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Transparent conductive thin films are a key building block of modern optoelectronic devices. A promising alternative to expensive indium containing oxides is aluminum doped zinc oxide (AZO). By correlating spectroscopic ellipsometry and photoluminescence, we analyzed the contributions of different optical transitions in AZO grown by atomic layer deposition to a model dielectric function (MDF) over a wide range of photon energies. The derived MDF reflects the effects of the actual band structure and therefore describes the optical properties very accurately. The presented MDF is solely based on physically meaningful parameters in contrast to empirical models like e.g. the widely used Sellmeier equation, but nevertheless real and imaginary parts are expressed as closed-form expressions. We analyzed the influence of the position of the Fermi energy and the Fermi-edge singularity to the different parts of the MDF. This information is relevant for design and simulation of optoelectronic devices and can be determined by analyzing the results from spectroscopic ellipsometry.
© 2015 Optical Society of America
1. Introduction
Zinc oxide and aluminum doped ZnO (AZO) thin films have gained great importance for application as a transparent conductive oxide, being a cheap alternative to commonly used indium tin oxide. AZO based electrodes have potential applications in many fields like solid state lighting [1, 2], flat panel displays [3] and solar cells [4, 5]. Furthermore, AZO is a promising candidate for the metal component of silicon based metal-insulator-semiconductor solar cells [6–8]. Modern optoelectronic devices, like third generation solar cells, rely on efficient photon management by introducing nanostructured components to increase for example the absorption of light [9–13]. Realizing such devices requires a technique of conformally coating three dimensional (3D), high aspect ratio nanostructures with precise control of layer thickness. Atomic Layer Deposition (ALD) is the method of choice assuring accurate control of layer thickness at angstrom precision and conformality, even in 3D micro-/nano-geometry.
ZnO and AZO thin films grown by ALD have been carefully characterized in view of their structural and electronic properties [14–18], as well as the doping mechanism [19, 20]. However, the link to optics, especially to the dielectric function, still remains unclear. By spectroscopic ellipsometry (SE), which analyzes the change of the polarization state of light when reflected at an interface, it is possible to accurately determine the thickness and optical dispersion of thin films. SE in the visible and ultraviolet spectral range is commonly used for this purpose. Very often an empirical model like a Sellmeier equation [21] is employed to describe the dielectric function in the visible. This is sufficient, if the only goal is determining the layer thickness and approximating the dielectric function in the spectral region where the material is transparent. Nevertheless, a SE spectrum allows additional physical parameters to be extracted, e.g. the position of the Fermi energy of degenerately doped semiconductors. Although this work is focused on analyzing ZnO and AZO grown by ALD, the developed model dielectric function (MDF) may as well be applied to doped ZnO thin films grown by other techniques like magnetron sputtering or chemical vapor deposition or to different transparent conductive oxides.
For analyzing thin films it is necessary to take the finite thickness and therefore the reflection from the backside of the layer into account. Especially when the penetration depth of the light in the part of the spectrum where the material is absorbing, is larger or of the order of the layer thickness, the dielectric function and the layer thickness have to be fitted simultaneously. For this case it is advantageous to use a model dielectric function (MDF), having the real and imaginary part expressed in closed-form. In this study, we employed a MDF, which is a sum of three independent models. The first is based on transitions at M0 critical points of the band structure as introduced by Adachi [22]. The second is a model introduced by Forouhi and Bloomer [23], describing transitions between arbitrary states. The third part is the Drude model [24], which takes the contribution of free electrons in the infrared into account. We correlated the results from room-temperature SE analysis and photoluminescence (PL). SE and PL are in parts complementary, since PL is based on emission and SE on absorption, we thereby verify the MDF. From the analysis of the three parts of the MDF, we obtain evidence, that the optical behavior of AZO deposited by ALD is at room temperature dominated by the direct band to band transition and the Fermi-edge singularity, which are both influenced by band gap renormalization (BGR) [25] and the Burstein-Moss effect [26, 27], and free electrons. Moreover, the real and imaginary part of the MDF is expressed in closed-form, without needing numerical evaluation of the Kramers-Kronig integrals, therefore facilitating numerical simulations especially in the time domain.
2. Experimental
ZnO and AZO films were deposited using a TFS200 ALD system (Beneq) on silicon wafers ((100) orientation, Silicon Materials). The substrates were cleaned by ultrasonication in acetone and isopropanol and subsequently dried in nitrogen. The natural oxide was removed from silicon substrates by dipping in 5% HF for one minute, rinsing in deionized (DI) water and blowing dry using nitrogen immediately before loading the substrates in the ALD system. Deposition was carried out at 200°C chamber temperature of the hot wall reactor. As carrier and purge gas, ultra pure nitrogen (≥ 99.9999%) was used at a flow rate of 300sccm. For deposition of ZnO, we used diethylzinc (TCO grade, DockChemicals) and DI water as precursors and for Al-doping trimethylaluminum (ALD grade, DockChemicals) and DI water.
Different doping concentrations were achieved by grouping ZnO and doping cycles into macrocycles with ratio ZnO:AlOx of y:1 (y=10, 20, 30, ..., 100). The number of complete macrocycles was chosen to always result in a layer thickness of approximately 100nm.
The aluminum content was measured by glow-discharge mass-spectroscopy (GD-MS) at AQura GmbH (Hanau, Germany) using the samples on silicon substrates. The measurement procedure is described in detail in reference [28].
Hall effect measurements in van der Pauw geometry [29] were carried out on layers deposited on glass substrates at room temperature using Hall-system (BioRad) and a picoamperemeter (Hewlett-Packard). The magnetic field was applied by a permanent magnet with magnetic flux density of 0.32T ± 0.032T.
Photoluminescence measurements were carried out using a LabRam HR 800UV spectrometer with a liquid nitrogen cooled CCD array detector (Horiba Jobin Yvon) and a He-Cd laser with 325nm excitation wavelength.
Spectroscopic ellipsometry was performed using an UVISEL (Horiba Jobin Yvon) phase modulated spectroscopic ellipsometer within a range of photon energy from 0.6eV–4.8eV in steps of 0.01eV. The angle of incidence was 70° and incident light was linearly polarized at an angle of 45° relative to the plane of incidence.
3. Data analysis
Analysis of SE results was performed by fitting the model described below to the quantities Is and Ic measured by SE. In the geometry used, they are related to the standard ellipsometric angles Ψ and Δ by [30]
For modeling the ALD deposited AZO layers in SE analysis, three distinct parts were considered: (i) the silicon substrate, (ii) the dense AZO layer itself and (iii) the surface roughness of the AZO layers, as schematically shown in Fig. 1. The roughness was modeled using a Bruggeman effective medium [31] with a filling fraction of 50%. (i) was assumed to be infinitely thick.Backside reflection from the silicon substrate can be neglected, because it is strongly absorbing above the silicon band gap of about 1.1eV and the backside is very rough, causing scattering of light out of the acceptance angle of the ellipsometer. The thicknesses of (ii) and (iii) were fitted. The roughness has to be taken into account, because AZO grown by ALD is nano-crystalline, which has been shown elsewhere [32] (cf. also SEM image in Fig. 2 and references [14–18]). As a consequence, it is suitable to not consider the birefringence of AZO in the model. The crystallites, which are several tens of nm in size, are randomly oriented, so the layers do not have a specified orientation. Therefore, the results should be seen as an average over all crystal orientations.
For modeling the dielectric function of the AZO layers we used a MDF combining three parts, εAdachi, εForouhi and εDrude, as summed up in Eq. (2)
where ε1 and ε2 denote the real and the imaginary part of the complex dielectric function, respectively.The first part, εAdachi, introduced by Adachi [22, 33], describes optical transitions from the valence band (VB) to the conduction band (CB) at 3D M0 critical points, which means from the maximum of the VB to the minimum of the CB. The imaginary part Im(εAdachi) of εAdachi is given by [33]
where μ is the joint density of states effective mass, PCV the momentum matrix element of a transition between the valence and conduction band and JCV(E) the joint density of states, which is given by the surface integral over a constant energy surface in the first Brillouin zone This model therefore explicitly describes direct interband transitions. The real part of εAdachi is then obtained by Kramers-Kronig integration. The result is given by with where A is the strength parameter, which is in principle the prefactor of Eq. (3). E1 is the band-to-band transition energy and Γ1 is the damping, resulting in lifetime broadening.The second part, εForouhi, was originally derived by Forouhi and Bloomer [23] for amorphous materials, assuming that localized atomic orbitals overlap, merge together and form a band. A finite lifetime of the excited state is assumed and momentum conservation is neglected.
εForouhi is easier to express in terms of real- and imaginary part of the refractive index, ñForouhi = nForouhi + ikForouhi and therefore , where
where f is the oscillator strength, E0 the cutoff energy (the band-gap in the derivation of Forouhi and Bloomer), E2 a parameter close to the transition energy, Γ2 the damping constant and n∞ a real refractive index constant greater than unity. It is worthwhile noting that the energy Emax, where kForouhi(Emax) is maximum is given by According to Forouhi and Bloomer, this corresponds to the energy difference between the initial orbitals.The third part of the model is the Drude model [24], describing the contribution of free electrons in the conduction band, determining the behavior of the material in the infrared part of the spectrum. εDrude is given by
where Ep is the plasma energy of the electron gas and ΓD the damping.We correlated the results from SE and PL by fitting the PL spectra using the results from the analysis of SE spectra. In degenerately doped semiconductors, the high energy part of the spectrum follows the Fermi-Dirac distribution of the conduction band. The distribution of the excess holes can be assumed to be δ like and therefore its influence on the spectral shape of the PL can be neglected [34]. Hence, in case of low excitation, which means the absorption of the excitation light does only create a negligible concentration of excess charge carriers, the PL intensity can be expressed as
where J(E) denotes the joint density of states of the bands and states involved in the optical transition. According to Eq. (3) J(E) ∝ ε2(E)E2, which leads to the following expression for fitting the PL spectra, with the proportionality factor D and the Fermi energy EF being the fitting parameters.4. Results
In Fig. 3 the charge carrier concentration and Hall mobility measured by Hall effect is shown as function of Al concentration, which was measured by GD-MS. The nominally undoped ZnO layer has a significant charge carrier concentration within the same order of magnitude as the lowly doped layers, which is about 1019 cm−3. This indicates that a large number of intrinsic donors are present in the layers. A doping efficiency, which is the fraction of donor atoms that contribute an electron to the conduction band, of the Al atoms between 19.7% for the lowest and 11.4% for the highest Al doping can be estimated. Low doping efficiencies for ALD deposited AZO are well known and have already been investigated [18, 20, 35].
Fig. 3 Charge carrier density, Hall mobility and resistivity of the AZO films determined by Hall measurements. The charge carrier density increases with increasing Al concentration, while the mobility decreases due to increased impurity scattering.
SE spectra were very accurately fitted, using the model explained in the previous section, especially in the near band edge region at about 3.3eV and the absorbing spectral region above the band gap. The results of the fit for all samples are shown in Fig. 4 and the resulting parameters of the MDF are given in Table 1 in the appendix. The blue-shift as well as the degeneration of the band edge feature around 3.3eV in the spectra for increased doping, can accurately be reproduced.
Fig. 4 Ellipsometry data and result of the model fit. Left: Is. Right: Ic. The band-edge features at about 3.3eV are excellently reproduced by the model. The spectra are vertically shifted by 1 for clarity.
The real part (ε1) and imaginary part (ε2) of the dielectric function of the AZO layers are obtained from the MDF and are plotted in Fig. 5(a) and (b), respectively. The blue-shift and degeneration of the band edge features are even more pronounced here. The peak in the imaginary part completely vanishes and the band edge feature in the real part smears out.
Fig. 5 (a) Real part of the dielectric function of the AZO layers obtained from the MDF. The spectra are vertically shifted by 1 for clarity. (b) Imaginary part of the dielectric function of the AZO layers. The contributions of the three different parts of the model are plotted in red (Adachi), blue (Forouhi) and green (Drude). With increasing Al concentration there is a blue-shift of the band-edge due to band-filling and also the Fermi-edge singularity degenerates. Due to the higher concentration of free charge carriers at high doping, the plasma frequency increases, resulting in higher absorption in the infrared. The spectra are vertically shifted by 2.
Figure 6 shows the measured PL spectra with the fit using the results from SE. For doping densities higher than 0.85at.%, the PL signal was too weak to be separated from the background. Additionally, at high doping, there are potentially more routes of non-radiative recombination, leading to increased heating of the samples, inducing deviations between SE and PL, which could not be accounted for. Nevertheless, especially the initial increase of the PL signal and peak position is resembled by the model very well, where the dispersion model has the most influence.
Fig. 6 PL spectra of AZO layers fitted using Eq. (15) and ε2 from the MDF. The PL signal becomes very weak at high doping, suggesting the existence of many channels of non-radiative recombination, leading to heating of the sample. This can result in deviations between the results of SE and PL.
5. Discussion
As shown before, the presented MDF delivers an accurate description for the optical behavior of AZO. Beyond that, it is based on physical parameters and, thus, relates the band structure to the interaction of the material with light. The results of both methods, SE and PL, are in very good agreement, validating our model. In the following, we will discuss the results for the three different parts of the dispersion model in more detail.
The imaginary part of the dielectric function in the infrared spectral region is determined by the free electrons in the conduction band. Their contribution is represented by the Drude model [24]. The plasma frequency increases with increasing charge carrier concentration, which is nicely visible in Fig. 5. In principle, the charge carrier concentration can directly be calculated from the plasma frequency. However, due to the deformation of the bands, which depends on the charge carrier concentration and the crystal quality (e.g. strain) [36], and a potential asymmetry of the bands, the effective mass of electrons is quite different from the value in the undoped case [37]. Hence, a quantitatively correct determination of the charge carrier concentration from the plasma frequency is at least very complicated and not part of this work.
The band to band transition energy obtained from εAdachi deviates from the intrinsic value of 3.37eV, because of BGR and band filling (Burstein-Moss effect) [25–27]. The Fermi energy of our AZO layers is supposed to be above the conduction band minimum, since they are degenerately doped. In this case the transition energy is determined by the position of the Fermi energy. Up to a charge carrier concentration of about 1019 cm−3 – 1020 cm−3, BGR is the dominant effect, causing a shrinkage of the band gap. At higher doping, the band filling overcompensates BGR causing the band gap to increase. The charge carrier concentration of our samples is above 1019 cm−3, which is apparent from Fig. 3, and therefore one expects an increase of the transition energy with increasing doping, which indeed is the case. Moreover, due to the position of the Fermi energy above the conduction band, the transition energy corresponds to the Fermi energy. This is also consistent with our results from PL (cf. Fig. 6), where the Fermi energy determines the decay of PL signal on the high energy side, as can be seen in Fig. 7(a). The deviation from the ideal case, which is indicated by the black line in Fig. 7(a) is within kBT at room temperature, given by the grey shaded area.
Fig. 7 (a) Comparison between the transition energy obtained from the Adachi model and the Fermi energy obtained from PL. The results from both methods are in very good agreement. The diagonal line is a guide to the eye, showing the ideal accordance. The shaded area indicates thermal energy at room temperature. (b) Energy where kForouhi is maximum.
As can be seen in Fig. 5, the contribution of the Forouhi-Bloomer model is due to bound states. This contribution is located at the fundamental absorption edge. In AZO, it has been theoretically and experimentally shown that Fermi-edge singularities [38] are persistent even at room temperature [39,40]. We believe, that the Forouhi-Bloomer model, which is rather generic since it was derived for the very general case of transitions between arbitrary states, resembles this feature very well. Nevertheless, the peak position of ε2,Forouhi can be calculated using Eq. (12), which corresponds to the peak of the absorption. The peak positions shown in Fig. 7(b) are in very good agreement with the findings of Makino et al. [39]. For higher charge carrier concentrations, the Fermi-edge singularity becomes less and less pronounced, as predicted by Mahan [38] and also shown by Makino et al. [39] and Schleife et al. [40].
6. Summary
We investigated the optical properties of AZO layers deposited by ALD by correlating spectroscopic ellipsometry and photoluminescence. A MDF combining three independent models, the Adachi, Forouhi and Drude model, describes the optical behavior of the material very accurately. The analysis using this model gave evidence that the optical properties in the near infrared, visible and near ultraviloet spectral range are determined by free electrons, the band to band transition and the Fermi-edge singularity, respectively. Correlating the results of SE and PL with one another validates the model. We showed that it is possible to extract physically meaningful parameters like the plasma energy, the Fermi energy and the position of the peak of absorption. Moreover both, real and imaginary part of the MDF, can be expressed in closed-form equations without necessitating numerical evaluation of the Kramers-Kronig integrals.
Appendix
Table 1. Result of least squares fit of the MDF to the SE data.
Acknowledgments
The authors would like to acknowledge the funding of the Deutsche Forschungsgemeinschaft (DFG) through the Cluster of Excellence Engineering of Advanced Materials. S. H. C. acknowledges the DFG for financial support within the research group “Dynamics and Interactions of Semiconductor Nanowires for Optoelectronics” ( FOR 1616). M. G. and S. H. C. also acknowledge the DFG GRK 1896 “In-Situ Microscopy with Electrons, X-rays and Scanning Probes”. The authors are thankful to Dr. Katja Höflich for fruitful discussions.
References
1. H. Kim, J. Horwitz, W. Kim, A. Mäkinen, Z. Kafafi, and D. Chrisey, “Doped ZnO thin films as anode materials for organic light-emitting diodes,” Thin Solid Films 420–421, 539–543 (2002). [CrossRef]
2. X. Jiang, F. L. Wong, M. K. Fung, and S. T. Lee, “Aluminum-doped zinc oxide films as transparent conductive electrode for organic light-emitting devices,” Appl. Phys. Lett. 83, 1875–1877 (2003). [CrossRef]
3. E. Fortunato, P. Barquinha, A. Pimentel, A. Gonçalves, A. Marques, L. Pereira, and R. Martins, “Recent advances in ZnO transparent thin film transistors,” Thin Solid Films 487, 205–211 (2005). [CrossRef]
4. J. Müller, B. Rech, J. Springer, and M. Vanecek, “TCO and light trapping in silicon thin film solar cells,” Sol. Energy 77, 917–930 (2004). [CrossRef]
5. M. Berginski, J. Hüpkes, M. Schulte, G. Schöpe, H. Stiebig, B. Rech, and M. Wuttig, “The effect of front ZnO:Al surface texture and optical transparency on efficient light trapping in silicon thin-film solar cells,” J. Appl. Phys. 101, 074903 (2007). [CrossRef]
6. M. Green, F. King, and J. Shewchun, “Minority carrier MIS tunnel diodes and their application to electron- and photo-voltaic energy conversion—I. Theory,” Solid. State. Electron. 17, 551–561 (1974). [CrossRef]
7. J. Shewchun, M. Green, and F. King, “Minority carrier MIS tunnel diodes and their application to electron- and photo-voltaic energy conversion—II. Experiment,” Solid. State. Electron. 17, 563–572 (1974). [CrossRef]
8. H. Kobayashi, H. Mori, T. Ishida, and Y. Nakato, “Zinc oxide/n-Si junction solar cells produced by spray-pyrolysis method,” J. Appl. Phys. 77, 1301–1307 (1995). [CrossRef]
9. E. Garnett and P. Yang, “Light trapping in silicon nanowire solar cells,” Nano Lett. 10, 1082–1087 (2010). [CrossRef] [PubMed]
10. G. Brönstrup, F. Garwe, A. Csáki, W. Fritzsche, A. Steinbrück, and S. Christiansen, “Statistical model on the optical properties of silicon nanowire mats,” Phys. Rev. B 84, 125432 (2011). [CrossRef]
11. S. W. Schmitt, F. Schechtel, D. Amkreutz, M. Bashouti, S. K. Srivastava, B. Hoffmann, C. Dieker, E. Spiecker, B. Rech, and S. H. Christiansen, “Nanowire arrays in multicrystalline silicon thin films on glass: A promising material for research and applications in nanotechnology,” Nano Lett. 12, 4050–4054 (2012). [CrossRef] [PubMed]
12. K. X. Wang, Z. Yu, V. Liu, Y. Cui, and S. Fan, “Absorption enhancement in ultrathin crystalline silicon solar cells with antireflection and light-trapping nanocone gratings,” Nano Lett. 12, 1616–1619 (2012). [CrossRef] [PubMed]
13. G. Shalev, S. W. Schmitt, H. Embrechts, G. Brönstrup, and S. Christiansen, “Enhanced photovoltaics inspired by the fovea centralis,” Sci. Rep. 5, 8570 (2015). [CrossRef] [PubMed]
14. P. Banerjee, W.-J. Lee, K.-R. Bae, S. B. Lee, and G. W. Rubloff, “Structural, electrical, and optical properties of atomic layer deposition Al-doped ZnO films,” J. Appl. Phys. 108, 043504 (2010). [CrossRef]
15. T. Dhakal, D. Vanhart, R. Christian, A. Nandur, A. Sharma, and C. R. Westgate, “Growth morphology and electrical/optical properties of Al-doped ZnO thin films grown by atomic layer deposition,” J. Vac. Sci. & Technol. A Vacuum, Surfaces, Film. 30, 021202 (2012). [CrossRef]
16. J. W. Elam, D. Routkevitch, and S. M. George, “Properties of ZnO/Al2O3 alloy films grown using atomic layer deposition techniques,” J. Electrochem. Soc. 150, G339–G346 (2003). [CrossRef]
17. R. M. Mundle, H. S. Terry, K. Santiago, D. Shaw, M. Bahoura, A. K. Pradhan, K. Dasari, and R. Palai, “Electrical conductivity and photoresistance of atomic layer deposited Al-doped ZnO films,” J. Vac. Sci. Technol. A Vacuum, Surfaces, Film. 31, 01A146 (2013).
18. Y. Wu, P. M. Hermkens, B. W. H. van de Loo, H. C. M. Knoops, S. E. Potts, M. a. Verheijen, F. Roozeboom, and W. M. M. Kessels, “Electrical transport and Al doping efficiency in nanoscale ZnO films prepared by atomic layer deposition,” J. Appl. Phys. 114, 024308 (2013). [CrossRef]
19. D.-J. Lee, H.-M. Kim, J.-Y. Kwon, H. Choi, S.-H. Kim, and K.-B. Kim, “Structural and electrical properties of atomic layer deposited Al-doped ZnO films,” Adv. Funct. Mater. 21, 448–455 (2011). [CrossRef]
20. Y. Wu, S. E. Potts, P. M. Hermkens, H. C. M. Knoops, F. Roozeboom, and W. M. M. Kessels, “Enhanced doping efficiency of Al-doped ZnO by atomic layer deposition using dimethylaluminum isopropoxide as an alternative aluminum precursor,” Chem. Mater. 25, 4619–4622 (2013). [CrossRef]
21. W. von Sellmeier, “Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Ann. der Phys. und Chemie 143, 272–282 (1871). [CrossRef]
22. H. Yoshikawa and S. Adachi, “Optical constants of ZnO,” Jpn. J. Appl. Phys. 36, 6237–6243 (1997). [CrossRef]
23. A. R. Forouhi and I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34, 7018–7026 (1986). [CrossRef]
24. P. Drude, “Zur Elektronentheorie der Metalle,” Ann. Phys. 306, 566–613 (1900). [CrossRef]
25. K. F. Berggren and B. E. Sernelius, “Band-gap narrowing in heavily doped many-valley semiconductors,” Phys. Rev. B 24, 1971–1986 (1981). [CrossRef]
26. T. S. Moss, “The interpretation of the properties of indium antimonide,” Proc. Phys. Soc. B 67, 775–782 (1954). [CrossRef]
27. E. Burstein, “Anomalous optical absorption limit in InSb,” Phys. Rev. 93, 632–633 (1954). [CrossRef]
28. S. W. Schmitt, C. Venzago, B. Hoffmann, V. Sivakov, T. Hofmann, J. Michler, S. Christiansen, and G. Gamez, “Glow discharge techniques in the chemical analysis of photovoltaic materials,” Prog. Photovoltaics Res. Appl. 22, 371–382 (2014). [CrossRef]
29. L. J. van der Pauw, “A method of measuring the resistivity and Hall coefficient on lamellae of arbitrary shape,” Philips Tech. Rev. 26, 220–224 (1958).
30. E. Garcia-Caurel, A. De Martino, J.-P. Gaston, and L. Yan, “Application of spectroscopic ellipsometry and Mueller ellipsometry to optical characterization,” Appl. Spectrosc. 67, 1–21 (2013). [CrossRef] [PubMed]
31. D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. 24, 636–679 (1935). [CrossRef]
32. M. Göbelt, R. Keding, S. W. Schmitt, B. Hoffmann, S. Jäckle, M. Latzel, V. V. Radmilović, V. R. Radmilović, E. Spiecker, and S. Christiansen, “Encapsulation of silver nanowire networks by atomic layer deposition for indium-free transparent electrodes,” Nano Energy 16, 196–206 (2015). [CrossRef]
33. S. Adachi, Optical properties of crystalline and amorphous semiconductors (Springer, New York, 1999), 1st ed.
34. B. G. Arnaudov, D. S. Domanevskii, A. M. Isusov, P. L. Gardev, and S. K. Evtimova, “Free electron recombination in degenerately doped and moderately compensated gallium arsenide,” Semicond. Sci. Technol. 5, 620–623 (1990). [CrossRef]
35. D.-J. Lee, J.-Y. Kwon, S.-H. Kim, H.-M. Kim, and K.-B. Kim, “Effect of Al distribution on carrier generation of atomic layer deposited Al-doped ZnO films,” J. Electrochem. Soc. 158, D277–D281 (2011). [CrossRef]
36. S. L. Chuang and C. S. Chang, “k·p method for strained wurtzite semiconductors,” Phys. Rev. B 54, 2491–2504 (1996). [CrossRef]
37. T. Pisarkiewicz, K. Zakrzewska, and E. Leja, “Scattering of charge carriers in transparent and conducting thin oxide films with a non-parabolic conduction band,” Thin Solid Films 174, 217–223 (1989). [CrossRef]
38. G. D. Mahan, “Excitons in degenerate semiconductors,” Phys. Rev. 153, 882–889 (1967). [CrossRef]
39. T. Makino, K. Tamura, C. H. Chia, Y. Segawa, M. Kawasaki, A. Ohtomo, and H. Koinuma, “Optical properties of ZnO:Al epilayers: Observation of room-temperature many-body absorption-edge singularity,” Phys. Rev. B 65, 121201 (2002). [CrossRef]
40. A. Schleife, C. Rödl, F. Fuchs, K. Hannewald, and F. Bechstedt, “Optical absorption in degenerately doped semiconductors: Mott transition or Mahan excitons?” Phys. Rev. Lett. 107, 236405 (2011). [CrossRef] [PubMed]
