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Preheating-temperature effect on structural and photoluminescent properties of sol-gel derived ZnO thin films

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Preheating-temperature effect on structural and photoluminescent

properties of sol

–gel derived ZnO thin films

C.H. Chia

a,n

, W.C. Tsai

a

, W.C. Chou

b

a

Department of Applied Physics, National University of Kaohsiung, Kaohsiung 81148, Taiwan bDepartment of Electrophysics, National Chiao Tung University, Hsinchu 30010, Taiwan

a r t i c l e i n f o

Article history:

Received 30 August 2013 Received in revised form 30 November 2013 Accepted 5 December 2013 Available online 14 December 2013 Keywords: ZnO Sol–gel Photoluminescence Exciton

a b s t r a c t

We investigated the evolution of structural and low-temperature photoluminescence properties of sol– gel derived ZnO thinfilms, as a function of heating temperature between 300 1C and 600 1C. The pre-heating crystallization plays an important role in the lineshape of PL spectra. The increase in pre-pre-heating temperature enables rearrangement of atoms at the pre-heating stage, leading to c-axis oriented growth offilms, modification in lineshape of impurity-defect emission, and reduction of deep-level emission intensity.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

ZnO thin films show a versatile combination of interesting optical, electrical and magnetic properties[1]. Consequently, this material has been attracting great attention due to its potential applications in ultra-violet optoelectronics devices[2,3], solar cells

[4], thinfilm transistors[5], gas sensors[6], and surface acoustic wave devices[7]and so on. In particular, wide band gap (3.37 eV) at room-temperature (RT) and large exciton binding energy (60 meV) [1] of ZnO enable the utilization of exciton-related transitions in ultra-violet light-emitting devices even at RT. It is therefore a promising semiconductor for photonics applications such as polariton lasers[8]and microcavity-based devices[9].

Sol–gel technique is much cheaper and easier to grow large area of ZnOfilms than much sophisticated techniques. The sol–gel derived ZnO thinfilms have already been realized in transparent p–n junction diodes[10]and transistors[4]. The ZnO-films synth-esis by sol–gel route includes three major steps: (i) solution preparation, (ii) coating and (iii) heat treatment. These three steps involve several parameters that influence the physical properties of the films. One of them is pre-heating temperature (Tph). The Tphis an important factor affecting the solvent

vapor-ization, decomposition of precursor-material, and crystallization process. Ohyama et al.[11]studied crystalline orientations of ZnO thinfilms, prepared by acetate solution, with Tphbetween 2001C

and 5001C. The authors found that Tph¼300 1C is the optimum

condition for preferred growth offilm with (0 0 2)-orientation. Kim et al.[12]also revealed that 2751C is the optimum Tphfor preferred

growth of film with (0 0 2)-orientation. Santos et al.[13], however, challenged low-Tphgrowth of ZnO thinfilm. The authors claimed that

highly c-axis oriented film has been obtained at Tph¼120 1C.

Although abundant literatures have been reported[11–16], the Tph

effect on low-temperature photoluminescence (PL) characteristics of sol–gel derived ZnO thin film is rarely studied. Moreover, to control the crystallization process at pre-heating stage, we adopted relatively high Tph, compared with the other studies[12–16]. For a large use of

sol–gel derived ZnO thin film for optoelectronic devices, a better understanding of the PL mechanisms in sol–gel ZnO thin film is necessary. Some literatures have reported the PL of sol–gel derived ZnO thinfilms[17–20]. However, neither of them discusses the Tph

-effect on the luminescent properties despite of this factor playing an important role in the crystallization of ZnO.

In this letter, we studied the evolution of PL from ZnO thinfilm grown by sol–gel spin-coating technique, as a function of Tph. The

focus of the present work is investigation of structural and PL proper-ties of the ZnO thinfilms. The emphasis has been given on the PL spectra of ZnO thinfilm, measured at low-temperature (T). We found that the low-T PL band of the ZnO thinfilms is characterized by the radiative recombination of donor-bound excitons and impurity-defect-related transitions. The latter transition is greatly influenced by the Tph.

2. Experimental details

The ZnO thin films were grown from aqueous solution pre-pared using zinc nitrate hexahydrate (Zn(NO3)2 6H2O) as the

Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/jlumin

Journal of Luminescence

0022-2313/$ - see front matter& 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jlumin.2013.12.006

nCorresponding author. Tel.:þ886 7 5919713; fax: þ886 7 5919357. E-mail address:[email protected] (C.H. Chia).

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starting material, isopropanol as the solvent, and monoethanola-mine (MEA) as the stabilizer. The molar ratio of zinc nitrate to MEA was kept at 2:1 and the molar concentration of Zn2þis 0.3 mol/L. The precursor solution was mixed thoroughly with a magnetic stirrer at 501C for an hour until the formation of a homogenous and transparent sol, which was aged for 7 days at RT before the coating of thin films. The ZnO thin films were formed by spin-coating the sol solution on the sapphire substrate at 2000 rpm (revolutions per minute) for 15 s. After the spin-coating process, the samples were subjected to pre-heating treatment. Four samples treated at pre-heating temperature Tph¼300, 400, 500,

6001C for 10 min were grown. The procedure was repeated 10 times in order to obtain desired thickness. The samples were finally annealed in air atmosphere in a furnace at T¼600 1C for 2 h. Structural properties of the films were examined by X-ray diffraction method (XRD) using a monochromatized X-ray beam CuKαwith wavelength of 0.154 nm. Field-emission scanning

elec-tron microscopy (SEM) was employed to investigate the morpho-logical properties of the samples. The PL spectra were measured by a 32-cm-long monochromator and a charge-coupled device cam-era. The pump source for the low-excitation spectra was the 310 nm-line of Xenon lamp. The excitation power for the PL measurement was kept at 2 mW/cm2. A closed cycle refrigerator

was used to perform the T-dependent measurement.

3. Results and discussion

Fig. 1shows the XRD patterns of the ZnO thinfilms. The pattern suggests all of thefilms have hexagonal wurtzite structure. Three diffraction signals with peaks located at 31.71, 34.41 and 36.21 are attributable to (1 0 0), (0 0 2) and (1 0 1) diffraction, respectively

[1]. The ZnO film of Tph¼300 1C is polycrystalline. As the Tph

increases from 3001C to 400 1C, the (1 0 1) diffraction intensity increases abruptly. However, the film tends to orient with c-axis perpendicular to the film surface as Tph reaches 5001C.

The (0 0 2) diffraction dominates the XRD spectra for the sample of Tph¼600 1C, indicating that the film with preferred c-axis

orienta-tion was obtained at high Tph. In order to describe the preferred

c-axis orientation, the relative intensities of (0 0 2) diffraction peak

I(2 0 0)/[I(1 0 0)þI(2 0 0)þI(1 0 1)] as a function of Tph are plotted in

the inset ofFig. 1. The relative (0 0 2)-diffraction intensity of the sample with Tph¼400 1C is smaller than that of Tph¼300 1C.

However, the relative intensity of (0 0 2)-diffraction increases as Tphexceeds 4001C and reaches close to unity as Tph¼600 1C. Since

the boiling points of isopropanol and MEA are 821C and 170 1C respectively, Tph of 3001C is adequate to vaporize the organic

residuals. However, the thermal decomposition of zinc nitrate was observed to be rapid at T above 3501C, which is the thermal decomposition-temperature of bulk zinc nitrate [21]. The film prepared at Tph¼400 1C, undergoes vaporization of organic

solvents and thermal decomposition of zinc nitrate abruptly and simultaneously, thus, preferentially c-axis oriented crystallization is obscured [12], leading to small relative (0 0 2)-diffraction intensity. Nevertheless, further increase of Tph enables atoms

acquiring enough thermal energy to rearrange, resulting in c-axis oriented growth.

Fig. 2shows the SEM image of ZnOfilms deposited on sapphire substrate. The microstructure of ZnO thin film transforms from particulate-like to porous plate-like as Tph increases. Thefilm of

Tph¼300 1C is homogeneous and slightly porous with crystalline

size ranging from 50 nm to 150 nm. The surface morphology changes abruptly as Tphreaches 4001C. A lot of large pores start

to form in the film prepared at Tph¼400 1C and the degree of

porosity persists up to Tph¼600 1C. As a result of simultaneous

chemical reactions, i.e., the quick evaporation of organic components

and thermal decomposition of zinc nitrate, the pores are left behind in the films during the crystallization process. The structural and morphological changes of the precursorfilm prepared at Tph4400 1C,

compared with that of Tph¼300 1C, also correlates to the PL profiles

demonstrated below.

Fig. 3reveals the normalized low-T PL spectra of the ZnO thin films prepared at various Tph. Here, the low-T refers to the lowest

temperature obtainable in our cryostat-system, i.e., 15 K. The radiative recombination of donor-bound exciton D1X (3.362 eV) dominates the low-T PL spectra. The chemical identity of the donor is probably the hydrogen atoms[22], which is an abundant element in our precursor-solvent. The full widths at half maximum (FWHM) of the D1X bands in all of the samples are about 7 meV, indicating good optical quality of the ZnOfilms. In the film of Tph¼300 1C, we identified an emission band located near 3.31 eV.

This PL band can decompose into three radiative recombinations originated from structural defects (SD, 3.332 eV), free-to-bound transitions (eA1, 3.317 eV) and donor-acceptor pair (DAP, 3.302 eV), according to literatures[23]. The SD-band is of excitonic nature and similar to the so-called Y-line commonly seen in ZnSe and ZnTe [22]. The eA1-band is the radiative recombination of an electron from the conduction band with a hole bound to an acceptor-like defect state, caused by stacking fault [23,24]. The DAP-band is attributable to the transitions between donor and acceptor levels in the forbidden band-gap. The large amount of

XRD Intensity (arb. units)

40 39 38 37 36 35 34 33 32 31 30

2

Θ

(Degree)

T

ph

= 600

°C

T

ph

= 500

°C

T

ph

= 300

°C

T

ph

= 400

°C

(100)

(002)

(101)

1.0 0.8 0.6 0.4 0.2 0.0

I

002

/ (

I

100

+

I

002

+

I

101

)

600 500 400 300

T

ph

(°C)

Fig. 1. XRD patterns of the ZnO thinfilms, as a function of Tph. Three diffraction peaks corresponding to (1 0 0), (0 0 2) and (1 0 1) plane were observed. The inset shows the relative intensities of (0 0 2) diffraction peak I(2 0 0)/[I(1 0 0)þI(2 0 0)þ I(1 0 1)] as a function of Tph.

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donors and acceptors in our samples is not surprising because of low-purity of chemical precursor materials. Small humps located at low-energy side of these emission lines are the one and two

LO-phonon sideband of eA1 and DAP transitions because the spectral separations between them are 72 meV and 144 meV, respectively[1]. Compared to thefilm of Tph¼300 1C, the intensity

of eA1-band (IeA) is unchanged and the intensity of DAP-band

(IDAP) decreases in the film of Tph¼400 1C. In addition, the

intensity of SD-band ISD becomes less pronounced. The decrease

in IDAPis simply because more impurity-atoms escape out of the

film treated at high-Tph. The SD-transitions originate from excitons

bound to the defects induced by irregularities in the crystal

[23,24]. As the irregularities disappear owing to the reconstruction of atomic arrangement under high-Tph, reduction of ISDis

expect-able. The microscopic nature of defect-states causing the eA 1-transitions is rather complicated and remains unknown. However, we noted that the total impurity-defect-related emission intensity IeAþIDAPþISDreduces as the Tphincreases from 3001C to 500 1C. It

has to remind that our samples were post-annealed at 6001C. Thus, once the structural defects formed at the low-Tph, they

cannot be annihilated by the next post-annealing treatment. For thefilm with Tph¼500 1C, the SD-transitions disappear because of

atomic reconstruction at the pre-heating process. The DAP and eA1-band merges into a single emission band with spectral peak about 3.308 eV. It is no doubt that the IeAis smaller in the sample

of Tph¼500 1C. Therefore, we can infer that the stacking faults,

leading to the occurrence of eA1-transition, could also be reduced by reconstruction of atomic arrangement under high-Tph. For the

film with Tph¼600 1C, the emission profile is similar to that of

Tph¼500 1C, except that the spectral peak of impurity-defect

emission slightly blueshifts. Owing to the low-purity of precursor materials, several kinds of impurity may exist in our sample.

1

μ

m

1

μ

m

1

μ

m

1

μ

m

Fig. 2. FESEM images of the ZnO thinfilms with Tphof (a) 3001C (b) 400 1C (c) 500 1C and (d) 600 1C. The porosity becomes severe as Tph4300 1C.

Normalized PL Intensity

3.40

3.30

3.20

3.10

3.00

Photon Energy (eV)

ZnO thin films

T = 15 K

T

ph

= 300

o

C

T

ph

= 400

o

C

T

ph

= 500

o

C

T

ph

= 600

o

C

D

o

X

SD

eA

o

DAP

DAP-LO+ eAo-LO DAP-2LO+ eAo-2LO

DAP + eA

o

Fig. 3. Low-T PL spectra of the ZnO thinfilms, as a function of Tph. The donor-bound exciton (D1X) dominates the PL spectra for all of the samples. The DAP, eA1 and SD refer to donor-acceptor-pair, electron-to-bound state, and structural defect, respectively.

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Purging out of particular donors or acceptors from the film prepared at different Tph, leads to energetic shifts of donor or

acceptor level. This leads to changes in spectral peak of DAP-emission. As will be verified later, this single impurity-defect emission is dominated by DAP-transitions. Therefore, the change in DAP-emission peak is the major reason for the spectral difference of impurity-defect emission between the samples of Tph¼500 1C and 600 1C.

To get further insight about the impurity-defect-related emis-sions from thefilms, we performed T-dependent PL measurement.

Fig. 4shows the evolutions of PL from two typical ZnO thinfilms (Tph¼300 1C and 500 1C), as a function of T. Because of the thermal

dissociation of donor-bound excitons, free excitonic emission (FX) located at 3.377 eV are observed at high T for both samples. For the film with Tph¼300 1C, we note that the PL-intensity of DAP-band

and SD-band quench rapidly and are hardly detectable above T440 K, consistent with the previous finding[23]. Furthermore, the eA1-band persists up to higher T and its band-profile starts to deviate from symmetric lineshape. The thermal distribution of electrons involved in the transitions at high T results in high-energy asymmetric tails [23], as can be seen in the spectrum measured at 80 K. These observations further confirms the assign-ment of DAP-, eA1- and SD-transitions. In the film of Tph¼300 1C,

we also detected spectral blueshift of the small hump (around 3.23 eV) as T increases. This is because the LO-phonon-sideband of

DAP-emission diminishes as T increases. At T¼15 K, a merged impurity-defect emission from thefilm of Tph¼500 1C is observed,

as shown inFig. 4(b). Similar to thefilm of Tph¼300 1C, the

DAP-emission disappears above 40 K and the eA1-emission remains. Therefore, we can conclude that the origin of the impurity-defect emission from thefilm of Tph¼500 1C is nothing else but DAP- and

eA1-transitions.

Fig. 5 illustrates the low-T PL spectra of the ZnO thinfilms, recorded at broad spectral range. The near-band-edge (NBE) emission is mainly due to the D1X-transitions mentioned above. The deep level (DL) emissions are attributable to defect-related transitions. The red emission centering around 1.8 eV could be attributed to the transition from oxygen vacancy level to the top of the valence of ZnO[25,26]. Moreover, the transitions between the zinc interstitial and oxygen interstitial may contribute the small hump centering about 2 eV at the DL emission band [26]. The intensity of DL emission increases abruptly as Tphincreases from

3001C to 400 1C. In accordance with the results of XRD and SEM, the vigorous and simultaneous chemical processes at 4001C induce metastable imperfections in the thinfilm, which cannot be annihilated by post-annealing treatment. This leads to larger DL emission intensity. These crystalline imperfections reduce as Tph

increases to 5001C, owing to reconstruction of atomic arrange-ment at the pre-heating stage. Therefore, the DL emission intensity is almost negligible in the thinfilm with Tph¼500 1C. However, the

DL emission intensity enhances again in thefilm with Tph¼600 1C

because the number of oxygen vacancy increases with increasing Tph[27].

4. Conclusion

In summary, we investigated the structural and low-T PL properties of sol–gel derived ZnO thin films, prepared from zinc nitrate solution. The emphasis has been given on the effects of pre-heating temperature on the PL spectra. Especially, the pre-pre-heating crystallization plays an important role in the lineshape of PL spectra. The donor-bound excitonic emission and relatively weak impurity-defect emission characterizes the low-T NBE-PL spectra. Deep level emissions due to oxygen vacancies, oxygen interstitials, and zinc interstitials are detected at orange-to-red spectral region. As pre-heating temperature reaches 4001C, vaporization

Normalized PL Intensity 3.40 3.38 3.36 3.34 3.32 3.30 3.28 3.26 3.24 3.22 3.20 3.18 3.16

Photon Energy (eV)

SD

eA

o

DAP

eA

o

-LO

DAP-LO

D

o

X

FX

15 K

20 K

30 K

40 K

50 K

60 K

80 K

Normalized PL Intensity 3.40 3.38 3.36 3.34 3.32 3.30 3.28 3.26 3.24 3.22 3.20 3.18 3.16

Photon Energy (eV)

D

o

X

FX

DAP

eA

o

15 K

20 K

30 K

40 K

50 K

60 K

80 K

Fig. 4. T-dependent PL spectra of the ZnO thin films with Tph of (a) 3001C (b) 5001C. The FX refers to free excitonic transition.

Normalized PL Intensity

4.0 3.6 3.2 2.8 2.4 2.0

Photon Energy (eV)

T

ph

= 300

o

C

T

ph

= 400

o

C

T

ph

= 500

o

C

T

ph

= 600

o

C

ZnO thin films

T = 15 K

DL

NBE

Fig. 5. Low-T PL spectra of the ZnO thinfilms measured at broad spectral range. The near-band-edge (NBE) PL is due to D1X-transitions.

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of organic residuals and thermal decomposition of zinc nitrate happens abruptly and simultaneously, leading to random oriented growth offilms, larger degree of porosity, change in lineshape of impurity-defect emission, and enhancement of deep-level emis-sion intensity. Further increase in pre-heating temperature enables rearrangement of atoms at the pre-heating stage, leading to c-axis oriented growth of films, further modification in line-shape of impurity-defect emission, and reduction of deep-level emission intensity.

Acknowledgement

This research was supported by National Science Council of Taiwan under Grant No. NSC-99-2112-M-390-001-MY3.

References

[1]Ü. Özgür, Ya.I. Alivov, C. Liu, A. Teke, M.A. Reshchikov, S. Doğan, A. Avrutin, S.-J. Cho, H. Morkoč, J. Appl. Phys. 98 (2005) 041301.

[2]Z.K. Tang, G.K.L. Wong, P. Yu, M. Kawasaki, A. Ohtomo, H. Koinuma, Y. Segawa, Appl. Phys. Lett. 72 (1998) 3270.

[3]X. Ma, P. Chen, D. Li, Y. Zhang, D. Yang, Appl. Phys. Lett. 91 (2007) 251109. [4]W.J. Jeong, S.K. Kim, G.C. Park, Thin Solid Films506–507 (2006) 180. [5]D. Gupta, M. Anand, S.W. Ryu, Y.K. Choi, S. Yoo, Appl. Phys. Lett. 93 (2008)

224106.

[6]S.T. Shishiyanu, T.S. Shishiyanu, O.I. Lupan, Sens. Actuators, B 107 (2005) 379. [7]S. Krishnamoorthy, A.A. Iliadis, Solid State Electron. 52 (2008) 1710. [8]M. Zamfirescu, A. Kavokin, B. Gil, G. Malpuech, M. Kaliteevski, Phys. Rev. B:

Condens. Matter 65 (2002) 161205.

[9]R. Shimada, J. Xie, V. Avrutin, Ü. Özgür, H. Morkoč, Appl. Phys. Lett. 92 (2008) 011127.

[10]X. Chen, K. Ruan, G. Wu, D. Bao, Appl. Phys. Lett. 93 (2008) 112112. [11] M. Ohyama, H. Kozuka, T. Yoko, Thin Solid Films 306 (1997) 78. [12]Y.S. Kim, W.P. Tai, S.J. Shu, Thin Solid Films 491 (2005) 153. [13]A.M.P. Santos, Edval J.P. Santos, Thin Solid Films 516 (2008) 6210. [14]L. Znaidi, Mater. Sci. Eng., B 174 (2010) 18.

[15]T. Ehara, T. Otsuki, J. Abe, T. Ueno, M. Ito, T. Hirayama, Phys. Status Solidi A 206 (2009) 2139.

[16]R. Brenier, L. Ortéga, J. Sol-Gel Sci. Technol. 29 (2004) 137.

[17] J. Petersen, C. Brimont, M. Gallart, O. Crégut, G. Schmerber, P. Gilliot, B. Hönerlage, C. Ulhaq-Bouillet, J.L. Rehspringer, C. Leuvrey, S. Colis, H. Aubriet, C. Becker, D. Ruch, A. Slaoui, A. Dinia, J. Appl. Phys. 104 (2008) 113539.

[18]P. Sagar, P.K. Shishodia, R.M. Mehra, H. Okada, A. Wakahara, A. Yoshida, J. Lumin. 126 (2007) 800.

[19]S. Mandal, M.L.N. Goswami, K. Das, A. Dhar, S.K. Ray, Thin Solid Films 516 (2008) 8702.

[20] N. Kumar, R. Kaur, R.M. Mehra, J. Lumin. 126 (2007) 784.

[21]first ed.,J.C. Bailar, A.F. Trotman-Dickenson, H.J. Emeleus, S.R. Nyholm (Eds.), Pergamon, Oxford, 1973.

[22] B.K. Meyer, H. Alves, D.M. Hoffmann, W. Kriegseis, D. Forster, F. Bertram, J. Christen, A. Hoffmann, M. Straßburg, M. Dworzak, U. Haboeck, A.V. Rodina, Phys. Status Solidi B 241 (2004) 231.

[23] M. Schirra, R. Schneider, A. Reiser, G.M. Prinz, M. Feneberg, J. Biskupek, U. Kaiser, C.E. Krill, K. Thonke, R. Sauer, Phys. Rev. B: Condens. Matter 77 (2008) 125215.

[24] D. Tainoff, B. Masenelli, P. Mélinon, A. Belsky, G. Ledoux, D. Amans, C. Dujardin, N. Ferodov, P. Martin, Phys. Rev. B: Condens. Matter 81 (2010) 115304. [25] C.H. Ahn, Y.Y. Kim, D.C. Kim, S.K. Mohanta, H.K. Cho, J. Appl. Phys. 105 (2009)

013502.

[26] N.H. Alvi, K. Ul Hasan, O. Nur, M. Willander, Nanoscale Res. Lett. 6 (2011) 130. [27] H.S. Kang, J.S. Kang, J.W. Kim, S.Y. Lee, J. Appl. Phys. 95 (2004) 1246.

數據

Fig. 1 shows the XRD patterns of the ZnO thin films. The pattern suggests all of the films have hexagonal wurtzite structure
Fig. 2. FESEM images of the ZnO thin films with T ph of (a) 300 1C (b) 400 1C (c) 500 1C and (d) 600 1C
Fig. 4 shows the evolutions of PL from two typical ZnO thin films (T ph ¼300 1C and 500 1C), as a function of T

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