Prediction of Surface Plasmon Resonsnce and More Accurate Representation of Absorption Features Both Below and Above the Bandgap in ZnO Nanorods array – Au Heterostructures

Well-oriented zinc oxide nanorods (ZnO NRs) arrays have been grown by low temperature chemical bath deposition on seeded substrates. A gold thin film has obliquely been deposited by DC magnetron sputtering on the ZnO NRs array. The structure, mophology/ chemical identity, vibrational identity have been studied by X-ray diffraction (XRD), field effectscanning electron microscope/ energy dispersive X-ray spectroscopy (FE-SEM/EDX) and Raman spectroscopy, respectively. The FCC structure of Au is formed on vertically oriented ZnO NRsarray. The wavelength dependent photocurrent of ZnO NRs array-Au heteronanostructure (HNS) was evaluated by photogain response under red, green and blue laser illuminations. Surface plasmon excitation activates selective response to green laser exposure. An analytical dispersion formalism has been constructed to fit experimental absorption spectrum over wide spectrum range and to extract precise bandgap energy, subband tailing, dielectric constant and carrier effective mass. The proposed model exploits the Frouhi–Bloomer (FB) parameterization and Gaussian oscillator dispersion to the complex dielectric function for Au decorated ZnO NRs array. Sharp variation in the optical absorption around the bandgap edge and the absorption behavior beyond the bandgap edge are covered as well. It is surprising that the surface plasmon resonance (SPR) is included without new formalism. The new model has been satisfactorily tested on CuO optical absorption.


Introduction
The optical constants of materials are necessary for different applications, such as design, characterization and/or process control of coatings for optics or semiconductor industry and probing into electronic structures [1][2][3][4]. To describe and parameterize the complex refractive index or the dielectric function of semiconductors, various models such as Forouhi and Bloomer (FB) [1], Campi and Coriasso [5], Tauc-Lorentz (TL) [6] and Cody-Lorentz [7] have been developed. To broaden the scope of materials and the spectral range, the models will need more parameters to fit.
Recently Franta et al. [3] developed a model which covers a broad spectral range which depends on a large number of parameters.
Optical constants of materials must satisfy some requirements that arise from fundamental properties of the radiation-matter interaction such as causality and linearity. Hence causality results in that the optical constants of a material versus photon energy can be extended with an analytic function to complex energies in the upper half plane, with a given parity and with a fast enough decay towards large energies [8]. The Kramers-Kronig (KK) analysis meets all these conditions and connects the imaginary to the real part of the optical constant complex function, and vice-versa. Most presented models optical-constant dispersion of semiconductors are defined with piecewise functions with a change in the functional behavior at the semiconductor bandgap and hence they are not analytic; an exception to this is FB model, even though this model does not work below the bandgap energy. Furthermore, divergences and parity violations are common problems among the mentioned models [3][4]8] One of the well-established and widely used dispersion models to describe the optical properties of the amorphous materials, is the Tauc-Lorentz (TL) model proposed by Jellison and Modine [6] which combine the Tauc expression for ε2 near the band edge with the imaginary part of the complex dielectric function for a single classical Lorenz oscillator [4,6]. Although The TL model severally covers as well smooth variation of band tailing for amorphous material, but for crystalline materials encounters problems due to sharp variations below the bandedge.
In addition, discontinuity in the bandgap energy point is an obvious disadvantage of the piecewise functions [2 -4]. Likhachev et al. [4] presented a modified TL model which leading to a more accurate representation of absorption features below the bandgap energy. A wellknown problem with the application of the TL model to thin films is that it explicitly assumes weak absorption below E g , i.e., ε 2 (E) = 0 for E ≤ Eg, which makes the modeling of the optical spectra of some materials (especially crystalline structures) in the vicinity of the absorption edge problematic [4].
The TL model is not fully analytic and presents mathematical shortcomings. [8]. Rodriguez De-Marcos et al. [8] proposed a procedure that transforms non-analytic TL model into analytic and self-consistent model which used to fit the optical constants of SiC films. They exemplified their model, by imposing the TL dispersion to the Urbach tail, for the optical constants of Si 3 N 4 .
In TL model the imaginary part of the dielectric constant ε2 turns zero below the material bandgap energy, whereas in practice, semiconductors and dielectrics are known to experience an exponential decay of the absorption coefficient, α, below the bandgap, which is called the Urbach tail. Valuable models were proposed to incorporate the Urbach tail to the TL model by Foldyna et al. [9] and a similar extension to the Cody-Lorentz model was performed by Ferlauto et al. [10]. However, the corrected models kept using the piecewise functions, so that the lack of analyticity is not solved. Likhacher et al. [4] and Franta et al. [11] reported that, the absorption below the band gap often does not correspond to the Urbach tail and the simple combination of Urbach subband tail with the TL oscillator(s) is insufficient near the band gap.
Li et al. [2] developed a model for describing the absorption coefficient based on optical transitions with a normalized average dipole matrix element. Their model was employed to fit the optical absorption edges of annealed amorphous Si:H films and subsequently to determine the values of the optical gap, tail breadth and mobility gap. The relationship between the optical gap and the band tail breadth reveals that the tail breadth depends linearly on the structural disorder [2]. Li et al. [2] and Franta et al. [3] used piecewise functions to model optical absorption and band parameters that requires various parameters and worry about the continuity of the absorption behavior.
Previous dispersion model were often concentrated on disordered and amorphous materials and glassy system [2,12,13]. formalism. An example has been provided to illustrate the practical use of the model.

Characterization
The crystalline structure was investigated using X-ray diffractometry (XRD, Bruker D8 ADVANCE). The surface morphology and chemical composition were studied by field emission scanning electron microscope and energy dispersive X ray spectroscopy (FE-SEM/EDX, Hitachi S-4800), respectively. The thickness was determined by cross sectional FE-SEM images, oscillating quartz crystal monitoring and reflectance ellipsometry (LEOI-44 model, wavelength 632.8nm of He-Ne laser). The optical absorption spectra were recorded by an UV-Vis-near IR spectrophotometer (Avantes-Spec2048). The Raman spectrum was measured at room temperature using a micro Raman spectrometer (Taksan, TakRam Model: P50C0R10) with 532 nm line of Nd: YAG laser as an exciting light source.

photogain measurement
The bare ZnO NRs and ZnO NRs-Au array heteronanostructure (HNS) were sandwiched between Ag and fluorine doped tin oxide (FTO) electrodes. Current -Voltage characteristic was measured under the red, green and red laser irradiation. A source measurement (model 2450 Keithley Instrument) was used to detect the output signals generated from the HNS.

Brief review of the dispersion models
The main models related to the topic of the present paper are briefly reviewed.

Lorentz oscillator dispersion and Drude model
The Lorentz oscillator model is one of the oldest classical dispersion models for the predicting the dielectric functions of materials, which assumes that the material can be described as a collection of non-interacting dipoles and damped harmonic oscillators, and it is still being used successfully to explain experimental results [4]. The electronic and vibrational dipoles are both examples of bound oscillators [17].
The imaginary part ε2 of the dielectric function is given by [4,17] where the fitting parameters are the j-th Lorentz oscillator position or natural resonant energy (E 0j ), the amplitude (A j ) and the broadening parameter of the bound electrons ( = ħ ) and damping rate.
Metals and doped semiconductors contain significant numbers of free electrons. These electrons are not bound to any atoms, and therefore do not experience any restoring forces when they are displaced. Hence the natural resonant energy for free electrons E 0 = 0. The free carrier contribution to the imaginary part of the dielectric function ε2D is given by Drude model [17,18] 2 ( ) = ∞ 2 3 + 2 (2)

Gaussian oscillator dispersion model
The atomic bond lengths and angles for amorphous and glassy materials are normally distributed around average values [4,19] hence, the Gaussian oscillator model may be a good approximation for amorphous and glassy materials. Likhachev et al. [4] first proposed the Guassin Oscillator dispersion may be used for modeling the optical functions of molecular resists or copolymers as well as for the dielectric functions of disordered materials in the infrared range.
A parameterized form for the imaginary part of the dielectric function ε2G is defined by where ∆ is the Guassian broadening parameter and EG is the peak position of Guassian oscillator dispersion.

Forouhi -Bloomer dispersion model
Easwarakhanthan et al. [20] reported Each term in the summations for k(E) and n(E) contributes either a peak or a shoulder to the spectra, respectively. Thus, the number of terms equals the number of discernible spectral peaks and shoulders. In Eq. 1, E g represents the bandgap energy which defined as that value of photon energy that k(E) exhibits an absolute minimum. The parameters A i , B i and C i are not mere fitting parameters; they depend on the electronic configuration of the material [1]. The quantities B 0i and C 0i are not independent parameters, they depend on A i , B i , C i and E g . Eqs 1 and 2 are valid over a very wide range of photon energies spanning ultraviolet, visible, and near infrared. Furthermore, they are related through the Kramers-Kronig relation [1].

Model construction
The total imaginary part of the dielectric function includes the bound electrons, band to band transition and free carriers contributions. So that 2 ( ) = 2 ( ) + 2 − ( ) + 2 ( ) where the first term is a single Lorentzian dispersion for the bound electrons [4,17] 2 ( ) = ( 0 2 − 2 ) 2 + 2 2 and the second terms is selected as the product of single FB terms and Gaussians peaks.  7) and (8): where ε ∞ represents the value of the real part of the dielectric function ε at infinite energy and it is an additional fitting parameter in the oscillator model, the P stands for the Cauchy principal part of the integral.
The total real part of the dielectric function includes where the first term is the Kramers-Kronig integration of a single Lorentzian dispersion for the bound electrons [17].
The second terms in Eq. 11 is the Kramers-Kronig integration of the product of single FB terms and Gaussians peaks 2 − ( ) which is obtained using residual (Res) calculus.
The third term is the real part of Drude dielctrice function [18]

Refractive index, dielectric constant and effective mass
The refractive index of the bare ZnO NRs array and ZnO NRs-Au heterostructure was calculated using the Moss relation [21], which is directly related to the fundamental energy band gap (Eg), The high-frequency dielectric constant ( ∞ ) was calculated from the following relation [25], The static dielectric constant ( ) of the films was calculated using a relation which expresses the energy band gap dependence of for semiconductor compounds in the following form [     The peaks between 942 and 1145 cm -1 may be attributed to the second order surface phonon modes. Breakdown of the translation symmetry in the bare ZnO NRs array cause to the surface phonon modes which highly localized near the boundary of the NRs [16]. Since the surface area is considerably increased due to the fabrication of ZnO NRs, the interaction of light with the surface should activate the surface phonon modes. This gives rise to two branches of surface modes between the TO and LO phonon frequencies of ZnO [26]. The frequency of the upper surface phonon mode (UM) is normally below the corresponding bulk E 1 (LO) value (591 cm -1 ) [29]. The lower surface phonon mode (LM) is clearly reported at around 475 cm -1 in the highly c-axis oriented and well-isolated vertically aligned ZnO NRs [16]. The oxygen deficiency results in a peak at 578 cm -1 that positioned between A 1 (LO) and E 1 (LO) optical phonon mode [16,[27][28]. The peaks around 287 and 970 cm -1 may be attributed to orders of surface phonon modesor oxygen deficiency. conductance and NPs parameters [16].

UV-Vis Near IR Absorption and Optical Modeling Results
To measure the UV−visible absorption spectra of bare ZnO NRs array and ZnO NRs -Au array HNS, they were deposited on quartz substrates. The bare ZnO NRs absorption spectra are shown in Fig. 4. The absorption edge above 3 eV is due to the interband absorption of ZnO. The     Table 2 shows the refractive index, high frequency and low (static) dielectric constants, bandgap energy and effective mass. The Dimitrov-Sakka relation predicts moderate value between that of Moss and Herve -Vandamme. ε ∞ increases as Au thickness increases while ε 0 , E g and * 0 shows decreasing terend versus increase in Au thickness.

Exemplification of the constructed model: CuO thin film
To test and generalize the constructed model, CuO thin film of 100 nm thickness has been evaluated. It is seen that the model result is well coincide with experimental data (Fig. 5). The predicted bandgap energy is 1.98 eV [21]. Also the model meet good fitting below and above band to band transition.

Photogain study
To evaluate the light sensitivity of ZnO NRs-Au 40nm array HNS, the photogain (photocurrent density per light power) was examined. Fig. 8 depicts the photogain -voltage characteristic of ZnO NRs-Au 40nm array HNS under the illumination of red, green and blue laser versus sweep voltage. The photosensitivity has higher value for green illumination in comparison to the red and blue light. SPR excitation in green region of electromagnetic spectrum spoil the carriers and surface charge transfer [16].

Funding: Not applicable
Conflicts of interest/Competing interests: This is our research paper. Without the participation of anyone and without oppressing anyone. We are fully responsible for it.