, inserting it into the Fresnel equations and comparing the resulting reflectance values with those obtained by measurement, an iterative procedure which is called dispersion analysis. In this contribution, we investigate how dispersion analysis performs if instead of the dielectric function, its inverse is
T.G. Mayerhöfer et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 168 (2016) 212–217
used and generated by Lorentz-type oscillators. Before we start we first motivate this idea by investigating how the real- and the imaginary part of the inverse dielectric function relates to the reflectance spectra of isotropic materials under high angles of incidence and p-polarized radiation with help of polycrystalline fresnoite (Ba2TiSi2O8) consisting of crystallites small compared to the wavelength. Subsequently we perform dispersion analysis on alum (KAl(SO4)2·12H2O) with particular emphasis on the performance of the inverse dielectric function modelling in dependence of the angle of incidence. We continue with investigating how inverse dielectric function modelling performs for a material with very strong oscillators and broad bands, namely NdGaO3 employing s-polarized light. In case of such materials often a semi-empirical 4 parameter model [16] is used instead of Lorentz-type oscillators to generate the dielectric function. The 4 parameter model allows e.g. to directly obtain values for the TO (transverse optic) and the LO frequency. These values can be readily compared to those obtained by the Lorentz-type modelling of dielectric and inverse dielectric functions. Finally, we show how the problem of determining the dispersion parameters of perpendicular modes in anisotropic crystals and layers can be solved using inverse dielectric function modelling. Typically, this problem is of interest e.g. whenever anisotropic crystals tend to grow plate-like. The modes perpendicular to the surface are then hard to analyze. Intuitively, one could assume that it is sufficient to record two spectra with parallel-polarized light with different angles of incidence. However, the perpendicular modes generate peaks close to their respective LO-position, and this position does depend not only on the TO-position, but also on the mode strength and on all contributions of higher frequency modes. As a consequence of these manifold mutual dependencies, the parameters are impossible to obtain by dispersion analysis of conventional reflectance spectra. Theoretically, this problem has been solved by recording both, an external and an internal reflection spectrum of the same sample, since the latter has a minimum close to the TO-position, which removes the ambiguity. [17] Unfortunately, internal reflection spectra are next to impossible to record with the required accuracy, even if progress has been made recently. [18] The use of inverse dielectric function modelling for the dispersion analysis of perpendicular modes removes this need for internal reflection spectra and allows to obtain the oscillator parameter of these modes in a nearly as simple way as parameters in conventional dispersion analysis as will be shown in the following. 2. Experimental IR reflection spectra were recorded with a Bruker IFS 66 (Ba2TiSi2O8), a Perkin-Elmer System 2000 FT-IR spectrometer (KAl(SO4)2·12H2O)
213
and a Bruker V80v FT-spectrometer (NdGaO3) with a resolution of at least 4 cm−1 in specular reflection. Two reflection units from Harrick Scientific (NdGaO3, fixed angle of incidence of 8°) and Seagull (Ba2TiSi2O8, KAl(SO4)2·12H2O, variable angle of incidence, 5°–85°) where used. To polarize the incidence beam we employed an Al/KRS-5 wire grid polarizer in the MIR region and a polyethylene polarizer for the FIR region. Dispersion analysis was performed with a homemade program using Mathematica 8. Polycrystalline randomly oriented fresnoite was obtained by crystallization of the corresponding glass. Details can be found in [19]. Single crystal of the KAl(SO4)2·12H2O alum was obtained by a slow evaporation from a water solution containing K2SO4 and Al2(SO4)3·18H2O in a molar ratio of 1:1. The NdGaO3 single crystal was grown by Czochralski method at the Leibniz-Institut für Kristallzüchtung, Berlin. [20]. 3. Results and discussion We start with comparing two spectra gained from the same sample of polycrystalline fresnoite with regard to the peak positions. The first spectrum was recorded with p-polarized radiation under an angle of incidence of 20°, whereas for the second spectrum an angle of incidence of 80° was employed. For comparison, in the first spectrum the real- and the imaginary part of the dielectric function obtained by KramersKronig analysis is shown, while in the second spectrum the real part and the negative imaginary part of the inverse dieletric function are also depicted. For these comparisons, we have chosen by intent a material consisting of randomly oriented anisotropic crystallites, which are small compared to the resolution limit of light. In contrast to cubic materials (whether single and polycrystalline), the forms of the reflectance bands do not show typical profiles of Lorentz-oscillators (this is also the reason why conventional dispersion analysis cannot be applied and a Kramers-Kronig analysis has to be performed instead [21]). In contrast, the band shapes are influenced by orientational averaging. As a result, the bands show a distinct maximum close to the TO frequency for small angles of incidence [21] (note that this characteristic property is lost, when the crystallites are no longer small compared to the resolution limit! [22,23]), which is obvious from the comparison with the imaginary part of the dielectric function. The shapes of the reflectance bands are in an obvious way influenced by both, the real and the imaginary part of the dielectric function. The same is true, but for the inverse dielectric function, at higher angles of incidence and p-polarization. Accordingly, all maxima of the reflectance bands are shifted towards their LO-positions, i.e. the maxima of the negative imaginary part of the inverse dielectric function. On closer inspection, e.g. of the band with the highest wavenumber at 1072 cm−1, it becomes obvious, that this
Fig. 1. Reflectance spectra as well as dielectric and inverse dielectric function of randomly oriented polycrystalline fresnoite consisting of optically small crystallites. Left side: Angle of incidence 20°, reflectance spectrum (black), real part (red/dark grey) and imaginary part (green/light grey) of the dielectric function. Right side: Angle of incidence 80°, reflectance spectrum (black), real part (red/dark grey) and negative imaginary part (green/light grey) of the inverse dielectric function.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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~ Þ (red/dark grey) and modelled spectra Fig. 2. Modelled and measured reflectance spectra of alum and differences between them. Measured spectra (black), modelled spectra using ε−1 ðν ~ Þ (green/light grey). Left side: Angle of incidence 20°, s-polarized. Right side: Angle of incidence 60°, p-polarized. (For interpretation of the references to color in this figure employing εðν legend, the reader is referred to the web version of this article.)
reflectance band is actually shifted beyond the maximum of −Imð1ε Þ, which is located at 1062 cm−1. This is a phenomenon which was realized earlier for the case of uniaxial materials having their optical axis perpendicular to the surface and is therefore not a specialty of polycrystalline samples [17,24]. The reason for this phenomenon is obviously the strong influence of Reð1ε Þ, the corresponding maximum of which is located beyond the LO position at 1074 cm−1. Accordingly, the reflectance band positions in spectra gained with p-polarization and at higher angle of incidence are no reliable indicators of the LO band positions! What becomes evident from Fig. 1, however, is that a direct use of the inverse dielectric function for the modelling should be of advantage, in particular, for p-polarized radiation at higher angles of incidence. We investigated this assumption on the example of the alum single crystal and found it fully confirmed. What is somewhat surprising, though, is that also at low incidence angles the modelling with the inverse dielectric function still has some advantage over dielectric function based modelling in terms of lower average deviation between measured and modelled spectrum. As pointed out, we generate the inverse dielectric function assuming a sum of Lorentz-terms and a constant background ε∞ summarizing the contributions from higher spectral ranges according to: N X ~ Þ ¼ ε −1 ε−1 ðν ∞ −
S2j;LO : 2 ~ 2 −iν ~ γ j;LO ~ j;LO −ν j¼1 ν
with the ordinary modelling following Eq. (3). This advantage is lowered to about 85% (85.4%) if s-polarized radiation and a comparably small angle of incidence are used (20°). If we analyze the insets of Fig. 2, it is obvious that most of the advantage of the inverse dielectric function modelling has its origin in the more accurate fitting of the reflectance due to the strong band around 2− 1100 cm−1, originating from the ν3(SO2− 4 ) mode [25] from the SO4 ions occupying major sites [26]. The frequency region from 1075 to 1200 cm−1 was fitted with two oscillators, as apart from the band at ca. 1100 cm−1 it has a shoulder at ca. 1125 cm−1 in the 20°, s-spectrum which develops nearly into a separate band in the 60°, p-spectrum. This second band, which also originates from a ν3(SO2− 4 ) mode, cannot be fitted satisfactorily in the latter spectrum with the conventional approach, but shows very good agreement using the inverse dielectric function modelling. The advantage is much smaller for the 20°, s-spectrum, but still noticeable. To investigate if the inverse dielectric function modelling is indeed able to capture strong bands more accurate, we decided to reinvestigate a material, which has even stronger bands, namely NdGaO3. Since NdGaO3 is orthorhombic, it has three principle spectra, which are gained with s-polarized light and orienting the polarization direction parallel to the respective crystallographic axis. The corresponding dispersion analysis can be found in [20] and was carried out
ð2Þ
Herein, Sj , LO represents the oscillator strengths of the jth LO mode, ~ j;LO its position and γj , LO its damping constant (all units in ν wavenumbers). Note that in Eq. (2) the individual summands are subtracted from the inverse dielectric background ε ∞ −1 in contrast to ~ Þ, the usual Lorentz-type modelling for εðν ~ Þ ¼ ε∞ þ ε ðν
N X
S2j;TO ; 2 ~ 2 −iν ~ γ j;TO ~ j;TO −ν j¼1 ν
ð3Þ
~ Þ, ε −1 ðν ~ Þ is with the same types of parameters, but for TO modes. As εðν unity at very high wavenumbers before the onset of absorption. In con~ Þ is reduced by every absorption, therefore ~ Þ, however, ε−1 ðν trast to εðν the sum of Lorentz-type contributions is subtracted from ε ∞ −1. The corresponding fits, together with the measured spectra, are shown in Fig. 2. While using the same amount of oscillators, the average deviation between measured and modelled spectrum at an angle of incidence of 60° is reduced to less than 50% (49.6%) for p-polarized radiation, if the inverse dielectric function modelling is used (Eq. (2)) in comparison
Fig. 3. Modelled and measured reflectance spectra of the principle spectrum of the c-axis of NdGaO3 and differences between them. Measured spectrum (black), modelled ~ Þ (red), modelled spectrum employing εðν ~ Þ (green) and modelled spectrum using ε −1 ðν spectrum using the 4 parameter model (blue). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
T.G. Mayerhöfer et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 168 (2016) 212–217 Table 1 Comparison of the oscillator parameter gained by the semiempirical 4 parameter model (Eq. (4)) with those from the Lorentz-models according to Eqs. (2) and (3). 4-Parameter model
1 2 3 4 5
Lorentz-model
~ TO ν [cm−1]
~ LO ν [cm−1]
γTO [cm−1]
γLO [cm−1]
~ TO ν [cm−1]
~ LO ν [cm−1]
γTO [cm−1]
γLO [cm−1]
592.6 342.5 290.5 270.7 171.9
659.8 528.1 309.3 285.0 194.9
22.7 13.3 6.3 8.4 10.7 ∑61.4
24.3 10.4 15.1 6.0 5.6 ∑61.4
596.3 346.3 289.0 272.1 167.6
659.1 527.0 310.3 284.4 195.1
23.6 15.7 12.2 4.1 11.7 ∑67.3
24.3 8.1 14.4 5.3 7.9 ∑60.0
and layers. A typical situation where this problem occurs are crystals that grow plate-like, can be split only in one direction as many layered materials the most prominent being probably graphite or are generally two-dimensional like graphene. In particular in the latter case it is obvious that information about the principal component of the dielectric tensor function perpendicular to the surface cannot be obtained in the conventional way not even if a microscope is used. The only possibility in such cases is to use p-polarized light and a non-zero angle of incidence. Assuming an optically uniaxial material with the optical axis oriented perpendicular to the surface, the reflectance is given by: [17,27, 28] 2 ε⊥ cos2 α − 1− ε1 sin2 α ∥ Rp ¼ 2 1 2 ðε⊥ cos α Þ þ 1− ε∥ sin α 1 2
1 2
conventionally according to Eq. (3) and, additionally, with the semi-empirical 4 parameter oscillator model: N
ε ¼ ε∞ ∏
~ 2 −iν ~ γ LO; j ~ 2LO; j −ν ν
ð4Þ
~ 2 −iν ~ γ TO; j ~ 2TO; j −ν j¼1 ν
The parameters used in this model should be directly comparable to those used in Eqs. (2) and (3). We reinvestigated all 3 principle spectra of NdGaO3, and found that the average deviation between measured and modelled spectrum is reduced to 78.1% relative to the conventional model for the a-axis spectrum, while it is increased to 100.4% in case of the b-axis spectrum if inverse dielectric function modelling is used (again in comparison with the conventional model). The average deviation for the c-axis spectrum lies with 87.4% of that reached with the conventional model in between. The corresponding measured and modelled spectra are depicted in Fig. 3. The performance of the different models is easier to evaluate near very broad bands like the one between 300 and 550 cm−1. In particular at the high-wavenumber side of this band and at the adjacent minima it is obvious that the 4 parameter model performs much better than the conventional model. The inverse dielectric function modelling leads to a closer resemblance between modelled and measured spectrum around this band than the conventional Lorentz model and performs only slightly inferior in comparison with the 4 parameter model (4P in Fig. 3), which gives the best correspondence with an average deviation of 73.8% of that achieved with the conventional model. It is now possible to compare the values obtained for the different parameters. This comparison is provided in Table 1. Overall there is a good agreement concerning the TO- and LOwavenumbers. In case of the damping constants the agreement is not as good as for the wavenumbers. The reason for this is that for the 4 paN
N
j¼1
j¼1
rameter model not to be unphysical, it is required that ∑ γ TO; j b∑ γLO; j. Such a constraint does not exist in case of the Lorentz-models. Our conclusion is that this constraint leads to errors concerning the damping constants if the 4 parameter model is employed, in particular for weaker oscillators which compensate with their values of γTO those of other oscillators with more weight in the average deviation. Nevertheless, the 4 parameter model still has its merits if more emphasis is placed on describing the dielectric function as good as possible than on determining the oscillator parameters. To complete the demonstrations of potential applications of inverse dielectric function modelling we focus in the last part of this paper on the dispersion analysis of perpendicular modes in anisotropic crystals
215
1 2
1 2
ð5Þ
Here, ε⊥ is the principle component of the dielectric tensor in the direction parallel to the surface and ε∥ the principle component perpendicular to the surface (we use the same notation as is common in the literature, even if this notation is not straightforward in our opinion). Note that Eq. (5) simplifies to Eq. (1) if ε∥ = ε⊥ = ε. It is therefore justified to Assume that inverse dielectric function modelling can be also helpful in case of reflectance from materials that can be described with Eq. (5) as it was for those the reflectance of which could be calculated by Eq. (1). To test this assumption, we specify the same simple test case as in [17] by assuming a material with one oscillator in ε⊥ and another oscillator in ε∥ with the parameters as given in Table 2 and taken over from [17] (the parameters were chosen in a way to avoid overlap of the cor~ ∥;LO , responding bands but otherwise arbitrarily, with the exception of ν ~ ∥;TO and S∥,TO and was determined by dielectric which is a function of ν function modelling). For the model oscillators Employing Eq. (5), we can then generate reflectance spectra for different angles of incidence. The results are displayed in Fig. 4. Obviously, the influence of the ∥-mode increases with increasing α and the corresponding band is located even beyond the LO wavenumber as mentioned earlier. What is therefore missing in the spectrum is any information about the TO position of the ∥-mode. Since the LO position on the other hand depends not only on the TO position, but also on the oscillator strength S∥,TO and on ε∞,∥ its location does not allow an unambiguous conclusion with regard to the TO position. Accordingly, the error function (the root mean square of the difference between experi~ ∥;TO , S∥;TO and ε∞;∥ mental and simulated spectrum) in dependence of ν does not show a pronounced minimum, even not in the ideal situation of the test case, where both modes are well-separated (cf. Fig. 5a) [17]. As solution, it was previously found feasible to record additionally an internal reflectance spec~ ∥;TO in contrast to the trum, which shows a distinct minimum close to ν external reflection spectrum and use both spectra for the fit. The drawback of this solution is, however, that it is next to impossible to record an internal reflection spectrum of the necessary quality. Therefore, in practice the problem of retrieving information about the transversal frequencies of modes with transition dipole perpendicular to the reflecting surface was unsolved until now. With inverse dielectric function modelling of ε ∥, it is not even necessary to use spectra measured at two different angles of incidence as one is completely sufficient which is demonstrated in Fig. 5b. In contrast to the error function shown in Fig.
Table 2 TO oscillator parameters employed to generate ε⊥ and ε∥ via Eq. (3) and LO oscillator parameters for the ‖-mode gained by inverse dielectric function modelling. j
~ j (cm−1) ν
γj (cm−1)
Sj (cm−1)
ε∞, j
⊥, TO ∥, TO ∥, LO
800 1000 1060.66
10 10 10
500 500 250
2.1 2.0 2.0
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Fig. 6. Comparison between the default Rp (black) using the starting (TO-) values from Table 2 and Rp simulated using Eq. (5) in combination with Eq. (2) for ε∥.
Fig. 4. Upper part: Parallel polarized reflectance spectra of a model material based on the principal dielectric functions generated by Eq. (3) and the values provided by Table 2. The model material is assumed to be semi-infinite like the incidence medium, which is characterized by an index of refraction n = 1. Lower part: Real (Re(ε ∥ ), black) and imaginary (Im(ε∥ ), red/dark grey) parts as well as dielectric loss (−Im(1/(ε ∥ ), green/ light grey) of the principal dielectric functions generated by Eq. (3). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
5a, the error function of the inverse dielectric function modelling is well-behaved in the sense that it shows a pronounced minimum. −1 ~ ∥;LO on ε∞,∥ , which is very obstructive There is also no dependence of ν ~ in case of ν ∥;TO and ε∞,∥ for the direct dielectric function fitting. Accordingly, it is possible within 2 iterations and less than a minute time to regain the original reflectance spectrum even in case of starting values that are far away from the true values (see Fig. 6). This gives confidence that even under worse conditions e.g. that the bands of the ⊥-modes overlap with those of the ‖-modes, inverse dielectric function modelling is successfully applicable (certainly, the ⊥-modes can always be isolated and separately modelled by applying s-polarized light). It is a matter of course that ε⊥ has been evaluated separately before the evaluation of ε∥. ~ ∥;TO , S∥,TO and γ∥,TO are unknown. AcStill, there is one step missing, as ν tually, for the example above this is not quite correct, since in case of Lorentz-type functions we know that γ∥,TO = γ∥,LO. Indeed, through the modelling we find that confirmed (cf. Table 2). Furthermore, from the ~ Þ which can be inverted to yield ε ∥ ðν ~ Þ (and, likefit we obtained 1=ε∥ ðν ~ Þ with help of Eq. wise, ε∥,∞). Subsequently, a direct modelling of ε∥ ðν (3), a procedure that is carried out routinely in ellipsometry [11], yields ~ ∥;TO , S∥,TO. Applying this procedure, we regain the missing parameters ν
the original parameters used to generate the reflectance spectrum within numerical precision. 4. Conclusion The main objective of this work was to investigate how and under which conditions and circumstances dispersion analysis can benefit from the application of inverse dielectric function modelling. We found that for isotropic materials, i.e. materials with cubic symmetry or randomly oriented polycrystalline materials with small crystallites, the reflectance with parallel polarized light depends with increasing angle of incidence decreasingly on the dielectric function and increasingly on the inverse dielectric function. From this finding, we concluded that application of a Lorentz-type modelling of this inverse dielectric function is of advantage for dispersion analysis. This assumption was proven correct in case of the reflectance spectra of KAl(SO4)2·12H2O alum. Even for dispersion analysis of s-polarized spectra, however, the use of inverse dielectric function modelling seems to allow a better fit of the experimental spectra and, thereby, a better description of the optical properties in many cases as was shown for the alum and NdGaO3. Finally, we have transferred the idea of inverse dielectric function modelling for isotropic samples to anisotropic samples with different principal dielectric functions parallel and perpendicular to the surface. The spectra of the latter are usually impossible to analyze with conventional dispersion analysis as the properties of the bands due to perpendicular components show complex dependencies among the different oscillator parameters. In contrast, the use of inverse dielectric function modelling resolves these complex dependencies and the dispersion
~ ∥, keeping the damping fixed at its true value of 10 cm−1 and the dielectric Fig. 5. Plot of the error function of the external reflection for different pairs of oscillator strength S∥ and position ν background at ε ∥;∞ =2.0. ~ ∥ relate to the TO-mode. a) Error function resulting from direct dielectric function fitting of two reflectance spectra with α = 60° and α = 70° simultaneously. Strength S∥ and position ν ~ ∥ relate to the LO-mode. b) Error function resulting from inverse dielectric function fitting of one reflectance spectrum with α = 60°. strength S∥ and position ν
T.G. Mayerhöfer et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 168 (2016) 212–217
parameter used in the direct modelling can be regained by inverting the inverse dielectric function and modelling the resulting dielectric function. Overall we conclude that dispersion analysis can profit in numerous ways by the use of inverse dielectric function modelling. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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Dispersion analysis with inverse dielectric function modelling Thomas G. Mayerhöfer a,c,⁎, Vladimir Ivanovski b, Jürgen Popp a,c a b c
Leibniz Institute of Photonic Technology (IPHT), Albert-Einstein-Str. 9, D-07745 Jena, Germany Institute of Chemistry, Faculty of Natural Sciences and Mathematics, Sts. Cyril and Methodius University, Arhimedova 5, 1000, Skopje, Macedonia Institute of Physical Chemistry, and Abbe Center of Photonics, Friedrich Schiller University, Jena D-07743, Helmholtzweg 4, Germany
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 5 February 2016 Received in revised form 24 May 2016 Accepted 31 May 2016 Available online 02 June 2016
We investigate how dispersion analysis can profit from the use of a Lorentz-type description of the inverse dielectric function. In particular at higher angles of incidence, reflectance spectra using p-polarized light are dominated by bands from modes that have their transition moments perpendicular to the surface. Accordingly, the spectra increasingly resemble inverse dielectric functions. A corresponding description can therefore eliminate the complex dependencies of the dispersion parameters, allow their determination and facilitate a more accurate description of the optical properties of single crystals. © 2016 Elsevier B.V. All rights reserved.
Keywords: Dispersion analysis Reflectance spectroscopy Infrared spectroscopy Dielectric function modelling Perpendicular modes
1. Introduction In his often-cited work “Infrared Absorption at Longitudinal Optic (LO) Frequency in Cubic Crystal Films” Berreman assumed “that the equivalent of longitudinal optic modes in an infinite medium actually are stimulated in films” [1]. Since then this assumption is the object of highly controversial discussions in literature and a number of theories on the microscopic level were provided to explain a potential stimulation of longitudinal optic modes. Among the manifold explanations are surface charges, [2] interference optical effects, [3] intensified-absorption-loss (an extension of the prior explanation) [4], surface polariton modes [5] and surface longitudinal optical mode absorption [6]. All these explanations are closely connected to films and their properties. It is less well known that also bulk samples of cubic materials in polycrystalline form and glasses show peculiarities which are closely related to the Berreman-effect in their reflectance spectra. [7–9] Indeed, it is easy to demonstrate that all polycrystalline materials, as long as they consist of crystallites which are randomly oriented and small compared to the wavelength as well as glasses show peaks close to LO frequencies when they are excited with parallel-polarized light at higher angles of incidence. According to the textbooks of optics, Rp (reflectance with ppolarized light) is given by [10] 2 2 ε cosα− ε− sin2 α ε cos2 α − 1− 1ε sin2 α Rp ¼ ¼ ; 2 2 1 2 ε cosα þ ε− sin α ðε cos α Þ þ 1− ε sin α 1 2
1 2
1 2
1 2
⁎ Corresponding author. E-mail address: [email protected] (T.G. Mayerhöfer).
http://dx.doi.org/10.1016/j.saa.2016.05.055 1386-1425/© 2016 Elsevier B.V. All rights reserved.
1 2
1 2
ð1Þ
wherein ε is the complex dielectric function of the glass or the polycrystalline material and α is the angle of incidence, when we assume incidence from vacuum. From the right form of Rp in Eq. (1) it is obvious that for higher angles of incidence the term εcos2α loses importance while at the same time the terms 1ε sin2 α gains. Therefore, for higher angles of incidence, Rp is increased if 1ε becomes large, which is near the LO frequencies, where the real part of ε tends to become zero. It seems therefore plausible that p-polarized reflectance spectra of materials which can be characterized by a scalar dielectric function can be better modelled by using 1ε instead of ε at higher angles of incidence. The basis
of this modelling is that 1ε can, like ε, be represented by a constant and a sum of Lorentz-type oscillators as was first noted by Humlíĉek [11]. Humlíĉek investigated the transverse and longitudinal vibration modes of α-quartz by employing “the most symmetric orientations” by infrared spectroscopic ellipsometry. By using this technique, he could directly gain the real- and the imaginary part of the two principal dielectric functions εa and εc of α-quartz. To obtain oscillator parameters, he then fitted not only these dielectric functions (a very common procedure), but also their negative inverse with help of Lorentz-type oscillators. The latter procedure was taken over in a number of other investigations [12–15]. As already stated we will employ this idea for dispersion analysis of reflectance spectra. In contrast to spectroscopic ellipsometry, it is not possible to gain the dielectric function directly from reflectance spectra. Instead, spectra are usually fitted by repeatedly generating an improved dielectric function, inserting it into the Fresnel equations and comparing the resulting reflectance values with those obtained by measurement, an iterative procedure which is called dispersion analysis. In this contribution, we investigate how dispersion analysis performs if instead of the dielectric function, its inverse is
T.G. Mayerhöfer et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 168 (2016) 212–217
used and generated by Lorentz-type oscillators. Before we start we first motivate this idea by investigating how the real- and the imaginary part of the inverse dielectric function relates to the reflectance spectra of isotropic materials under high angles of incidence and p-polarized radiation with help of polycrystalline fresnoite (Ba2TiSi2O8) consisting of crystallites small compared to the wavelength. Subsequently we perform dispersion analysis on alum (KAl(SO4)2·12H2O) with particular emphasis on the performance of the inverse dielectric function modelling in dependence of the angle of incidence. We continue with investigating how inverse dielectric function modelling performs for a material with very strong oscillators and broad bands, namely NdGaO3 employing s-polarized light. In case of such materials often a semi-empirical 4 parameter model [16] is used instead of Lorentz-type oscillators to generate the dielectric function. The 4 parameter model allows e.g. to directly obtain values for the TO (transverse optic) and the LO frequency. These values can be readily compared to those obtained by the Lorentz-type modelling of dielectric and inverse dielectric functions. Finally, we show how the problem of determining the dispersion parameters of perpendicular modes in anisotropic crystals and layers can be solved using inverse dielectric function modelling. Typically, this problem is of interest e.g. whenever anisotropic crystals tend to grow plate-like. The modes perpendicular to the surface are then hard to analyze. Intuitively, one could assume that it is sufficient to record two spectra with parallel-polarized light with different angles of incidence. However, the perpendicular modes generate peaks close to their respective LO-position, and this position does depend not only on the TO-position, but also on the mode strength and on all contributions of higher frequency modes. As a consequence of these manifold mutual dependencies, the parameters are impossible to obtain by dispersion analysis of conventional reflectance spectra. Theoretically, this problem has been solved by recording both, an external and an internal reflection spectrum of the same sample, since the latter has a minimum close to the TO-position, which removes the ambiguity. [17] Unfortunately, internal reflection spectra are next to impossible to record with the required accuracy, even if progress has been made recently. [18] The use of inverse dielectric function modelling for the dispersion analysis of perpendicular modes removes this need for internal reflection spectra and allows to obtain the oscillator parameter of these modes in a nearly as simple way as parameters in conventional dispersion analysis as will be shown in the following. 2. Experimental IR reflection spectra were recorded with a Bruker IFS 66 (Ba2TiSi2O8), a Perkin-Elmer System 2000 FT-IR spectrometer (KAl(SO4)2·12H2O)
213
and a Bruker V80v FT-spectrometer (NdGaO3) with a resolution of at least 4 cm−1 in specular reflection. Two reflection units from Harrick Scientific (NdGaO3, fixed angle of incidence of 8°) and Seagull (Ba2TiSi2O8, KAl(SO4)2·12H2O, variable angle of incidence, 5°–85°) where used. To polarize the incidence beam we employed an Al/KRS-5 wire grid polarizer in the MIR region and a polyethylene polarizer for the FIR region. Dispersion analysis was performed with a homemade program using Mathematica 8. Polycrystalline randomly oriented fresnoite was obtained by crystallization of the corresponding glass. Details can be found in [19]. Single crystal of the KAl(SO4)2·12H2O alum was obtained by a slow evaporation from a water solution containing K2SO4 and Al2(SO4)3·18H2O in a molar ratio of 1:1. The NdGaO3 single crystal was grown by Czochralski method at the Leibniz-Institut für Kristallzüchtung, Berlin. [20]. 3. Results and discussion We start with comparing two spectra gained from the same sample of polycrystalline fresnoite with regard to the peak positions. The first spectrum was recorded with p-polarized radiation under an angle of incidence of 20°, whereas for the second spectrum an angle of incidence of 80° was employed. For comparison, in the first spectrum the real- and the imaginary part of the dielectric function obtained by KramersKronig analysis is shown, while in the second spectrum the real part and the negative imaginary part of the inverse dieletric function are also depicted. For these comparisons, we have chosen by intent a material consisting of randomly oriented anisotropic crystallites, which are small compared to the resolution limit of light. In contrast to cubic materials (whether single and polycrystalline), the forms of the reflectance bands do not show typical profiles of Lorentz-oscillators (this is also the reason why conventional dispersion analysis cannot be applied and a Kramers-Kronig analysis has to be performed instead [21]). In contrast, the band shapes are influenced by orientational averaging. As a result, the bands show a distinct maximum close to the TO frequency for small angles of incidence [21] (note that this characteristic property is lost, when the crystallites are no longer small compared to the resolution limit! [22,23]), which is obvious from the comparison with the imaginary part of the dielectric function. The shapes of the reflectance bands are in an obvious way influenced by both, the real and the imaginary part of the dielectric function. The same is true, but for the inverse dielectric function, at higher angles of incidence and p-polarization. Accordingly, all maxima of the reflectance bands are shifted towards their LO-positions, i.e. the maxima of the negative imaginary part of the inverse dielectric function. On closer inspection, e.g. of the band with the highest wavenumber at 1072 cm−1, it becomes obvious, that this
Fig. 1. Reflectance spectra as well as dielectric and inverse dielectric function of randomly oriented polycrystalline fresnoite consisting of optically small crystallites. Left side: Angle of incidence 20°, reflectance spectrum (black), real part (red/dark grey) and imaginary part (green/light grey) of the dielectric function. Right side: Angle of incidence 80°, reflectance spectrum (black), real part (red/dark grey) and negative imaginary part (green/light grey) of the inverse dielectric function.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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~ Þ (red/dark grey) and modelled spectra Fig. 2. Modelled and measured reflectance spectra of alum and differences between them. Measured spectra (black), modelled spectra using ε−1 ðν ~ Þ (green/light grey). Left side: Angle of incidence 20°, s-polarized. Right side: Angle of incidence 60°, p-polarized. (For interpretation of the references to color in this figure employing εðν legend, the reader is referred to the web version of this article.)
reflectance band is actually shifted beyond the maximum of −Imð1ε Þ, which is located at 1062 cm−1. This is a phenomenon which was realized earlier for the case of uniaxial materials having their optical axis perpendicular to the surface and is therefore not a specialty of polycrystalline samples [17,24]. The reason for this phenomenon is obviously the strong influence of Reð1ε Þ, the corresponding maximum of which is located beyond the LO position at 1074 cm−1. Accordingly, the reflectance band positions in spectra gained with p-polarization and at higher angle of incidence are no reliable indicators of the LO band positions! What becomes evident from Fig. 1, however, is that a direct use of the inverse dielectric function for the modelling should be of advantage, in particular, for p-polarized radiation at higher angles of incidence. We investigated this assumption on the example of the alum single crystal and found it fully confirmed. What is somewhat surprising, though, is that also at low incidence angles the modelling with the inverse dielectric function still has some advantage over dielectric function based modelling in terms of lower average deviation between measured and modelled spectrum. As pointed out, we generate the inverse dielectric function assuming a sum of Lorentz-terms and a constant background ε∞ summarizing the contributions from higher spectral ranges according to: N X ~ Þ ¼ ε −1 ε−1 ðν ∞ −
S2j;LO : 2 ~ 2 −iν ~ γ j;LO ~ j;LO −ν j¼1 ν
with the ordinary modelling following Eq. (3). This advantage is lowered to about 85% (85.4%) if s-polarized radiation and a comparably small angle of incidence are used (20°). If we analyze the insets of Fig. 2, it is obvious that most of the advantage of the inverse dielectric function modelling has its origin in the more accurate fitting of the reflectance due to the strong band around 2− 1100 cm−1, originating from the ν3(SO2− 4 ) mode [25] from the SO4 ions occupying major sites [26]. The frequency region from 1075 to 1200 cm−1 was fitted with two oscillators, as apart from the band at ca. 1100 cm−1 it has a shoulder at ca. 1125 cm−1 in the 20°, s-spectrum which develops nearly into a separate band in the 60°, p-spectrum. This second band, which also originates from a ν3(SO2− 4 ) mode, cannot be fitted satisfactorily in the latter spectrum with the conventional approach, but shows very good agreement using the inverse dielectric function modelling. The advantage is much smaller for the 20°, s-spectrum, but still noticeable. To investigate if the inverse dielectric function modelling is indeed able to capture strong bands more accurate, we decided to reinvestigate a material, which has even stronger bands, namely NdGaO3. Since NdGaO3 is orthorhombic, it has three principle spectra, which are gained with s-polarized light and orienting the polarization direction parallel to the respective crystallographic axis. The corresponding dispersion analysis can be found in [20] and was carried out
ð2Þ
Herein, Sj , LO represents the oscillator strengths of the jth LO mode, ~ j;LO its position and γj , LO its damping constant (all units in ν wavenumbers). Note that in Eq. (2) the individual summands are subtracted from the inverse dielectric background ε ∞ −1 in contrast to ~ Þ, the usual Lorentz-type modelling for εðν ~ Þ ¼ ε∞ þ ε ðν
N X
S2j;TO ; 2 ~ 2 −iν ~ γ j;TO ~ j;TO −ν j¼1 ν
ð3Þ
~ Þ, ε −1 ðν ~ Þ is with the same types of parameters, but for TO modes. As εðν unity at very high wavenumbers before the onset of absorption. In con~ Þ is reduced by every absorption, therefore ~ Þ, however, ε−1 ðν trast to εðν the sum of Lorentz-type contributions is subtracted from ε ∞ −1. The corresponding fits, together with the measured spectra, are shown in Fig. 2. While using the same amount of oscillators, the average deviation between measured and modelled spectrum at an angle of incidence of 60° is reduced to less than 50% (49.6%) for p-polarized radiation, if the inverse dielectric function modelling is used (Eq. (2)) in comparison
Fig. 3. Modelled and measured reflectance spectra of the principle spectrum of the c-axis of NdGaO3 and differences between them. Measured spectrum (black), modelled ~ Þ (red), modelled spectrum employing εðν ~ Þ (green) and modelled spectrum using ε −1 ðν spectrum using the 4 parameter model (blue). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
T.G. Mayerhöfer et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 168 (2016) 212–217 Table 1 Comparison of the oscillator parameter gained by the semiempirical 4 parameter model (Eq. (4)) with those from the Lorentz-models according to Eqs. (2) and (3). 4-Parameter model
1 2 3 4 5
Lorentz-model
~ TO ν [cm−1]
~ LO ν [cm−1]
γTO [cm−1]
γLO [cm−1]
~ TO ν [cm−1]
~ LO ν [cm−1]
γTO [cm−1]
γLO [cm−1]
592.6 342.5 290.5 270.7 171.9
659.8 528.1 309.3 285.0 194.9
22.7 13.3 6.3 8.4 10.7 ∑61.4
24.3 10.4 15.1 6.0 5.6 ∑61.4
596.3 346.3 289.0 272.1 167.6
659.1 527.0 310.3 284.4 195.1
23.6 15.7 12.2 4.1 11.7 ∑67.3
24.3 8.1 14.4 5.3 7.9 ∑60.0
and layers. A typical situation where this problem occurs are crystals that grow plate-like, can be split only in one direction as many layered materials the most prominent being probably graphite or are generally two-dimensional like graphene. In particular in the latter case it is obvious that information about the principal component of the dielectric tensor function perpendicular to the surface cannot be obtained in the conventional way not even if a microscope is used. The only possibility in such cases is to use p-polarized light and a non-zero angle of incidence. Assuming an optically uniaxial material with the optical axis oriented perpendicular to the surface, the reflectance is given by: [17,27, 28] 2 ε⊥ cos2 α − 1− ε1 sin2 α ∥ Rp ¼ 2 1 2 ðε⊥ cos α Þ þ 1− ε∥ sin α 1 2
1 2
conventionally according to Eq. (3) and, additionally, with the semi-empirical 4 parameter oscillator model: N
ε ¼ ε∞ ∏
~ 2 −iν ~ γ LO; j ~ 2LO; j −ν ν
ð4Þ
~ 2 −iν ~ γ TO; j ~ 2TO; j −ν j¼1 ν
The parameters used in this model should be directly comparable to those used in Eqs. (2) and (3). We reinvestigated all 3 principle spectra of NdGaO3, and found that the average deviation between measured and modelled spectrum is reduced to 78.1% relative to the conventional model for the a-axis spectrum, while it is increased to 100.4% in case of the b-axis spectrum if inverse dielectric function modelling is used (again in comparison with the conventional model). The average deviation for the c-axis spectrum lies with 87.4% of that reached with the conventional model in between. The corresponding measured and modelled spectra are depicted in Fig. 3. The performance of the different models is easier to evaluate near very broad bands like the one between 300 and 550 cm−1. In particular at the high-wavenumber side of this band and at the adjacent minima it is obvious that the 4 parameter model performs much better than the conventional model. The inverse dielectric function modelling leads to a closer resemblance between modelled and measured spectrum around this band than the conventional Lorentz model and performs only slightly inferior in comparison with the 4 parameter model (4P in Fig. 3), which gives the best correspondence with an average deviation of 73.8% of that achieved with the conventional model. It is now possible to compare the values obtained for the different parameters. This comparison is provided in Table 1. Overall there is a good agreement concerning the TO- and LOwavenumbers. In case of the damping constants the agreement is not as good as for the wavenumbers. The reason for this is that for the 4 paN
N
j¼1
j¼1
rameter model not to be unphysical, it is required that ∑ γ TO; j b∑ γLO; j. Such a constraint does not exist in case of the Lorentz-models. Our conclusion is that this constraint leads to errors concerning the damping constants if the 4 parameter model is employed, in particular for weaker oscillators which compensate with their values of γTO those of other oscillators with more weight in the average deviation. Nevertheless, the 4 parameter model still has its merits if more emphasis is placed on describing the dielectric function as good as possible than on determining the oscillator parameters. To complete the demonstrations of potential applications of inverse dielectric function modelling we focus in the last part of this paper on the dispersion analysis of perpendicular modes in anisotropic crystals
215
1 2
1 2
ð5Þ
Here, ε⊥ is the principle component of the dielectric tensor in the direction parallel to the surface and ε∥ the principle component perpendicular to the surface (we use the same notation as is common in the literature, even if this notation is not straightforward in our opinion). Note that Eq. (5) simplifies to Eq. (1) if ε∥ = ε⊥ = ε. It is therefore justified to Assume that inverse dielectric function modelling can be also helpful in case of reflectance from materials that can be described with Eq. (5) as it was for those the reflectance of which could be calculated by Eq. (1). To test this assumption, we specify the same simple test case as in [17] by assuming a material with one oscillator in ε⊥ and another oscillator in ε∥ with the parameters as given in Table 2 and taken over from [17] (the parameters were chosen in a way to avoid overlap of the cor~ ∥;LO , responding bands but otherwise arbitrarily, with the exception of ν ~ ∥;TO and S∥,TO and was determined by dielectric which is a function of ν function modelling). For the model oscillators Employing Eq. (5), we can then generate reflectance spectra for different angles of incidence. The results are displayed in Fig. 4. Obviously, the influence of the ∥-mode increases with increasing α and the corresponding band is located even beyond the LO wavenumber as mentioned earlier. What is therefore missing in the spectrum is any information about the TO position of the ∥-mode. Since the LO position on the other hand depends not only on the TO position, but also on the oscillator strength S∥,TO and on ε∞,∥ its location does not allow an unambiguous conclusion with regard to the TO position. Accordingly, the error function (the root mean square of the difference between experi~ ∥;TO , S∥;TO and ε∞;∥ mental and simulated spectrum) in dependence of ν does not show a pronounced minimum, even not in the ideal situation of the test case, where both modes are well-separated (cf. Fig. 5a) [17]. As solution, it was previously found feasible to record additionally an internal reflectance spec~ ∥;TO in contrast to the trum, which shows a distinct minimum close to ν external reflection spectrum and use both spectra for the fit. The drawback of this solution is, however, that it is next to impossible to record an internal reflection spectrum of the necessary quality. Therefore, in practice the problem of retrieving information about the transversal frequencies of modes with transition dipole perpendicular to the reflecting surface was unsolved until now. With inverse dielectric function modelling of ε ∥, it is not even necessary to use spectra measured at two different angles of incidence as one is completely sufficient which is demonstrated in Fig. 5b. In contrast to the error function shown in Fig.
Table 2 TO oscillator parameters employed to generate ε⊥ and ε∥ via Eq. (3) and LO oscillator parameters for the ‖-mode gained by inverse dielectric function modelling. j
~ j (cm−1) ν
γj (cm−1)
Sj (cm−1)
ε∞, j
⊥, TO ∥, TO ∥, LO
800 1000 1060.66
10 10 10
500 500 250
2.1 2.0 2.0
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Fig. 6. Comparison between the default Rp (black) using the starting (TO-) values from Table 2 and Rp simulated using Eq. (5) in combination with Eq. (2) for ε∥.
Fig. 4. Upper part: Parallel polarized reflectance spectra of a model material based on the principal dielectric functions generated by Eq. (3) and the values provided by Table 2. The model material is assumed to be semi-infinite like the incidence medium, which is characterized by an index of refraction n = 1. Lower part: Real (Re(ε ∥ ), black) and imaginary (Im(ε∥ ), red/dark grey) parts as well as dielectric loss (−Im(1/(ε ∥ ), green/ light grey) of the principal dielectric functions generated by Eq. (3). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
5a, the error function of the inverse dielectric function modelling is well-behaved in the sense that it shows a pronounced minimum. −1 ~ ∥;LO on ε∞,∥ , which is very obstructive There is also no dependence of ν ~ in case of ν ∥;TO and ε∞,∥ for the direct dielectric function fitting. Accordingly, it is possible within 2 iterations and less than a minute time to regain the original reflectance spectrum even in case of starting values that are far away from the true values (see Fig. 6). This gives confidence that even under worse conditions e.g. that the bands of the ⊥-modes overlap with those of the ‖-modes, inverse dielectric function modelling is successfully applicable (certainly, the ⊥-modes can always be isolated and separately modelled by applying s-polarized light). It is a matter of course that ε⊥ has been evaluated separately before the evaluation of ε∥. ~ ∥;TO , S∥,TO and γ∥,TO are unknown. AcStill, there is one step missing, as ν tually, for the example above this is not quite correct, since in case of Lorentz-type functions we know that γ∥,TO = γ∥,LO. Indeed, through the modelling we find that confirmed (cf. Table 2). Furthermore, from the ~ Þ which can be inverted to yield ε ∥ ðν ~ Þ (and, likefit we obtained 1=ε∥ ðν ~ Þ with help of Eq. wise, ε∥,∞). Subsequently, a direct modelling of ε∥ ðν (3), a procedure that is carried out routinely in ellipsometry [11], yields ~ ∥;TO , S∥,TO. Applying this procedure, we regain the missing parameters ν
the original parameters used to generate the reflectance spectrum within numerical precision. 4. Conclusion The main objective of this work was to investigate how and under which conditions and circumstances dispersion analysis can benefit from the application of inverse dielectric function modelling. We found that for isotropic materials, i.e. materials with cubic symmetry or randomly oriented polycrystalline materials with small crystallites, the reflectance with parallel polarized light depends with increasing angle of incidence decreasingly on the dielectric function and increasingly on the inverse dielectric function. From this finding, we concluded that application of a Lorentz-type modelling of this inverse dielectric function is of advantage for dispersion analysis. This assumption was proven correct in case of the reflectance spectra of KAl(SO4)2·12H2O alum. Even for dispersion analysis of s-polarized spectra, however, the use of inverse dielectric function modelling seems to allow a better fit of the experimental spectra and, thereby, a better description of the optical properties in many cases as was shown for the alum and NdGaO3. Finally, we have transferred the idea of inverse dielectric function modelling for isotropic samples to anisotropic samples with different principal dielectric functions parallel and perpendicular to the surface. The spectra of the latter are usually impossible to analyze with conventional dispersion analysis as the properties of the bands due to perpendicular components show complex dependencies among the different oscillator parameters. In contrast, the use of inverse dielectric function modelling resolves these complex dependencies and the dispersion
~ ∥, keeping the damping fixed at its true value of 10 cm−1 and the dielectric Fig. 5. Plot of the error function of the external reflection for different pairs of oscillator strength S∥ and position ν background at ε ∥;∞ =2.0. ~ ∥ relate to the TO-mode. a) Error function resulting from direct dielectric function fitting of two reflectance spectra with α = 60° and α = 70° simultaneously. Strength S∥ and position ν ~ ∥ relate to the LO-mode. b) Error function resulting from inverse dielectric function fitting of one reflectance spectrum with α = 60°. strength S∥ and position ν
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parameter used in the direct modelling can be regained by inverting the inverse dielectric function and modelling the resulting dielectric function. Overall we conclude that dispersion analysis can profit in numerous ways by the use of inverse dielectric function modelling. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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