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Accepted Manuscript Title: Optical Properties of ZnO Semiconductor Nanolayers Authors: Igor Khmelinskii, Vladimir I. Makarov PII: DOI: Reference:

S0025-5408(18)31835-X https://doi.org/10.1016/j.materresbull.2018.09.030 MRB 10197

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MRB

Received date: Revised date: Accepted date:

12-6-2018 29-8-2018 17-9-2018

Please cite this article as: Khmelinskii I, Makarov VI, Optical Properties of ZnO Semiconductor Nanolayers, Materials Research Bulletin (2018), https://doi.org/10.1016/j.materresbull.2018.09.030 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Optical Properties of ZnO Semiconductor Nanolayers Igor Khmelinskii (1) and Vladimir I. Makarov (2)

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University of the Algarve, FCT, DQF and CEOT, 8005-139, Faro, Portugal University of Puerto Rico, Rio Piedras Campus, PO Box 23343, San Juan, PR 00931-3343, USA

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Corresponding Author: Dr. Vladimir Makarov Department of Physics, UPR, Rio Piedras Campus, PO Box 23343 San Juan PR 00931 USA

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GRAPHICAL ABSTRACT 0.7 (a)

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OPtical Density

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Emission Intensity (a.u.)

(b) 30

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Energy (10 , cm )

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Graphical abstract: (a) absorption and (b) emission spectra of ZnO nanolayers with different thickness

Highlights 1. We report a deposition technology of high-quality uniform ZnO thin films;

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2. We report discrete bands in absorption and emission spectra of ZnO films deposited on CaF2, interpreted in terms of one-dimensional quantum confinement;

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3. We report very long exciton lifetimes in ZnO films, as compared to other ZnO nanostructures.

Abstract

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Presently we explore absorption and emission spectra of ZnO semiconductor nanolayers 4.1 –

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17.3 nm thick. We report that their absorption spectra have discrete structure, with the transition band density increasing with the nanolayer thickness. The emission spectra

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recorded at 4.1 and 9.3 nm thickness have resolved band structure, with the bands partially

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overlapping in the 9.3 nm sample. On the other hand, the emission spectra are strongly overlapped in the 13.1 and 17.3 nm samples. We used our modeling approach that considers

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electronic states in a one-dimensional infinite potential well, calculating the relative electron mass of 0.205, and the starting quantum number for the absorption transitions of 7, 8, 9 and 9, for the respective samples. We also discuss the present results using the traditional approach

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of solid-state physics, considering potential surfaces in the linear momentum space.

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Keywords: ZnO; absorption spectrum; emission spectrum; quantum confinement; thin film;

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exciton

I.

Introduction

Presently physical properties of nanostructured systems attract a lot of attention, due to their unique physical and chemical properties [1-3]. Nanostructured materials have promising applications in optoelectronics [4-9], nanophotonics [10-14], and optical communications 2

[15-19]. Quantum confinement (QC) occurs when at least one of the system dimensions becomes smaller than the de Broglie wavelength of the electrons [20-22]. QC may thus occur in tree dimensions (3D), two dimensions (2D) or one dimension (1D) only. The 1D QC is created in nanolayers, where both longitudinal (plasmon/polaron) and transverse (exciton) oscillations may propagate. Quantum confinement results in significant changes in the electronic structure of nanostructured materials, very interesting for both fundamental studies and engineering applications, including development of novel optical devices. Earlier we

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reported the analysis of spectral properties (generation of exitons) of metal and

semiconductor nanolayers, determined by 1D QC [23-28]. We also reported superemission

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(SE) in metal nanolayers [9-12] that may be used to build lasers. Other authors reported SE

within the QC – generated level system in homogeneous nanolayers and nanotubes [29-31]. One-dimensional QC in conductive nanolayers always operates along the direction normal to the layer surface. We found [23-25] that the absorption spectra of conductive nanolayers have

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a discrete structure, with the energies of the absorption band maxima very well described by

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the simple model of electron in a rectangular potential box with infinite walls [32]. The



1 2mn  m2 2 fa



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 n, n  m  3029.2

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absorption band maxima are given by the expression [32]

(1)

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where f = meff/me is the relative electron mass, meff is the effective electron mass, which is an empirical parameter depending on the material and less than me, the free electron mass, a is the nanolayer thickness expressed in nm, n is quantum number of the starting electronic state, cm-1.

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m is the quantum number difference in the transition, and the transition energy is expressed in

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The QC observed in metals is understood as quantization of the continuous electron energy spectrum upon a transition from bulk material to a nanostructure. However, the same effect in semiconductor materials requires additional experimental and theoretical studies, as it was

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explored in a limited number of semiconducting nanostructured systems. These include freestanding nanolayers fabricated of InAs [29], and deposited nanolayers of Si and SnO2 [16,17]. In particular, optical absorption properties of free-standing InAs nanomembranes 3 to 19 nm thick were investigated by Fourier transform infrared spectroscopy [29], with the room-temperature absorption growing stepwise towards higher energies. These spectra were 3

interpreted as arising from the interband transitions between the subbands of the twodimensional InAs nanomembranes. The absorbance change associated with each step was ca. 1.6%, independent on the membrane thickness. These experimental results are consistent with the theoretically predicted absorbance quantum: AQ = πα/nc for each set of the interband transitions in a 2D semiconductor, where α is the fine structure

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constant and nc is an optical local field correction factor [29]. The absorbance quantization

appears to be universal in 2D systems including quantum wells and grapheme, with the

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experimental results and their theoretical interpretation reported in [29]. Comparing the

experimental and modeling results obtained for deposited Si and SnO2 [24,25] and freestanding InAs [21], we formulate several research tasks: (i) compare the absorption/emission spectra of free-standing and deposited semiconductor thin films; (ii) develop theoretical

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approaches to the analysis of the absorption/emission spectra; (iii) explore superemission in

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such films, and (iv) further applications of the studied material using new properties coupled with QC phenomenon. The present paper uses ZnO as a model system in such a study, aiming

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at a deeper understanding of the electronic structure and properties of semiconductors. ZnO is a very interesting semiconductor material, with numerous studies devoted to its

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optical properties [33-41]. Structural, thermal, morphological and optical properties of ZnO nanoparticles were investigated by X-ray diffraction (XRD), differential scanning calorimetry (DSC), scanning electron microscopy (SEM), energy dispersive spectroscopy (EDS), Ultraviolet (UV) –Visible (Vis) spectroscopy and Raman spectroscopy [33]. Thin films of

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intrinsic and Al-doped ZnO were prepared by the sol-gel technique with spin coating onto glass substrates [34]. All of those films exhibited a transmittance above 80–90% along the

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visible range up to 650 nm and a sharp absorption onset about 375 nm corresponding to the fundamental absorption edge at 3.3 eV. Intense UV photoluminescence was observed for undoped and 1% Al-doped ZnO films [34]. Changes in structural, morphological, and optical

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properties of sol-gel derived ZnO and ZnO:Fe nanoparticles occurred due to dense electronic excitations produced by heavy ion irradiations using 200 MeV Ag+ ion beams [35,36]. UVVis measurements found that the band gap increased on Fe doping [35], attributed to 4s–3d and 2p–3d interactions and the Burstein-Moss band filling effect. Photoluminescence (PL) intensity was enhanced and two new emission bands at ca. 480 nm (due to surface defects) 4

and at 525 nm (due to O vacancies) were observed in ion-irradiated nanoparticles [35]. A review of current research on the optical properties of ZnO nanostructures was compiled earlier [37]. The optical band gaps of the ZnO thin films deposited on Si substrate were determined using UV spectra, varying from 3.24 to 3.29 eV in function of the oxygen ratio [38]. Other authors explored graphene-ZnO sandwich assemblies and other nanostructured devices containing ZnO [39-41]. However, there are no published absorption/emission spectra of ZnO thin films demonstrating QC, although such data are important for

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understanding whether QC is inherent to all semiconductors including ZnO, or it is dependent on the electronic structure of specific materials.

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Presently we investigate this latter issue, and research an adequate theoretical description for the recorded spectra. In particular, we report absorption and emission spectra of ZnO semiconductor nanolayers 4.1, 9.3, 13.1 and 17.3 nm thick, and discuss theoretical

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approaches that may be used to analyze these spectra. The absorption spectra of these thin films have discrete structure, with the band density increasing with the nanolayer thickness.

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The thinner films have a resolved discrete structure, although the emission bands in the 9.3

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nm film are already partially overlapped. On the other hand, the emission bands in the thicker

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samples are overlapped quite strongly. We analyzed the absorption spectra in the frameworks using our earlier modeling approach [23-25], calculating the relative electron mass of 0.2051

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in ZnO films, independent on the film thickness, while the n value increased with the film thickness, equal respectively to 7, 8, 9 and 9, for the four samples. We also interpret the same results using the classic solid-state physics ideas [29]. Presently we also report an extension to our theoretical approach, with a detailed treatment of the interaction of the nanolayer with

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the electromagnetic field, producing excitons, leaving any additional issues related to QC in

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semiconductors for future studies.

II.

Experimental

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Fluorite (CaF2) substrates sized 2512.5 mm2 and 1 mm thick (Esco Optics) were used to deposit the samples. We used commercial ZnO (Sigma/Aldrich) to deposit nanocrystalline films using a commercial sputtering/thermo-evaporation Benchtop Turbo deposition system (Denton Vacuum). The ZnO nanolayers were deposited using thermo-evaporation, with the substrate at 375ºC in all experiments. The films were annealed for 2 hours at 900oC and 5

atmospheric pressure of pure nitrogen gas. Film thickness was controlled by XRD [42], on an XPert MRD system (PANalytic), calibrated using standard nanofilms of the same material. The estimated absolute uncertainty of the film thickness was 0.2%; the relative uncertainties were much smaller, determined by the shutter opening times of the deposition system. The method for measuring the thin film thickness is based on recording the scattering/interference pattern of X-rays. The X-ray photons reflected from each of the atomic layers of the thin film

the number of atomic/molecular layers in the film and thus its thickness.

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contribute to the total reflectivity. The observed reflections may be therefore used to calculate

Absorption and emission spectra were recorded on a Hitachi U-3900H UV-Visible

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Spectrophotometer and Edinburgh Instruments FS5 Spectrofluorometer, respectively. The absorption spectra in the middle- and near-IR were recorded on a PF2000 FTIR spectrometer (Perkin Elmer). Absorption spectra are presented as the difference of the transmission and the

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reflection spectra. The spectral peak maxima were precisely located using PeakFit software (Sigmaplot). The polynomials were fitted and the fitting uncertainties estimated using the

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LINEST function in Excel (Microsoft). Photo-induced response measurements were

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performed using a high-pressure Xe lamp (1000W, Ariel Corporation, Model 66023), a

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monochromator (Thermo Jarrell Ash, Mono Spec/50), a DET10A Biased Si detector (THORLABS, supplied with the spectral calibration curve), a model 2182A nanovoltmeter

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(Keithley Instruments) connected to a computer by a GPIB interface, and home-made software in the LabView programming environment (National Instruments). The light of the Xe lamp was filtered by interference filters with the pass-band centered at 223 nm, and used to excite emission of the ZnO layers. Emission spectra recorded in IR and UV/VS spectral

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regions were sewed and presented as common one. We recorded time-resolved emission using the fundamental harmonics of a dye laser for

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excitation: LPD-2000 (-Physics) with Coumarin-4 dye (with frequency multiplication on a BBO crystal; -Physics). The Coumarin-4 dye fundamental could be tuned in the 370-580

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nm spectral range (frequency-doubled to obtain 220-290 nm radiation), and pumped by the fourth harmonics of a Nd-YAG laser (266 nm, Surelight-II, Continuum Inc.). The laser pulse duration was 7-10 ns. The dye laser radiation was defocused onto the entire surface of the respective sample. The emission was collected by a 30-cm spherical CaF2 lens and detected by a photodiode (PD1; DET10A Biased Si Detector from THORLABS) or a photomultiplier (PMT-H9305-03, Hamamatsu), after passing through a neutral density filter. The data 6

acquisition system contained a PC computer, a digital oscilloscope (WaveSurfer 400 series, LeCroy), two digital delay generators (DG-535, Stanford Research), a photo-detector (PD: DET10A Si Detector from THORLABS) or a photomultiplier (PMT-H9305-03, Hamamatsu), two boxcar integrators (SR-250, Stanford Research), a fast amplifier (SR-240, Stanford Research), and a computer interface board (SR-245 Stanford Research). For the spectroscopy experiments, the LIF signal was monitored by the digital oscilloscope and averaged, typically using 5 laser pulses per frequency step. The output energy was controlled

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and the monochromator scan operated using a PD and PCI-6034E DAQ I/O board (National

Instruments), with the control code in the LABVIEW environment running on a second Dell

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PC. The presently used experimental methods produced 2.5 ns time resolution.

All of the measurements were performed in an optical cryostat (Optistat DN-V2, Oxford

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Instruments) at 77K, unless expressly indicated otherwise.

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III. Results

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a) Absorption and emission spectra of ZnO nanolayers Absorption and emission spectra were recorded for the ZnO layers 4.1, 9.3, 13.1, and 17.3 nm

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thick, with the results shown in Fig. 1.

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The absorption spectra of Fig. 1 were analyzed using commercial PeakFit software package,

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with the respective results listed in Tab. 1.



Table 1. Absorption band maxima and widths. 7

13.1 nm film Maximum, Width, cm-1 cm-1 1635 873 3442 823 5422 817 12393 874 15061 913 17901 874 20913 892 24097 911 27454 874 30982 816 34683 853 38556 875 42601 907 46818 892 51207 896

17.3 nm film Maximum, Width, cm-1 cm-1 937 911 1973 894 3108 879 4342 923 5675 911 7106 932 13817 879 15742 823 17765 879 19887 877 22107 845 24427 859 26845 893 29362 872 31977 813 34691 876 37504 811 40416 894 43426 003 46535 891 49742 859

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9.3 nm film Maximum, Width, cm-1 cm-1 2902 934 6147 897 13661 943 17930 952 22540 921 27493 917 32786 911 38421 931 44398 931 50716 927

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4.1 nm film Maximum, Width, cm-1 cm-1 13179 1484 28115 1477 44808 1481

Fig. 2 shows the band maxima in function of the quantum number increment of the respective

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resulting absorption band.

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We fitted the plots of Fig. 2 using Eq. (1), with the fitting parameters n and f listed in Tab. 2.

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Table 2. Values of n (the starting quantum number for the transitions) and f (the relative electron mass) fitting parameters for the band maxima of Table 1 in function of the quantum number increment. Eq. (1) was used to fit the data. Fitting parameters 8

4.1

9.3

Sample thickness, nm 13.1

17.3

n f

7 0.2050

8 0.2053

9 0.2049

9 0.2052

The emission spectra of the two thinner films were analyzed in the entire recorded range, while the spectrum of the 13.1 nm was only simulated below 28500 cm-1. Note that resonance emission at the excitation wavelength was cut off using an appropriate optical filter, which passed longer wavelengths only. We did not analyze the emission of the 17.3 nm sample, as

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the respective spectrum lacked discernible structure. Tab. 3 lists the emission bands that were identified.

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Table 3. Emission bands of the three thinner ZnO films; the emission spectrum of the thickest

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Bandwidth, cm-1 1821 1793 1893 1826 1783 1812 1769 1791 1804

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Bandwidth, cm-1 3175 3241

9.3 nm Maximum, cm-1 41496 38251 34665 30737 26468 21858 16905 11612 5977

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4.1 nm Maximum, cm-1 31629 16693

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film lacked a resolved structure.

13.1 nm Maximum, cm-1 27540 24700 21688 18504 15147 11619 7918 4045

Bandwidth, cm-1 1721 1678 1709 1721 1734 1689 1703 1817

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expression

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Considering the data of Tab. 3, we conclude that the band maxima may be described by the





1 2mn  m 2 , 2 fa

(2)

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 em   exc  n, n  m   exc  3029.2

where exс is the photon energy corresponding to the maximum of the absorption band used for excitation. The following absorption bands were used for excitation: (1) n = 7  n = 10 (44808 cm-1); (2) n = 8  n = 18 (44398 cm-1); (3) n = 9  n = 25 (46818 cm-1); and (4) n = 9  n = 31 (43426 cm-1). Using Eq. (2) and the spectral band data, we fitted the emission 9

band maxima in function of the quantum number decrement for the two intermediate samples. The data points and the respective fitting curves are shown in Fig. 3.



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Note that the fitting parameters n (8 and 9, respectively) and f = 0.20480.0003 agree quite well with the same parameters obtained for the absorption spectra and listed in Tab. 2.

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Below we shall discuss the bandwidth formation mechanism in thin films. b) Time-resolved experiments

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Time-resolved experiments were carried out for all samples of interest using the same excitation wavelengths provided by a pulsed laser. The data recording was started with 20 ns

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delay after the maximum of the excitation laser pulse, to reduce scattered laser light and fast

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resonance emission. The emission waveforms are shown in Fig. 4a.

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The recorded waveforms were fitted with good accuracy by an exponential function t

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

I em t   Aeme

 em

(3)

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where A0 is the signal amplitude and em is the emission lifetime, with the values of the latter

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listed in Tab. 4.

Table 4. Measured emission lifetimes in ZnO thin films.

Sample

4.1 nm

9.3 nm

13.1 nm

17.3 nm

em, ns

38.22.1

43.62.3

51.13.3

31.52.1

10

las, ns

7.30.2

7.50.4

7.10.3

7.70.4

prop,1, ns

12.20.9

11.10.8

13.10.8

9.70.7

prop,2, ns

37.22.2

41.72.1

48,92.7

29.82.2

These emission lifetimes are longer in comparison with the data reported earlier [43-46] in other nanostructured systems, which fell into the 1 – 20 ns time range. Time-resolved

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experiments on nanostructured ZnO systems also reported much shorter lifetimes in the 1 –

10 ps time range [47-52]. However, we found no reported emission lifetimes in ZnO nanofilms. Note also that earlier reported emission lifetimes in Co metal nanofilms are even

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longer than the currently reported lifetimes in ZnO [47]. We shall discuss these results in more detail below.

We also studied the dynamics of exciton propagation along the ZnO film. In these

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experiments, the film was excited at 22 mm2 area in its center, while the emission was

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collected from a rectangle at the edge of the film, sized 12.52 mm2. To take into account the

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eventual propagation of the excitation light along the CaF2 substrate, we used an clean CaF2

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substrate for reference, subtracting reference signal from the sample signal. The recorded emission signal dynamics from the films is shown in Fig. 4b. We fitted the plots by a tri-

I prop,em t   A1e

 t t  0   las

   

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exponential function: 2

t    t  prop, 2 prop,1   A2 e e  

   

(4)

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With the values of the fitting parameters las, prop,1 and prop,2 are also listed in Tab. 4. We

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conclude that prop,2  em, las value coincides with the laser pulse duration, while prop,1 may be interpreted as the exciton wave package propagation along the 10.5 mm of ZnO film, corresponding to the propagation velocity of ca. 108 cm/s, about 300 times slower than the

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speed of light in vacuum. Presently we only report this experimental result, without its analysis, which we shall provide in a follow-up publication.

IV. Discussion

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We shall discuss quantization in the absorption and emission spectra and the mechanisms of optical transitions, the mechanisms of spectral bandwidth formation, and the mechanism describing the emission lifetimes in ZnO nanofilms, starting from the first issue. i. Quantization in the absorption and emission spectra and the mechanism of optical transitions The analysis we presented for the absorption and emission spectra is based on a simple

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physical model, with quantum confinement (QC) operating in the direction perpendicular to the film surface, with the system states quantized in this direction. We described the ZnO

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film as a rectangular potential box with infinite walls. This remains an acceptable approximation while the energy of excited states considered remains significantly below the

ionization energy of the material. In practice, Eqs. (1-2) describe the respective absorption and emission band positions with the errors below 1 cm-1 for the states that are several eV

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below the ionization energy, with only one fitted parameter (the effective electron mass). We

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analyzed the results reported earlier by Fang, et al. (2013) [29] using the same model to simulate their reported absorption spectra, for example, the spectrum of the 19 nm thick InAs

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thin film (see Fig. 5a).

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We took the band position maxima in the 19 nm InAs film from the earlier reported data [7].

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The spectrum was simulated as a superposition of the discrete component defined by QC and of the continuous spectrum defined by the dielectric constant of the semiconductor material.

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The continuum contribution is proportional to

h exc [53], where exc is the frequency of the

electromagnetic field interacting with the semiconductor. Our simulated spectrum reproduces

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the earlier reported spectrum [7] with good accuracy, if we disregard the details of the spectral structure attributed to different types of holes. Fig. 5b shows the band maxima in function of the quantum number increment, that may be fitted to a second-order polynomial, Eq. (1), with the fitting parameters of n = 1 and f = 0.125. In other words, a simple QC model may be used to successfully describe the experimental spectra. This model, however, imposes certain selection rules that define the allowed transitions. Indeed, only transitions with an odd 12

quantum number increment are allowed in the electric dipole approximation, represented as n + 2m-1  n, for integer m values. We may obtain this result analyzing the interaction of our system with the electromagnetic field. The respective matrix elements may be written as [53]:

Vij  

     e e 2  i p  Ak '  j    i p  sk 'eik 'r  j mc m k

(5)

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where Ak’ is the k’-th component of the vector potential

   A   Ak 'eik 'r ,

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k'

p is the electron linear momentum, sk’ is the unit vector directed along the vector potential component Ak’, k’ is the wave vector of the respective component, r is the radius-vector of the electromagnetic wave distribution in space, k’ is the frequency of the k’-th component of

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the vector potential. The wavefunctions in the k-space are given by:

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 2kn2 En  ; n  i or j  integer values. 2m

1  2 Sin  n  kl  2  n  kl

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

1 l n  n  p   n  x e ikx dx    n  kl 2 

and integration is performed over the ki-space. Here, the kn 

(6)

pn values are defined by the 

zeroes of the function (5). Taking into account the perturbations due to the finite amplitude of

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the potential walls (the amplitude of 12 eV seems to produce the best model for the experimental spectra, see Appendix) and using the definition of the optical transition

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efficiency: 2

Pij  Vij ,

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(7)

we calculated the relative band intensities for the 9.3 nm ZnO thin film using the assignment presented above, with the results shown in Fig. 6.

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The results of Fig. 6 qualitatively reproduce the intensity distribution in the experimental absorption spectrum of Fig. 1a(2). The results obtained for the emission spectrum using the band assignment based on Eq. (2) are also shown in Fig. 6b, being qualitatively similar to the experimental emission spectrum of Fig. 1b(2). ii. Emission lifetimes We already noted that the presently measured emission lifetimes in ZnO thin films (31 – 51

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ns) are anomalously long compared to earlier reports on other ZnO nanostructures, all in the picosend time range [32-37]. This discrepancy may be explained by significant differences

described in the simple case by a biexponential function [39]: t

f

 As e



t

s

(8)

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I em t   Af e



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between the respective systems. The emission dynamics of nanostructures is typically

where Af and As are the amplitudes of the fast- and slow-decaying components of the

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emission signal, and f and s are the respective lifetimes. Presently we assume that the

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relationships Af >> As and f << s are satisfied, proposing two different mechanisms for the

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emission: (1) the short component describes fast dephasing of the initially prepared wave package, while the long component describes independent emission dynamics of the

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quasistationary states forming the initial wave package. In an ideal case, with no radiationless relaxation operating, we obtain Af f = Ass.

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(2) the initially prepared excited state is transformed into an exciton (a bound electron-hole pair), which recombines on a longer time scale, emitting a photon. We presume that the first

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mechanism is not operating, as the excited state quickly transforms into an exciton. The second mechanism may be presented as follows:



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1. NL + hexc  NL*; Wexc t   W0 t  1  e D 2. NL*  NL + hem; 1/em 3. NL*  (h+…e-); k1 4. (h+…e-)  NL + hrec; krec,em

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

5. (h+…e-)  NL; krec,nr where NL is the ZnO nanolayer, NL* is the excited state in the ZnO nanolayer, (h+…e-) is the exciton, W0(t) is a Gaussian function describing the laser pulse shape: W0 t   W0e

 t t0    0

  

2

,

(9)

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with its parameters defined in section II, and t0 is an instant of time corresponding to the peak of the laser pulse, D is the optical density, determined by the absorption spectra of Fig. 1a,

em is the excited state lifetime, k1 is the rate constant of exciton formation, krec,em is the

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radiative exciton recombination rate constant defining the recombination emission rate of the slow component, krec,nr is the nonradiative exciton recombination rate constant. This scheme produces

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1/f = 1/em + k1 and  s1  krec, em  krec, nr  .

The time resolution of our measurement system is about 10 ns, therefore we are can’t observe

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the fast decay component and the rising part of the slow emission component. In order to

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1/em << k1 and krec, nr  krec,em  k1 ,

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explain the slow emission component that we observe, we have to assume that

or the excited states generate excitons with unitary yield, and the radiationless recombination is negligible. In the following subsection, we shall analyze the excited state transformation

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into the exciton, and the exciton recombination, using the ab initio approach. iii. Ab initio analysis of the electronic structure of ZnO monoatomic layer

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The monoatomic-layer ZnO electronic structure was investigated using DFT using the plane wave pseudopotential method implemented in the generalized gradient approximation (GGA)

A

[49] as parameterized by Perdew, Burke and Ernzerhof [40] and as implemented in the SIESTA package [41,42]. The calculations were performed using Heyd-Scuseria-Ernzerhof hybrid density functional, which mixes the exact non-local exchange of Hartree-Fork potential with the exchange and correlation potential from GGA [39,43-45]. The effects of mixing ratio between GGA and HF exchanges were controlled by the calibration of the structural parameters using the database on the compounds of interest. The norm-conserving 15

pseudopotentials including the Zn 3d state in its valence shell and the wavefunctions with the energy cutoff at 80 Ry were chosen. In the current study, we analyzed hexagonal structure of 2D ZnO films [46-48]. We began with calculating the state structures and energies of 2D ZnO in the bonding and conductive zones. The calculated state energy diagrams of the ground state – excited state and ground state – exciton systems are shown in Fig. 7.

IP T



SC R

Here,  is the Brillouin zone center. The density function method can’t analyze the excited

states in ZnO directly, while neither MPT nor coupled-cluster methods could be used due to insufficient computational resources. Therefore, we calculated the state structure for the originally prepared system configuration using the density function method (see Fig. 7a),

U

while the state configuration for the exciton (electron-hole pair) was calculated as average

N

between those of positively and negatively charged ZnO systems (see Fig. 7b). The

A

wavefunctions in the latter case were represented as combinations of the positively and negatively charged systems. We tested this approximate approach against the results obtained

M

by MP2 for H2 molecules, with the resulting discrepancy of ca. 10%. The energy gap at the  point is 3.87 eV in the ground-state – excited state system, and only 2.02 eV in the ground

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sate – exciton system. Thus, the stabilization energy of the exciton with respect to the excited state is 1.85 eV. The energy gap for the excited state system is in an acceptable agreement with the earlier reported data [46-49].

We next used the results of ab initio calculations as input data to calculate the rate constant of

EP

the NL*  (h+…e–) process, using homemade FORTRAN software. We used the Golden Fermi rule to calculate the rate constant [50]: 

CC

2 2 2 2 VExc ,h...e    EExc  Eh...e  dEh...e   VExc ,h...e  h...e  EExc    0

(10)

A

k1  k f 

where VExc, h...e   exc Vna   h...e

16

(11)

Here

 h...e  EExc 

is the exciton state density at the energy level of the excited state created

upon absorption of light, Vˆna is the nonadiabatic perturbation coupling the excited state and the exciton states [51-57], exc and (h…e) are the wavefunctions of the excited state and the exciton. We calculated the state density according to:

 2 1 d k  ei  Ei  Eexc  4 3 i 

(12)

IP T

h...e  EExc  

Where ei is the eigenvector (normalized to unit length) corresponding to the energy

SC R

eigenvalue Ei. We calculated k1 = 3.71011 s-1, corresponding to the exciton formation time of 2.7 ps, in acceptable agreement with the emission lifetimes in nanostructured ZnO [32-37]. We next calculated the radiative recombination rate constant of the exciton, once more using

U

the golden Fermi, see Eq.(5): 2 2 Vrad  rad Eh...e         h...e  p  sk 'eik 'r  G

A

(13)

M

Vrad

N

krec,em 

where G is the ground state wavefunction. Here we obtained the rate constant of 43 s-1,

TE D

corresponding to the natural emission lifetime of 23 ns. Using the same approach as presented in Eqs.(9-11), we calculated the nonradiative recombination rate of the exciton. The resulting value is 79 s-1; combining this values with

EP

the ratiative rate constant, we obtain ks = 122 s-1. Therefore, we have the slow component lifetime of 8.2 ns, and its emission yield of 0.35. Note that the calculated lifetime of the slow

CC

emission is shorter than the measured value. This is attributable to the simple model we used, which only has a monoatomic layer of Zn and O atoms, while the real film that has dozens of

A

such layers.

Thus, taking into account the results presented in this subsection, we conclude that the measured emission lifetimes are quite reasonable, and the observed emission is attributable to exciton recombination.

V.Conclusion 17

We performed detailed measurements of optical properties of ZnO nanofilms. We found that their absorption spectra (for all films, 4.1 to 17.3 nm thick) and emission spectra (for 4.1, 9.3 and 13.1 nm nanofilms the spectra are at least partially resolved) have discrete structure, with the energies of the respective states proportional with very high precision to the quantum number squared n2. We interpreted these spectra in terms of quantum confinement in the direction perpendicular to the flat film. We also report the emission lifetimes in the nanosecond range, much longer than the values reported earlier for bulk ZnO and ZnO

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nanostructures. To explain these anomalously long emission lifetimes, we carried out a detailed theoretical analysis, attributing such long lifetimes to the exciton recombination.

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However, additional studies are need to fully understand the underlying physics, preferably with much higher time resolution, that would allow to monitor the faster initial processes,

U

including the exciton formation.

N

Acknowledgements: the authors are grateful for the financial support of the Institute of Functional Nanomaterials of PR and PR NASA EPSCoR grant (NASA Cooperative

M

A

Agreement NNX15AK43A) for V.M.

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21

Appendix We consider the electron in a rectangular potential box schematically shown in Fig. A.1.

E

IP T

U2 U1

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a

U

Figure A.1. Rectangular potential box with finite walls.

N

Three regions are should be taken into account for E < U1, U2.

x0

 1  c1e k x 1

1 2mU 1  E  h

(A.1)

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k1  

M

A

The first region is

0 xa

EP

The second region is

CC

 2  c2 Sin  k 2 x   

A

k2  

i 2mE h

And the third

22

(A.2)

xa

 3  c3 e k x 3

k3  

(A.3)

1 2mU 3  E  h

Taking into account boundary conditions

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d d ln  1 x  x  0  ln  2 x  x  0 dx dx d d ln  2 x  x  a  ln  3 x  x  a dx dx

SC R

(A.4)

 '1 0  '2 0   1 0  2 0

(A.5)

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 ' 2 a   '3 a    2 a   3 a 

M

The second condition in Eq.(A.4) produces

A

N

2m 1 2 k2 Ctg    k1  U1  k 2  2mU1  E  2 h h

U

The first condition in Eq. (A.4) produces

2m 1 2 k 2 Ctg a k 2     k3  U 2  k2   2mU 2  E  2 h h

(A.6)

1 2mU1  E  h

CC

kCtg  

EP

Introducing k2  k , we obtain

(A.7)

A

or

Sin   

kh 2mU1

 kh     Arc sin  2 mU 1   23

(A.8)

and

kCtg ak     

1 2mU 2  E  h

(A.9)

or

 kh ka     Arc sin  2mU 2 

(A.10)

   

     Arc sin kh   n   2mU  1   

(A.11)

U

 kh ka   Arc sin  2mU 2 

IP T

kh 2mU 2

SC R

Sin ka     

N

where the n term describes the periodic properties of the Sin function. Therefore finally we

    Arc sin kh   2mU 2  

M

 kh ka  n  Arc sin  2mU 1 

A

obtain the transcendent equation:

(A.12)

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 E   E    Arc sin  2mE a  n  Arc sin   U  U 1 2    

   

EP

Assuming U1 = U2 = U0, we obtain

(A.13)

CC

 E  2mE a  n   Arc sin  2  U0 

A

The last equation may be rewritten as follows:

2mU 0

E a  n U0 2

 E    Arc sin   U0 

Defining the parameter 24

(A.14)

z

E U0

(A.15)

We rewrite Eq. (A.14) as follows:

2

 Arc sinz 

(A.16)

IP T

z  n

where   2mU0 a

SC R

(A.18)

or

(A.19)

N

U

z  n  Sin  z  2 

This transcendent equation may be solved numerically or by graphical analysis; however, for

M

(A.20)

EP

TE D

1 z  n  2 Sin    z   z  4  2  1 2    z  1 4 1  2ma2 EU 0  2 2 2ma E  Vˆpert   U0

A

z << 1, Eq. (A.19) may be simplified:

CC

The latter relationship was used as a perturbation mixing different states represented by

A

Eq.(5); this perturbation was transformed into the k-space as follows:

Vˆpert  p  

1 1 2ma Sin ka Vˆperteikx dx    k 2   2 U 0 

The perturbed function is represented as follows:

25

(A.21)

 n ' p   n  p  

 n  p Vˆ  p  m  p 

(A.22)

En  Em

Finally, the relative transition probabilities were calculated using the relationship

  e  'i p  Ak '  ' j . mc

(A.23)

A

CC

EP

TE D

M

A

N

U

SC R

IP T

V 'ij  

26

EP

TE D

M

A

N

U

SC R

IP T

Figure captions

Figure 1. Absorption (a) and emission (b) spectra of ZnO layers: (1) 4.1 nm thick (excitation

CC

at 223 nm); (2) 9.3 nm thick (excitation at 225 nm); (3) 13.1 nm thick (excitation at 235 nm);

A

and (4) 17.3 nm thick (excitation at 230 nm).

27

IP T SC R U N A M TE D

Figure 2. Band maxima in function of the quantum number increment for the four ZnO film

A

CC

EP

samples: (1) 4.1 nm thick, (2) 9.3 nm thick, (3) 13.1 nm thick, and (4) 17.3 nm thick.

28

IP T SC R U N A M TE D

Figure 3. The emission band maxima in function of the quantum number decrement for the

A

CC

EP

ZnO thin films: (2) 9.3 nm film, and (3) 13.1 nm film.

29

IP T SC R U N

A

Figure 4. Emission kinetics for the ZnO thin films (1) 4.1 nm think (excited at 223 nm); (2) 9.3 nm thick (excited at 225 nm); (3) 13.1 nm thick (excited at 235 nm); and (4) 17.3 nm

M

thick (excited at 230 nm) (a) and propagated emission (1) 4.1 nm think (excited at 223 nm); (2) 9.3 nm thick (excited at 225 nm); (3) 13.1 nm thick (excited at 235 nm); and (4) 17.3 nm

A

CC

EP

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thick (excited at 230 nm) (b).

30

IP T SC R

U

Figure 5. (a) The simulated absorption spectrum and (b) dependence of the band position

A

CC

EP

TE D

M

A

N

maxima on the quantum number increment in the 19 nm thick InAs thin film [7].

31

Figure 6. The relative transition probabilities in the 9.3 nm ZnO thin film: (a) absorption and

CC

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M

A

N

U

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(b) emission.

Figure 7. Calculated state diagrams of three bonded and two unbounded states: (a) ground

A

state –excited state system; (b) ground state – exciton system for a hexagonal ZnO film.

32