Journal of Non-Crystalline Solids 373-374 (2013) 28–33

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Journal of Non-Crystalline Solids

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Vibrational density of states of TiO2 nanoparticles

Kulbir Kaur Ghuman ⁎, Navdeep Goyal, Satya PrakashDepartment of Physics, Panjab University, Chandigarh 160014, India

⁎ Corresponding author. Tel.: +91 1 647 993 4473.E-mail address: kulbirghuman@gmail.com (K.K. Ghu

0022-3093/$ – see front matter © 2013 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.jnoncrysol.2013.04.022

a b s t r a c t

a r t i c l e i n f o

Article history:Received 21 February 2013Received in revised form 16 April 2013Available online xxxx

Keywords:Vibrational density of states;Amorphous;TiO2

The lattice dynamics (LD) and molecular dynamics (MD) simulations are carried out to investigate the pho-non dispersion relations of rutile supercell and vibrational density of states (VDOS) of 3 nm size rutile andamorphous TiO2 nanoparticles using the Matsui and Akaogi (MA) rigid ion model. The calculated phonon dis-persion relations and optical mode frequencies at symmetry points agree evenly with the experimental dataexcept in the low frequency range. The VDOS of rutile and amorphous nanoparticles get extended in the 0–2 THz and in the 24–32 THz frequency range as compared to bulk VDOS due to structural changes in the coreand surface regions of rutile and amorphous nanoparticles. The Ti atom's contribution to VDOS is in the rangeof acoustic and low frequency optical modes while the O atom's contribution is significant in the entire fre-quency range due to mass and number differences. The VDOS of amorphous phase nanoparticles can be wellrepresented by the step function to study the thermodynamical properties of nano phase TiO2 and may be amodel for future experiments.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

The technologically important properties of titanium dioxide(TiO2) are intimately related to its dynamical properties. The excep-tionally high c-axis dielectric constant and its anomalous increasewith decrease of temperature and its frequency dependence is relatedto transverse optical (TO) mode A2u where the Ti and O ions aredisplaced along the c-axis. The frequency of the A2u mode decreaseswith a decrease in temperature but never becomes completely softeven at 0 K which leads to incipient ferroelectric transition [1]. Thedynamical properties of crystalline rutile are extensively investigatedby Raman and infrared spectroscopy [2] and coherent inelastic neu-tron scattering [3]. The rigid ion and shell models are used to explainthe Raman spectra and measured phonon dispersion relations andpolar modes are identified [4–6]. The splitting between infrared ac-tive longitudinal and transverse optical modes, phonon dispersion re-lations, Raman and infrared mode frequencies and partial vibrationaldensity of states (VDOS) of rutile super cell are also studied usingdensity functional theory (DFT) [7–9].

Nano size TiO2 is of particular interest for future technology as ithas high surface area, chemical stability and semiconducting proper-ties. However a systematic study of vibrational properties of nano sizeTiO2 does not exist in the literature except the one carried out byOgata et al. [10]. These authors used modified variable chargeinteratomic potential to study the structural and elastic propertiesof nano size TiO2 cluster of 1050 atoms and the VDOS of a cluster of672 atoms at 100 K using molecular dynamics (MD) simulations.

man).

rights reserved.

Periodic boundary conditions were used and hence the shape factorwas not accounted for. Mitev et al. [11] used the Swami and Gale var-iable charge potential [12] and Matsui–Akaogi (MA) force field [13] toinvestigate the atomic structure, atomic vibrations and atomic chargefluctuations of two dimensional slabs of thickness 3–35 Å using MDand lattice dynamics (LD) simulations at 300 K. The non-electronicproperties were better reproduced by MA force field.

Earlier we investigated the structural properties of 3 nm sizeamorphous TiO2 nanoparticles generated by different heating andquenching rates using MA force field and MD simulations [14,15].As discussed in Ref. [14], sample S1 was prepared by heating the op-timized rutile sample up to 4000 K at the rate of 2.0 × 1014 K/s for20 ps and then quenched at the rate of 1.84 × 1014 K/s for 20 ps asit reached at 315 K. Similarly sample S2 was prepared by heatingand quenching in two steps with different heating and quenchingrates. However sample S3 was the spherical core of 3 nm diameterof a spherical crystal of 6 nm diameter melted at 4000 K and thencooled up to 0 K. The 3 nm spherical core was again heated up to400 K and quenched to 315 K. The samples S1 and S2 were of samedensity with 454 units of TiO2 while sample S3 had 426 units ofTiO2 and its density was reduced by about 7%. The agreement be-tween the calculated and measured reduced structure factors sug-gested that atomic arrangements in the laboratory samples may besimilar to those predicted by MD simulations.

As the dielectric and optical properties of nanosize TiO2 dependupon its phonon structure, we found it essential to investigate the vi-brational spectra of nanosize rutile and differently prepared amor-phous nanoparticles of TiO2 by LD and MD simulations respectivelyusing MA force field. The relative shifting and broadening of opticaland acoustic modes and the structural changes in the VDOS of the

02468

10 12 14 16 18 20 22 24 26

A Γ M X Γ Z R

Fre

quen

cy (

TH

z)

q

Fig. 1. Phonon dispersion relations for primitive super cell of rutile calculated by LDsimulations. The blue dots represent the experimental data for bulk rutile [3]. (For in-terpretation of the references to color in this figure legend, the reader is referred to theweb version of this article.)

29K.K. Ghuman et al. / Journal of Non-Crystalline Solids 373-374 (2013) 28–33

nanoparticles are studied. The calculated Raman and infrared modesand shifting and splitting of acoustic and optical modes agree evenlywith the existing experimental data. The extensions in partial andtotal VDOS of rutile and amorphous nanoparticles are found due tostructural changes in the core and surface regions of nanoparticles.The present study may be used to fabricate nanoparticles of desiredvibrational properties.

The paper is structured as follows: The computational details aregiven in Section 2. The results are discussed in Section 3 and conclu-sions are drawn in Section 4.

2. Computational details

2.1. Lattice dynamics

The rutile nanoparticle is obtained by cutting a sphere of 3 nm di-ameter from the center of a large tetragonal crystal structure, gener-ated by repeated unit cell of TiO2 in three dimensions with latticeparameters a = b = 4.594 Å, c = 2.959 Å and u = 0.305 Å [12].The excess atoms are removed from the surface to obtain charge bal-ance and charge neutrality.

The LD calculations are carried out for the primitive super cell ofTiO2 with the help of the GULP [16,17] program. The structure is re-laxed by minimizing the energy with respect to geometrical variablesat constant volume. The LD simulations are carried out at 0 K withbox size of 5 cm−1. The mass weighted force constant matrices aregenerated and these are diagonalized to obtain the eigen values fordifferent values of wave vectors q

→along principal symmetry direc-

tions in the first Brillouin zone.

2.2. Molecular dynamics

MD simulations are carried out using classical MD packageDLPOLY which involves Verlet integration algorithm [18] with atime step of 0.0005 ps. The rutile nanoparticle is optimized by ‘zero’temperature MD simulation in NVE ensemble, then it is heated upto 315 K for 10 ps and finally equilibrated for 50 ps. Periodic bound-ary conditions are not used. The atomic positions of Ti atoms remainunchanged while O atoms get squeezed or expanded by less than 1%of interatomic distance in the optimized rutile primitive super celland the elastic constants are well reproduced. The three amorphousTiO2 nanoparticles of 3 nm size were obtained by different heatingand quenching rates as discussed in Section 1 [14]. The velocitiesare collected at the intervals of 0.01 ps for a period of 50 ps at zeropressure and 315 K. The velocity auto correlation function (VACF)g(t) is generated as

g tð Þ ¼ ∑ivi tð Þvi 0ð Þ� �

∑ivi 0ð Þvi 0ð Þ� � ; ð1Þ

where vi(t) is the velocity of the ith particle at time t, b > denotes theaverage over ensemble and sum is over all the oscillating atoms. TheVDOS g(ω) is obtained by calculating the time Fourier transform ofg(t) given as

g ωð Þ ¼ 12∫g tð Þeiωtdt: ð2Þ

If the velocities of atoms are projected onto a plane wave, one mayget the information about the individual frequencies. For the vibra-tional mode with wave vector q and polarization vector p, the projec-tion of velocities of ith atom at the averaged position ri0 on to a planewave is given as:

vpq tð Þ ¼ ∑Ni¼1 p

→⋅ vι→

tð Þe−iq→⋅r0j : ð3Þ

The corresponding peak positions in the VDOS are obtained bycomputing the function:

gpq ωð Þ ¼ ∫dtb vpq tð Þ��v 0ð Þ >

b vpq 0ð Þ��vpq 0ð Þ >eiωt

: ð4Þ

The function gqp(ω) will have the sharp peaks at the phonon posi-tions with wave vector q and the polarization vector p if the system iscrystalline.

For a non-crystalline system, the vibrational modes are no longerplane waves. However, if the system is nearly crystalline, vibrationalmodes can be described by phonon like wave packets. These wavepackets will appear in gqp(ω) as broadened peaks whose maximamay be used to determine the main frequencies of vibrationalmodes. As the system continues deviating from the crystalline struc-ture, the peaks may continue broadening and even disappear if thesystem becomes amorphous.

The finite size TiO2 nanoparticles, discussed here, are non-crystalline.The atomic network is strained and over and under coordinated struc-tures are in the core and surface regions, respectively [15]. Therefore,there will be broadening and shifting of phonon peaks in the VDOSof nanoparticles, as compared to the crystalline structure. Theseaspects will be discussed in Section 3.2.

3. Results and discussion

3.1. Phonon dispersion relations

The phonon dispersion relations are obtained by diagonalizing thedynamical matrix for 200 equidistant q vectors along A → Γ, 132 qvectors along Γ → M and 65 q vectors along M → X, X → Γ, Γ → Zand Z → R directions in the Brillouin zone. The lines are drawnthrough the calculated points and no interpolation is carried out.These results along with the experimental data for the bulk [5] areshown in Fig. 1. The calculated phonon frequencies for acousticmodes are higher than the experimental values near the zone bound-ary. However, the slopes of the calculated acoustic modes agree withthe experimental results as the elastic constant data are used in theevaluation of MA potential parameters. The calculated phonon bandwidth and dispersive character of optical phonon modes in the higherfrequency range agrees with the experimental data. However in thelower frequency range the calculated frequencies are higher thanthe experimental data.

The above deviations are because MA potential is axially symmet-ric and does not include the tensorial forces due to quasi localized d

Table 2Comparison of calculated (Calc.) phonon frequencies for primitive supercell of rutileand experimental (Exp.) data for bulk rutile (in THz) at symmetry points X, M, Z, R

30 K.K. Ghuman et al. / Journal of Non-Crystalline Solids 373-374 (2013) 28–33

electrons of Ti atoms. The zone boundary degeneracy criterion is alsonot satisfied, although the general features of phonon dispersion arewell reproduced. The DFT based calculations due to Sikora [8] includethese electronic effects. However Han et al. [9] have to correct thelongitudinal optical mode frequencies in the long wave length limitto include these effects. Our results are similar to those obtained byHan et al. [9] except near the zone boundary.

The calculated optical mode frequencies at Γ point are comparedwith the experimental data [5,19,20] and other calculations [6,8] forthe bulk rutile in Table 1. All the optical modes at q = 0 are non de-generate except the LO infrared mode which is doubly degenerate.A2u and Eu polar modes get split into longitudinal optical (LO) andtransverse optical (TO) modes respectively with different frequenciesdue to macroscopic electric field associated with LO and TO modes asdiscussed by Dayal et al. [4]. Here Eg is non-polar infrared inactivemode and B2g, B1g and A1g are Raman active modes. The A2g isnon-polar infrared inactive and B1u is Raman inactive mode. In theLO modes with A2u and Eu symmetries, the atoms move parallel andperpendicular to the c-axis respectively and these modes are infraredactive. The A2u (LO) modes are active for light polarized parallel to thecrystallographic c-axis and Eu (TO) modes are active for light polar-ized perpendicular to the c-axis.

The experimental data for phonon frequencies at Γ point are dis-persed from 3.39 THz to 24.78 THz, while the calculated results arewithin the range of 5.22 to 24.37 THz. The calculated results forhigh energy Raman and infrared modes agree well with neutron scat-tering and optical data. However, in the intermediate energy rangethe deviations between the calculated and experimental results arewithin 15% and in the lower energy range the calculated results arehigher than the experimental values. These deviations are due tothe reason that the electronic contribution is not well accounted forin the MA potential. The results due to Katiyar et al. [6] and Sikora[8] are also tabulated in Table 1. Katiyar et al. [6] results arenon-degenerate, Sikora [8] got the B2g and A2u degenerate modes at24.02 THz while we obtained this degeneracy in E2

u(LO) and B1u(2)

modes.The optical and acoustic mode phonon frequencies at symmetry

points X, M, Z, R and A along with available experimental data forthe bulk are tabulated in Table 2. The results at M and X symmetrypoints are tabulated by matching the degeneracy of calculated andexperimental data. The results at R and A symmetry points are ar-ranged using compatibility relations with Z and Γ-points respectively.The calculated acoustic mode frequencies at X1(LA), X1(TA) and Z1

Table 1Comparison of the calculated frequencies (in THz) of primitive supercell at Γ point withexperimental (Exp.) data for bulk [3,19,20] and other calculations (Calc.) [6,8]. Here Iand R stand for infrared and Raman modes respectively.

Kosternotation

Mullikennotation

Presentcalculation

Exp. Other Calc.

(Optical)[19,20]

(Neutron)[3]

[6] [8]

Optical modes

Γ4+ B2g (R) 24.37 24.78 24.72 24.78 24.02Γ5+ Eu3(LO) (I) 24.29 24.18 25.24 23.04 23.63Γ1− A2u(LO) (I) 18.90 24.33 – 24.24 24.02Γ2+ A2g 18.25 – – 17.19 12.38Γ1+ A1g (R) 17.94 18.36 18.30 18.36 18.46Γ5+ Eu3(TO) (I) 17.16 15.00 14.81 15.87 14.95Γ5− Eg (R) 17.20 13.41 13.34 13.41 14.16Γ5+ E2u(LO) (I) 13.57 13.74 12.85 14.04 13.04Γ4− B1u

(2) 13.57 – 12.18 14.7 12.52Γ5+ Eu2(TO) (I) 13.42 11.64 – 10.50 11.79Γ5+ Eu1(LO) (I) 12.73 11.19 11.23 10.47 10.57Γ5+ Eu1(TO) (I) 11.34 5.49 5.66 5.37 4.31Γ1− A2u(TO) (I) 10.79 5.01 5.18 11.88 5.74Γ3+ B1g (R) 7.45 4.29 4.25 4.29 3.96Γ4− B1u

(1) 5.22 – 3.39 6.99 3.53

(LA) are in agreement with the experimental data while the calculat-ed results for X2(TA), M (LA, TA) and Z2(TA) are higher than the ex-perimental values. There is no data for comparison at R and Apoints. The calculated optical mode frequencies in high and low ener-gy ranges agree with the experimental data except at degeneratepoints X1 and M5,6 while in the intermediate energy range, calculatedresults are higher than the experimental data by about 15%. Theremay also be some uncertainties in the measurements. The reasonsfor deviations are the same as discussed above. However, the phononband width in the calculated and experimental results is nearly thesame, therefore the calculated results may be used to explain thethermal properties of TiO2.

3.2. Vibrational density of states

The calculated VDOS using LD simulations for rutile super cell iscompared with the neutron scattering data in Fig. 2. The acousticmode phonon band between 2 and 4 THz is more pronounced inthe neutron scattering data as compared to super cell results althoughthe peak positions for longitudinal and transverse acoustic modes andlow frequency Raman modes are well represented in the super cellresults and agree with the experimental findings [3]. In the frequencyrange 5–8.5 THz, the super cell results show the larger peak heightsfor acoustic and optical modes as compared to the experimentaldata. In the 8.5–10 THz region, the VDOS for the super cell has de-creased while the peak positions remain nearly unaltered.

The VDOS beyond 10 THz is mainly due to optical modes. In the10–11 THz frequency range, the calculated results for super cell andneutron data are similar although the split peaks at 10.35 and10.5 THz of super cell results are shifted towards low energy as com-pared to experimental results. In the vicinity of 12 THz, the neutronresults are higher than the super cell results although the peak posi-tions for optical modes in both the VDOS are the same. The opticalphonon peak at 13.8 THz of super cell is larger than the neutrondata and beyond 14.7 THz the VDOS of super cell is consistentlyhigher than the neutron data as the high energy optical modes arenot included in VDOS from neutron data. In the calculated resultsthe peaks are well split and shifted towards higher frequencies.

and A of Brillouin zone. LA and TA denote longitudinal and transverse acousticmodes respectively.

XCalc.

XExp. [3]

MCalc.

MExp. [3]

ZCalc.

ZExp. [3]

RCalc.

ACalc.

3.91 X1: 25.38 22.49 M1,2:23.26 21.60 – 21.49 22.7423.91 X1: 24.38 22.49 M1,2:23.26 21.60 – 21.49 22.7419.64 – 20.95 – 17.24 – 18.12 19.2119.64 – 20.95 – 19.09 – 20.16 19.2116.65 – 16.74 – 19.09 – 20.16 16.7716.65 – 16.74 – 17.24 – 18.12 17.2316.04 X2: 13.53 16.51 M9

+: 13.52 11.49 Z2: 12.31 11.05 17.2313.58 X1: 11.76 13.67 M9

+: 9.44 17.18 – 15.24 16.7710.68 X2: 9.03 10.25 M5,6: 9.40 17.18 – 15.24 12.4216.04 – 16.51 – 11.49 – 11.05 12.7113.58 X1: 9.13 9.23 M9

+: 7.81 10.63 – 10.37 9.797.99 X1: 8.07 13.67 M9

+: 9.44 10.63 Z4:9.72 10.37 12.7110.68 X2: 9.03 10.25 M5,6: 9.40 8.00 – 8.32 9.797.99 X1: 8.07 9.23 M9

+: 7.81 8.00 Z3: 9.04 8.32 8.375.15 X2: 3.08 5.31 M5,6: 3.06 13.36 Z1: 12.39 14.19 8.37

Acoustic Modes6.24 X1: 5.82

(LA)3.39 M9

+: 2.94(LA)

7.79 – 6.46(TA)

5.02(TA)

6.24 X1: 5.82(TA)

3.39 M9+: 2.94

(TA)7.79 Z2: 3.69

(TA)6.46(TA)

5.02(TA)

5.15 X2: 3.08(TA)

5.31 M5,6: 3.06(TA)

13.36 Z1: 12.39(LA)

14.19(LA)

12.42(LA)

0

0.01

0.02

0.03

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

VD

OS

(1/

TH

z)

Ferquency (THz)

Expt.Calc. : rutile supercell

Calc. : rutile nanoparticle

Fig. 2. The calculated (Calc.) VDOS of rutile supercell, 3 nm size rutile nanoparticle byLD simulations and the results from experimental data (Expt.) [3]. The calculated VDOSare smoothened with the cubic-splines. The description is given in the legend.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

VD

OS

(1/

TH

z)

Ferquency (THz)

MDLD

Fig. 3. The calculated VDOS of 3 nm size rutile nanoparticle by LD and MD simulations.The LD and MD data are smoothed using acsplines with a uniform smoothing weight of103 and 106 respectively. The description is given in the legend.

31K.K. Ghuman et al. / Journal of Non-Crystalline Solids 373-374 (2013) 28–33

It is interesting to compare the VDOS of rutile nanoparticle of 3 nmsize consisting of 1362 atomswith those of super cell and neutron data.These results are shown by dashed line in Fig. 2. There is a pile up of ad-ditional modes between 0 and 2 THz. The analysis reveals that thesemodes are due to atoms in the vicinity of the surface. Singh et al. [21]and Nakhmonson et al. [22] identified these modes as localized modesin amorphous Si. The VDOS in the low frequency region of the nanopar-ticle is nearly linear as compared to bulk and super cell. The peaks forlongitudinal and transverse acoustic modes in the vicinity of 0–8 THzget shifted towards low energy as discussed in Section 2.2 and alsofound by Meyer et al. [23] for silicon nanoparticles.

In the 8–14 THz range, the TO and LO phonon peak heights getsignificantly reduced due to finite size of the nanoparticle althoughthe peak positions remain nearly the same as that of super cell be-cause the short range order persists in the rutile nanoparticle asgiven in Section 2.2. In the 14.4–21 THz range the VDOS of rutilenanoparticle is higher than the experimental results for bulk TiO2

and lower than that of the super cell and the peak positions of TOmodes get shifted towards higher energy. The Raman modes at17.20 THz and 17.94 THz also get significantly broadened due to fi-nite size disorder. The LO peak heights in the vicinity of 20 and22.5 THz are reduced and get shifted to higher energy. The rutilenanoparticle VDOS crosses the bulk and super cell VDOS at 22.6 and23.9 THz respectively and gets extended up to 32 THz. Thus we findthat finite size disorder, mainly in the vicinity of surface, significantlyalters the force constants and hence the VDOS as compared to bulkTiO2 [27].

3.3. The VDOS through LD and MD simulations

It is essential to check the closeness of VDOS obtained by LD andMDsimulations for rutile nanoparticle. This may establish that the VDOSobtained by MD simulations is reliable; therefore the calculated resultsfor the amorphous system may also be reliable. The ensemble averageof velocity auto correlation function as given in Eq. (1) is carried outby averaging over 5000 bins obtained by dumping the velocities at theinterval of 0.01 ps. These results are used to calculate g(ω) as given inEq. (2) to obtain VDOS of rutile nanoparticle. These results are furtherscaled and smoothed using acsplines with uniform smoothing weightand are shown by the solid line in Fig. 3. Similar scaling is carried outfor the results obtained by LD simulations as discussed in Section 2.1and these are shown by the dashed line in Fig. 3.

We find that in the low and high frequency ranges both results arein close agreement. However in the intermediate frequency range,the characteristics of both results are similar although the VDOSobtained by MD simulations is higher than that obtained by LD

simulations. These characteristics are similar to those obtained formono atomic nanoparticles at low temperature [24,25]. Ogata et al.[10] have also shown the similar closeness of LD and MD simulationresults of rutile cluster of 672 atoms using variable charge potential.However Kong [26] has shown that MD simulation results are closerto the experimental data as the sample temperature is accountedfor in the calculations. In view of this the VDOS of amorphous TiO2

nanoparticle obtained by MD simulations may be the model resultsfor future experiments. In the following we carry out the comparativestudy of partial VDOS of rutile and amorphous nanoparticles of TiO2.

3.4. Partial VDOS of rutile and amorphous nanoparticles

The total and partial VDOS of rutile nanoparticle obtained by MDsimulations are shown in Fig. 4(a). It is evident that the Ti atoms con-tribute mainly in the range of acoustic and low energy optical modes,while the O atoms contribute to the entire range of frequencies. TheVDOS in the high frequency range is mainly due to O atoms. The TOphonon peaks around 10 and 17.21 THz, LO phonon peak at 14 THzand Raman mode peak at 24 THz are mainly due to O atoms. This isdue to the reason that the Ti atom is nearly four times heavier thanthe O atom and in the MA force field Ti–Ti interaction is nearlythree times larger than that of O–O interaction. Further the numberof O atoms in the sample is twice that of Ti atoms. The phonon disper-sion relations of TiO2 are explained by Traylor et al. [3] assuming thatTi atoms are rigidly bound and do not contribute to the dynamics ofTiO2. Ogata et al. [10] found two prominent peaks, one in the acousticregion and other in high energy optical region in the rutile cluster of672 atoms at 100 K. However we do not find such singular peaks inthe acoustic and optical regions and such peaks are also not foundin the experimental data for the bulk [3].

The VDOS of three differently prepared amorphous nanoparticlesS1, S2 and S3 are shown in Fig. 4(b), (c) and (d) respectively. In thesamples S1 and S2, prepared by different heating and quenchingrates, the fine structure of phonon peaks for acoustic and opticalmodes persists nearly up to 20 THz as shown in Fig. 4(b) and (c).This indicates that a significant part of amorphous sample remain tet-rahedrally coordinated [14]. The partial VDOS of O atoms increases upto 8 THz and then remains nearly constant up to 25 THz and then de-creases sharply. However the partial VDOS of Ti atoms increases up to5 THz and remains nearly constant up to 12 THz and then decreasesprogressively. The structure of partial VDOS of Ti and O atoms in dif-ferent frequency ranges remains the same as for the rutile nanoparti-cle. The structure of partial VDOS of Ti and O atoms for sample S3 isthe same as for the samples S1 and S2 as shown in Fig. 4(d), howeverthe phonon peak structure in the partial and total VDOS is sharper in

0

0.2

0.4

0.6

0.8

VD

OS

(1/

TH

z)

a) Rutile nanoparticle Tot. VDOS Part. VDOS OPart. VDOS Ti

0

0.2

0.4

0.6

0.8

VD

OS

(1/

TH

z)

b) Amorp. Sample S1

0

0.2

0.4

0.6

0.8

VD

OS

(1/

TH

z)

c) Amorp. Sample S2

0

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30

VD

OS

(1/

TH

z)

Ferquency (THz)

d) Amorp. Sample S3

Fig. 4. The calculated partial (Part.) and total (Tot.) VDOS of 3 nm size rutile nanopar-ticle (a) and 3 nm size amorphous nanoparticles S1 (b), S2 (c) and S3 (d) by MD sim-ulations. The data are smoothed using acsplines with a uniform smoothing weight of106. The description is given in the legend.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

VD

OS

(1/

TH

z)

Ferquency (THz)

Rutile nanoparticleAmorp. S1Amorp. S2Amorp. S3

Fig. 5. The comparison of calculated total VDOS of 3 nm size rutile nanoparticle andamorphous TiO2 nanoparticles S1, S2 and S3 by MD simulations. The data aresmoothed using acsplines with a uniform smoothing weight of 106. The descriptionis given in the legend.

32 K.K. Ghuman et al. / Journal of Non-Crystalline Solids 373-374 (2013) 28–33

the acoustic and low energy optical phonon regions and is similar tothat of rutile nanoparticle. This indicates that sample S3 is less amor-phous as compared to S1 and S2 and thus one may conclude that thefine structure of VDOS depends upon the sample preparationprocedure.

The relative comparison of total VDOS of rutile nanoparticle andamorphous nanoparticles S1, S2 and S3 is shown in Fig. 5. The peakedstructure of VDOS of rutile nanoparticle gets dispersed in the finestructure in the amorphous phase and gets extended in the low andhigh frequency ranges as compared to the rutile phase. This is dueto structural changes in the atomic arrangements in the core and sur-face regions in amorphous nanoparticles which alter the atomic forceconstants [14,15,27]. The VDOS of amorphous nanoparticles is fairlyflat in the high energy acoustic and optical phonon frequency rangeand can be well represented by the step function in the computationof thermodynamical properties. We do not find the gap in the VDOSin the vicinity of 20 THz as reported by Sikora [8]. There exists nosuch gap in the bulk TiO2 [3]. However no experimental data is avail-able for amorphous TiO2 nanoparticle to compare our results.

4. Conclusions

The MA rigid ion model with effective charges on Ti and O atoms isused to investigate vibrational properties of rutile supercell andrutile and amorphous TiO2 nanoparticles. The phonon band widthand dispersive character of optical phonon modes agree with the

experimental data in the higher frequency range. However, in the in-termediate energy range the deviations between the calculated andexperimental results are within 15% and in the lower energy rangethe calculated results are higher than the experimental values. Asthe model potential parameters are determined with the help of staticstructural properties of material, the vibrational frequencies obtainedwith such model potentials often deviate from the experimental data.

There is pile up of additional modes and broadening of Ramanmodes in the low energy range in the VDOS of rutile nanoparticleand it crosses the VDOS of bulk and supercell at higher energies andgets extended up to 32 THz. This is due to finite size disorder whichalters the atomic force constants. The Ti atoms contribute to theacoustic and low energy optical modes, while O atoms contribute inthe entire frequency range of VDOS as the Ti atom is heavier thanthe O atom, the number of O atoms is twice that of Ti atoms and inthe MA force field Ti–Ti interaction is nearly three times larger thanthat of O–O interaction.

The peaked structure of VDOS of rutile nanoparticle gets dispersedin the fine structure in amorphous phase and gets extended in thelow and high frequency ranges. The comparison of VDOS of samplesS1, S2 and S3 indicates that sample S3 is less amorphous as comparedto S1 and S2 and the fine structure of VDOS does depend upon thesample preparation procedure. The VDOS of amorphous nanoparticlemay be a model for future experiments and can be well representedby a step function for the investigation of its thermodynamical prop-erties. The calculated results may further be improved if shell–shelland shell–ion interactions are included in the calculations.

Acknowledgments

The authors are thankful to Dr. R. Meyer, Dr. G. S. S. Saini and TimTeatro for fruitful discussions. The financial assistance from UGC NewDelhi for SRF and Emeritus Fellowship is acknowledged.

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