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Structural, electronic, vibrational and dielectric properties of selected high-shape K

semiconductor oxides

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2014 J. Phys. D: Appl. Phys. 47 413001

(http://iopscience.iop.org/0022-3727/47/41/413001)

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Journal of Physics D: Applied Physics

J. Phys. D: Appl. Phys. 47 (2014) 413001 (14pp) doi:10.1088/0022-3727/47/41/413001

Topical Review

Structural, electronic, vibrational anddielectric properties of selected high-Ksemiconductor oxidesL M R Scolfaro1, H W Leite Alves2, P D Borges3, J C Garcia4,6

and E F da Silva Jr5

1 Department of Physics, Texas State University, 78666 San Marcos, TX, USA2 Departamento de Ciencias Naturais, Universidade Federal de Sao Joao Del Rei, CP 110, 36301-160Sao Joao Del Rei, MG, Brazil3 Instituto de Ciencias Exatas e Tecnologicas, Universidade Federal de Vicosa, 38810-000 Rio Paranaiba,MG, Brazil4 Escola Politecnica, Universidade de Sao Paulo, CP 61548, CEP 05424-970, Sao Paulo, SP, Brazil5 Departamento de Fısica, Universidade Federal de Pernambuco, Cidade Universitaria, 50670-901Recife, PE, Brazil

E-mail: ls61@txstate.edu, hwlalves@ufsj.edu.br, pdborges@gmail.com, jcott@usp.br, eron@ufpe.brand j c459@txstate.edu

Received 5 March 2014, revised 20 June 2014Accepted for publication 10 July 2014Published 10 September 2014

AbstractThe semiconductor oxides SnO2, HfO2, ZrO2, TiO2 and SrTiO3 are interesting materials forapplications as high-K dielectric gate materials in silicon-based devices and spintronics,among others. Here we review our theoretical work about the structural, electronic andvibrational properties of these oxides in their most stable structural phases, including dielectricproperties as derived from the electronic structure taking into account the lattice contribution.Finally, we address the recent role played by the presence of transition metal atoms insemiconductor oxides, considering in particular SnO2 as an example in forming dilutedmagnetic alloys.

Keywords: high-K , semiconductor oxides, ab initio calculations, dielectric properties,electronic properties, vibrational properties, structural properties

(Some figures may appear in colour only in the online journal)

1. Introduction

Semiconducting oxides involving group IV-A and IV-Belements in the form MO2 (M = Sn, Zr, Hf and Ti) aswell as SrTiO3, are interesting materials for applications asgas sensors, transparent conducting electrodes, solar cells,high-Kdielectric gate materials in silicon-based devices and,recently, in spintronics as diluted magnetic materials [1, 2].They are wide band gap materials that present excellent

6 Present address: Department of Physics, Texas State University, 78666 SanMarcos, TX, USA.

chemical stability in hostile environments and, due to theirrefractory characteristics, some of their optical properties,such as the dielectric constant and the refraction index, aredependent on the electron–lattice interaction effects exhibitedby these semiconducting oxides.

In the present work, we present the results of ab initiocalculations performed within the density functional theory(DFT) of the band structures, vibrational modes, opticaland dielectric properties of SnO2, HfO2, ZrO2, TiO2

and SrTiO3, for the most stable structural phases of thecompounds, and make an analysis of the trends observed

0022-3727/14/413001+14$33.00 1 © 2014 IOP Publishing Ltd Printed in the UK

J. Phys. D: Appl. Phys. 47 (2014) 413001 Topical Review

Table 1. Calculated lattice constants, internal parameter, angle, bulk modulus and crystal densities. Experimental values are also shown forcomparison.

a Expt. b Expt. c Expt. c/a Expt. βcal βexpt u Expt. B0 Expt. Dens. Expt.(Å) (Å) (Å) (Å) (Å) (Å) (GPa) (GPa) (g cm−3) (g cm−3)

r-SnO2 4.829 4.737a 3.327 3.188a,b 0.670 0.673a 0.307 0.307a 193 203d 7.07 7.02c

a-TiO2 3.67 3.782e 9.47 9.502e 0.206 0.208e 243 179f 3.8941(2)f

r-TiO2 4.66 4.587e 2.97 2.954e 0.652 0.644e 0.305 0.305e 226 211g

m-ZrO2 5.12 5.15h 5.16 5.21h 5.33 5.31h 99.6 99.23h,i 157i 233.1i 4.42 4.37m-HfO2 5.291j 5.116k 5.405j 5.172k 5.366j 5.295k 97.92j 99.11k 251i 6.90 7.58c-SrTiO3 3.95 3.905l 170 4.94 5.12

a [30]; b [31]; c [32]; d [33]; e [34]; f [35]; g [36]; h [37]; i [38]; j [39]; k [40]; l [41].

in a variety of calculated physical properties. Knowledgeof these properties is helpful for the conception ofnew electronic/optoelectronic/photovoltaic devices based onfunctional oxide interfaces [2]. In the case of applyingoxides in new spintronics devices, transition metal (TM) dopedoxides such as ZnO, TiO2, In2O3 and SnO2, also knownas diluted magnetic oxides (DMOs), have received a greatdeal of attention due to their use as functionalities in carriersand optical control [2]. SnO2 has extraordinary propertiesas an oxidation catalyst or gas sensor, due to its chemicalsensitivity and structural stability. Since the discovery ofroom temperature ferromagnetism in Co-doped SnO2 with ahigh value for the magnetic moment (m = 7.5µB) for 5%impurity concentration [3], the interest in the study of SnO2

based DMOs has increased [4–8]. The major challenge hasbeen how to grow DMOs with a high Curie temperature andwith controllable magnetic properties [7–11].

Native defects such as oxygen vacancies (VO) have beenobserved in the pure material, and the presence of the complexpair TM–VO may contribute to the ferromagnetic (FM)characteristics seen in TM-doped SnO2 [9, 12, 13]. The roleplayed by VO in tuning the magnetic properties of Co-dopedinsulating SnO2 has also been demonstrated [14]. Similarbehaviour has been reported in other oxides (ZnO, HfO2,TiO2, In2O3) [15, 16]. These results have shown that ratherthan free carriers, the VO concentration and distribution canbe directly connected to room temperature ferromagnetism.However, the origin and mechanism of ferromagnetism havebeen quite controversial in these systems, and recent reviewson the subject can be found in the literature [1, 2]. AlthoughDMOs are far from being applied in practical devices, theongoing experimental and theoretical research can pave theway to breakthroughs in basic research, and the developmentof future spintronic devices.

In order to improve our understanding of the origin offerromagnetism in DMOs, a section is included in this paperreporting our ab initio results for the 3d series of TMs inSnO2,which show the occurrence of spin-crossover (SCO)transitions in these DMOs, attributed to the presence ofmagnetic metastability (MM). Some of the obtained resultshave been reported in our previous work [17–19]. Similarbehaviour, i.e the observation of SCO was previously observedin a GaN derived diluted magnetic semiconductor [20]. SCOis the phenomenon associated with the change in the spin stateof a system where a transition occurs from a low-spin (LS)ground state electron configuration to a high-spin (HS) one,

and vice versa. Technological applications of SCO materialssuch as in memory devices, switches, optical displays havebeen proposed [21]. Our results for the 3d TM derived DMOs,using SnO2 as a prototype, can be used to drive further researchin oxide based spintronics.

2. Methodology and computational methods

All calculations were carried out using the (spin) DFT and thegeneralized gradient approximation (GGA) for the exchange–correlation term [22]. For the structural properties, weemployed the projector augmented wave method, implementedin the Vienna Ab-initio simulation package (VASP-PAW)[23]. Band structures and optical properties were obtainedthrough the ab initio full-potential linear augmented plane-wave (FLAPW), within a full relativistic approach, includingthe spin–orbit (SO) coupling effects (WIEN2k code) [24, 29].Convergence in total energy was achieved when it differedby less than 10−6 Ry between two self-consistent iterationsby using a plane-wave cutoff of RKmax = 8, and the chargedensity was expanded up to a cutoff Gmax = 14 (a.u.)−1. Theintegration in the Brillouin zone (BZ) was sampled employingup to 2000 k-point meshes for optical properties calculations,and using the Monkhorst–Pack scheme [28].

In the case of lattice dynamics properties, we have usedthe plane-wave description of the wave function, and thepseudopotential method (abinit code [25]). Troullier–Martinspseudopotentials were used, and the phonons were obtainedby means of the density functional perturbation theory. Theresults all converged well, with a cutoff energy of 120 Ry forthe plane-wave expansion of the wave functions, and we useda (6, 6, 6) Monkhorst–Pack mesh to sample the BZ. Fromthe obtained phonon frequency results, we also calculated thefrequency dependence of both the real and the imaginary partsof the dielectric permittivity tensor (as developed by Rignaneseet al [26] and Gonze et al [27]).

For the SnO2 DMOs, the calculations were performedincluding only the scalar relativistic effects within the VASP-PAW method. To describe the DMO alloys, we used a 72atom supercell (24 Sn and 48 O atoms) and a (4, 4, 4) mesh ofMonkhorst–Pack k-points for integration in the BZ. All thecalculations were done with a 490 eV energy cutoff in theplane-wave expansions and the systems were fully relaxed untilthe residual forces on the ions were less than 10 meV Å−1.

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J. Phys. D: Appl. Phys. 47 (2014) 413001 Topical Review

Figure 1. Energy band structure along several high-symmetry directions in the first Brillouin zone and total density of states (TDOS) in:(a) SnO2, (b) r-TiO2, (c) ZrO2, (d) HfO2 and (e) SrTiO3. The energy zero was set at the top of the valence-band, shown by a dashedhorizontal line. The main character of the peaks in the TDOS is emphasized.

3. Results

3.1. Structural properties

Tin dioxide crystallizes with the rutile structure while titaniumdioxide presents two phases, the most stable rutile and ametastable phase, anatase. The most stable phases of HfO2 andZrO2 at up to room temperature, correspond to the monoclinicphase. Strontium titanate is a material with a perovskite cubicstructure. Through total energy and forces minimization, weobtain the lattice constants, with an internal parameter u, and

angle β. We present our calculated values for the structuralproperties for the various oxides studied in table 1, and comparethem with the available experimental values. Good agreementis also observed when our predictions are compared with otherab initio DFT calculations (not shown).

3.2. Electronic properties

Figure 1 depicts the band structures and total density ofstates, as obtained from full relativistic calculations, using

3

J. Phys. D: Appl. Phys. 47 (2014) 413001 Topical Review

the WIEN2k code, for the most stable phases of the studiedsemiconductor oxides, i.e. r-SnO2, r-TiO2, m-HfO2, m-ZrO2

and SrTiO3 (see in [42], our own work).The consequences of the interaction between the (metal)

M(d, s) electrons with O(p) electrons are reflected in theirband structures. The following important features can bedistinguished from the calculations [56]: (i) O(2s) states formhyperdeep bands around 16 eV below the top of the valencebands; (ii) their valence bands have a predominant O(2p)character with small contribution of the M(s) states; (iii) theelectronic states close to the bottom of their conduction bands(CB) have a character resulting from the combination of boththe antibonding M(s) and O(2p) states, with an increasingcontribution of the antibonding M(d) state when we go from Tito Hf and to SrTiO3. In SnO2, as the Sn electronic configurationhas not got a partially filled d state (as observed for TiO2,ZrO2 and HfO2), but yet sp3 hybridization is observed, and itsvalence band has contributions of both Sn-s and O-p atomicorbitals. Similar behaviour is seen in other recent ab initioDFT calculations (see [42] and references therein).

The scalar relativistic effects are shown to be relevant forthe energy levels in the gap region, however the SO coupling isnot found to significantly affect that region, except for SrTiO3.SnO2 has a direct band gap at � of 0.60 eV. In the study of the tindioxide-based diluted magnetic alloy (DMA) (described laterin section 3.6), we included an on-site Coulomb correlationinteraction U to the Sn (d) states. By this approach, for theSnO2 band gap we obtained the value Egap = 1.65 eV. Inrecent works, very good agreement between the experimentalband gap value and theoretical ones has been obtained usingthe newest approaches, such as HSE03+G0W0, LDA-1/2, andTran–Blaha modified Becke–Johnson (TB-mBJ), carried outby Schleife et al, Kufner et al and Dixit et al, respectively [42].For m-ZrO2, we have obtained an indirect gap, � → X,of 3.58 eV and a direct band gap, at �, of 3.64 eV, and thisis very close to the value of the indirect one. HfO2 and r-TiO2 are also indirect gap materials (� → X) with bandgapvalues of 3.98 eV and 3.05 eV, respectively. In general, theseresults show very similar behaviour when compared to othertheoretical predictions [42].

For SrTiO3, we observe that the inclusion of relativisticcorrections is especially important for the Ti (d) antibondingstates, which form the low-lying states of the CB. It is also anindirect band gap material, with the top of the valence band atthe R-point and the minimum of the CB at �. The calculatedvalue for the indirect (R → �) band gap was 1.78 eV, andfor the direct band gap (� → �) was 2.14 eV. Due to SOinteraction, we observe a SO splitting energy�so of theR and�

states of the valence band maximum and of the minimum of theCB at the �-point. The values found are �vb

so (R) = 18.3 meV,�vb

so (�) = 60.7 meV and �vbso (�) = 28.5 meV, with vb (cb)

standing for valence (conduction) band. The value obtainedby us for �cb

so (�) is similar to that recently reported by Zhonget al [42].

It is worth mentioning the work reported by Gruninget al and by Jiang et al [42] in which the authors investigatedthe electronic properties of ZrO2 and HfO2 using the many-body perturbation theory in the GW approach, based on DFT

calculations, and showed that the GW approach was able todescribe the quasiparticle band structure of widegap oxideswith empty d states and/or fully filled semicore f states. Bothoxides are different from conventional sp semiconductorssince the CBs are mainly of d character, for which the GWapproximation has not yet been fully established. However, theelectron–hole and the electron–phonon interaction may play aconsiderable role in ZrO2 and HfO2.

3.3. Vibrational related properties

The lattice contribution to the static dielectric constant ofthese oxide materials is a very important issue in explainingtheir observed high values, once the electronic counterpartcontributes with values of around 4–5, and not exceeding8 [44]. This means that some of their optical properties, suchas refraction indices, are dependent on the electron–latticeinteraction effects exhibited by these compounds. In this case,the lattice contribution is directly proportional to the oxidelarge Born effective charges and is inversely proportional tothe TO phonon frequencies,

K = Ne2Z∗2T

mωTO, (1)

as pointed out by Robertson [44].Figure 2 shows our calculated phonon dispersions for

rutile SnO2, and for both of the monoclinic ZrO2 and HfO2

along high symmetry lines of their respective BZs. Our resultsare in good agreement with both the available experimentaldata and theoretical results [39, 45]. In table 2, we depictsome calculated phonon frequencies for the active infrared (IR)modes at the centre of the BZ, compared with the availableexperimental data, and the calculated averaged Born effectivecharges, Z∗, for the studied oxides as well. The results forTiO2 and SrTiO3 were extracted from the latest DFT resultspublished in the literature [46].

Considering equation (1), we have to focus on lower TOphonon frequencies together with higher Z∗ values in theavailable data. From figure 2, as well as observed in table 2, wenote that these materials have more IR-active modes than thegroup IV or III–V semiconductors. From table 2, the lowestTO frequency found is ∼100 cm−1 (SrTiO3), followed by alarge number of IR-active modes in the range of 175–250 cm−1

observed in the other studied oxides.Based only on the number of IR-active TO modes with

frequencies smaller than 250 cm−1, and the value of thesmallest one as well, both TiO2 and SrTiO3 have the highest Kvalue, followed by HfO2 and ZrO2. However, due to the factthat the former oxides are not considered as wide band gapones when compared with HfO2 and ZrO2, these latter werechosen to replace SiO2 in the MOSFET devices.

The Born effective charges tensor, Z∗, is defined as theinduced polarization due to the unit cell displacement in agiven phonon mode. Thus, materials with high Z∗ valuesrespond well to an applied electric field when they are placed asdielectrics, as depicted in equation (1). Again, as observed forthe TO phonon frequencies, the highest Z∗ value found is 7.11(SrTiO3), followed by the values obtained for TiO2 in the rutilephase, and both the monoclinic HfO2 and ZrO2. This observedfeature reinforces the high-K character of these oxides.

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J. Phys. D: Appl. Phys. 47 (2014) 413001 Topical Review

Figure 2. Phonon dispersions along several high-symmetry directions in the first Brillouin zone and total density of states in (a) ZrO2,(b) HfO2 and (c) SnO2.

Table 2. IR-active vibrational modes and effective charges.

A2u Expt. Eu Expt. Eu Expt. Eu Expt. Z∗ Z∗

(cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (M) (O)

SnO2-r 457.4 477.0a 200.3 244.0a 270.5 293.0a 584.1 618.0a 4.34 −2.16 — — — —TiO2-a 401.9 366.9b 239.8 261.8b 325.6 365.9b — — 6.33 −3.22 — — — —TiO2-r 191.5 182.8b 175.5 188.8b 394.6 388.3b 505.3 494.0b 6.71 −3.37 — — — —

Au Expt. Au Expt. Au Expt. Au Expt. Au Expt. Au Expt. Z∗ Z∗

(cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (M) (O)ZrO2-m 174.6 180a 242.0 250.0a 247.0 257.0a 343.0 — 399.0 415.0a 473.0 — 5.37 −2.69HfO2-m 129.6 — 175.6 — 230.0 228.0a 356.0 — 403.0 420.0a 470.0 445.0a 5.44 −2.72

Bu Expt. Bu Expt. Bu Expt. Bu Expt. Bu Expt. Bu Expt.(cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1)

ZrO2-m 220.0 224.0a 300.0 — 313.0 324.0 a 354.0 346.0a 413.0 — 485.1 510.0a

HfO2-m 219.0 — 245.0 — 320.0 332.0a 350.0 352.0a 413.0 — 478.0 —

T1u Expt. T1u expt. T1u Expt. T1u Expt. Z∗ Z∗ Z∗

(cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (M) (Sr) (O)SrTiO3 135 91b 188.5 175.0b 238.5 265.0b 569.2 545.0b 7.11b 2.30 b −3.76b

a [45]. b [46].

3.4. Complex dielectric function

The dielectric tensor components are obtained within therandom phase approximation (RPA) sum rules in the linearresponse regime. The electronic contributions for the realε1 (ω) and imaginary ε2 (ω)parts of the complex dielectricfunction ε (ω) = ε1 (ω) + iε2 (ω) can be obtained fromthe band structure directly through the WIEN2K [24, 29]and VASP packages [23]. The real part ε1 (ω) is obtainedfrom the imaginary part ε2 (ω) by using Kramers–Kronig

relations. Once the imaginary part ε2 (ω) is known from theelectronic structure calculations, the real part ε1 (ω) is thengiven by [23, 24, 29]

ε1 (ω) = 1 +

(2

π

) ∫ ∞

0dω′ ω

′2ε2(ω′)

ω′2 − ω2(2)

All theoretical values for the band gap energy are smallerthan the experimental ones, which is due to the well-knownunderestimation of the energy values of CB states in ab initio

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J. Phys. D: Appl. Phys. 47 (2014) 413001 Topical Review

(a)

(b)

(c)

(d)

(e)

Figure 3. The real (ε1) and imaginary (ε2) parts of the complex dielectric tensor diagonal components (εxx , εyy , εzz) versus the incidentphoton energy. The calculated spectrum for ε = ε1 + iε2 was shifted to higher energies, by matching its energy threshold to the experimentalvalue of the gap energy E

exptg . The panels show the calculated spectrum ε = ε1 + iε2 in (a) tetragonal rutile structure of tin dioxide (r-SnO2),

(b) tetragonal rutile structure of titanium dioxide (r-TiO2), (c) monoclinic zirconia structure (m-ZrO2), (d) monoclinic hafnia structure(m-HfO2) and (e) the cubic perovskite structure of the strontium titanate (c-SrTiO3). The experimental values of the energy band gap (Eexpt

g )

of oxides in (a) r-SnO2 (Eexptg ∼ 3.6 eV) as obtained in [43], (b) r-TiO2 (Eexpt

g ∼ 3.51 eV) as obtained in [53] through optical measurements,(other values can be found in [47]), (c) m-ZrO2 (5.83 eV) as obtained in [48], (d) m-HfO2 (Eexpt

g ∼ 5.8 eV) as obtained in [49] and(e) c-SrTiO3 (Eexpt

g ∼ 3.2 eV) as obtained in [51]. The experimental data of the dielectric function in: (a) the experimental data of Yuberoet al [52] is depicted by the red dotted–dashed line, (b) the spectroscopic ellipsometry measurements of Jellison et al [53] are depicted bythe red and blue square–solid line, (c) the VUV optical measurements of the optical conductivity by French et al [48] are depicted by the reddotted–dashed line, (d) the transmission and ellipsometry spectroscopic measurements of Lim et al [49], and those due to Edwards [50], aredepicted by inverted red closed triangles and green closed dots, respectively, and (e) the experimental data of Cardona [51] is depicted by thered dotted–dashed line.

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J. Phys. D: Appl. Phys. 47 (2014) 413001 Topical Review

Figure 4. The real (ε1) and imaginary (ε2) parts of the lattice contribution for the complex dielectric tensor diagonal components (εxx , εyy ,εzz) versus the phonon frequency for the m-ZrO2.

Figure 5. Limit to zero frequency of the real (ε1) part of the latticecontribution for the complex dielectric tensor diagonal componentsfor m-HfO2, m-ZrO2, SnO2 and anatase TiO2.

calculations within the local-DFT. For this comparison, thewhole calculated spectrum for ε (ω) was shifted to higherenergies, by matching the energy threshold to the experimentalvalue of the gap energy. The calculated frequency behaviourof the imaginary part of the dielectric function ε (ω) in thelow energy range is in good agreement with the experimentalmeasurements performed by far ultraviolet (UV) spectroscopicellipsometry for energies smaller than 10 eV.

In figure 3(a) we show the theoretical and experimentalcomplex dielectric function for bulk rutile SnO2: theexperimental data was extracted from reflection electronenergy loss spectroscopy by Yubero et al [52]. Recentexperimental data by Feneberg et al [52] investigated the linearoptical properties of rutile SnO2 by spectroscopic ellipsometryin the energy range up to 20 eV, and this demonstrates that theelectron–hole interaction in SnO2 is strong, and the excitoniceffects have to be included in the calculations.

In figure 3(b), we show the experimental complexdielectric function for bulk rutile TiO2, determined by thespectroscopic ellipsometry measurements of Jellison et al[53]. The overall shape of ε1 and ε2 is in agreement withphoton energies of up to 6 eV by using a shifted operator.Since the optical absorption measurements create an electron–hole pair, the agreement between experimental results andtheoretical calculations can only be achieved when takingthe exciton into account. Chiodo et al [52] computedthe TiO2 properties, based on the GW approximation and theBethe–Salpeter equation approaches (BSE), to improve thestandard mean field RPA calculation with the Perdew–Burke–Ernzerhof (PBE) functional. The inclusion of the quasiparticledescription RPA/GW improves the electronic gap description,however, it does not affect the overall shape of the spectrumand it acts as a scissors operator. The absorption edge of theRPA/GW calculation is shifted to higher energies in regardto RPA/PBE. The optical TiO2 properties calculated using theBSE/GW approach, truly improves the result of the RPA/GWcalculations.

In figure 3(c), we show the calculated imaginary (ε2) partof the ZrO2 dielectric function. This is in agreement with thedata reported by French et al [48], as obtained from vacuumultraviolet spectroscopy measurements (VUV). Figure 3(d)demonstrates an agreement of the calculated ε2 with theexperimental data reported by Lim et al for energies smallerthan 9.5 eV, as well as with those performed by Edwardset al [50], all of which use UV ellipsometry spectroscopy.For comparison, the whole calculated spectrum for ε2 was

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J. Phys. D: Appl. Phys. 47 (2014) 413001 Topical Review

Figure 6. Pressure dependence of the real (ε1) part of the lattice contribution for the complex dielectric tensor for: (a) m-ZrO2, (b) m-HfO2

and (c) SnO2.

Figure 7. TM-doped tin dioxide diluted magnetic alloys for (a) Sn0.96TM0.04O2 where a single tin atom was replaced with a TM atom and(b) Sn0.96TM0.04O1.98(VO)0.02 where a complex pair TM–VO was formed after a single oxygen atom has been removed. TM, Sn and O arerepresented by black, grey and white solid spheres, respectively.

shifted to higher energies by matching the energy thresholdto the HfO2 experimental gap energy value. The calculatedfrequency behaviour of the HfO2 imaginary part in the lowenergy range is in agreement with the measurements of Lim

et al [49] by UV spectroscopic ellipsometry, as well as withthose of Edwards [50]. New approaches beyond DFT havebeen employed to compute the optical properties of HfO2 andZrO2.

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J. Phys. D: Appl. Phys. 47 (2014) 413001 Topical Review

Figure 8. Total energy, per cell, versus magnetic moment for TM-doped tin dioxide diluted alloys considering (a) Sn0.96V0.0402,(b) Sn0.96Cr0.04O2, (c) Sn0.96Mn0.04O2, (d) Sn0.96Fe0.04O2, (e) Sn0.96Co0.04O2 and (f ) Sn0.96Ni0.04O2. The total energy of the ground statefor each system is set to zero.

Kobayashi et al [52] studied the electronic structure anddielectric properties of cubic zirconia (ZrO2). They calculatedthe band structure by using the LDA and GW approximationbased on the LMTO method, and reported that the Zr (d) bandslocated at the bottom of the CBs shift rigidly to the upperenergy side, and the band gap widened from 3.3 to 5.3 eV. Theelectronic structure is more sensitive than the band gap to theoptical properties, since the interband optical transitions aregenerated between the occupied and unoccupied bands in awide energy range.

In figure 3(e), we compare our calculations and theexperimental data of Cardona [51] up to photon energies of

16 eV. Guo and Liu [52] investigated the SrTiO3 electronicstructures and optical dielectric functions by using the Tran–Blaha modified Becke–Johnson approach (TB-mBJ) for theexchange potential. The energy gap was improved by atleast 40% over previous LDA and GGA results, and remainsreasonable in contrast to the larger gap values obtained fromthe GW approach. This substantial improvement was achievedbecause the energy levels of the empty Ti (d) states, in regardsto the filled O (p) states, are correctly calculated using TB-mBJ. Cappellini et al [52] computed the SrTiO3 propertieswithin the DFT in the local density approximation (LDA) andin Hedin’s GW scheme for self-energy correction by using a

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J. Phys. D: Appl. Phys. 47 (2014) 413001 Topical Review

dielectric function model, which approximately includes localfield and dynamical effects. The deep valence states are shiftedby the GW method to higher binding energies, in agreementwith photoemission spectra measurements.

We conclude that the overall shapes of the dielectricfunctions are in accordance with experimental results, and thelowest critical points observed in figure 3 are primarily dueto transitions from the O (p) valence bands to the d states inTiO2, ZrO2, HfO2 and SrTiO3 CBs. We observe O (p) to Sn(s) transition in SnO2.

3.5. Lattice contribution to the dielectric function

With the knowledge of the TO frequencies, we then canevaluate the lattice contribution to the low frequency regionof the frequency dependent dielectric function. In figure 4,the results for the frequency dependence of the dielectricpermittivity tensor components of m-ZrO2 are shown and, infigure 5, we depict the behaviour of the real component ofthis tensor when the frequency, ω, tends to zero for m-HfO2,m-ZrO2, SnO2 and anatase TiO2.

As described in our previous work [55], the results werefitted by the Lorentz–Drude formula, which is defined as

ε(latt)1 (ω) = ε1 (∞) +

n∑i=1

ω2pi (ω

2TOi − ω2)[

(ω2TOi − ω2)

2 − γ 2i ω2

] , (3)

where n is the total number of oscillators, ωpi is the plasmafrequency for ith oscillator, and γi is its damping parameter,which is directly related to sample size. The imaginarycomponent of this tensor can be obtained by applying a simpleKramers–Kronig transformation. From equation (3), it isclear that, if the material has large number of TO modeswith low frequencies, the value of ε1(0) will be higher. So,one feature of a high-K material is the presence of plasmaoscillations, of somewhat high effective charges (Z∗), whena low frequency TO phonon is turned on. These oscillationsenhance its dielectric capabilities, described in equation (1).

Another example holds for SnO2 [54]. In this case thevalues εlatt

1⊥(0) = 14.6 and εlatt1‖ (0) = 10.7 were obtained

for directions perpendicular and parallel to the tetragonal c-axis, respectively, in excellent agreement with the availableexperimental data as obtained from far-IR spectroscopy data,14.2±2 and 9.0±0.5 [58], and also with the measured averagevalue of 12.5 reported in [59]. So, the obtained averagedvalue for rutile SnO2 was 13.29. The same is observed foranatase TiO2 for which we found the averaged value of 52.73,and for both ZrO2 and HfO2, of around 20.7, reinforcing ourstatements [55, 56].

Finally, in figure 6 we show some aspects of the pressuredependence of the static dielectric function. From figure 6,we note that the value for the static dielectric function,ε1(0), diminishes with increasing pressure, as described in ourprevious work [57]. This indicates that, when deposited overa non-mismatched substrate with a smaller lattice parameterthan that of the high-K oxide, the resulting dielectric constantwill certainly have smaller values.

Table 3. Calculated spin-crossover energies (SEs) for TM-dopedSnO2 DMAs and TM-doped plus a nearby oxygen vacancy SnO2

DMAs; mGS is the magnetic moment for the ground state in eachsystem.

System m (µB/cell) mGS (µB/cell) SE (meV)

Sn0.96Cr0.04O2 0/2 2 144Sn0.96Cr0.04O1.98(VO)0.02 0/2 4 32.0

4/2 171Sn0.96Mn0.04O2 1/3 3 80.0Sn0.96Mn0.04O1.98(VO)0.02 5/3 3 97.0Sn0.96Fe0.04O2 0/2 2 18.0

4/2 9.00Sn0.96Fe0.04O1.98(VO)0.02 2/4 4 124

6/4 76.0Sn0.96Co0.04O2 3/1 1 17.0Sn0.96Co0.04O1.98(VO)0.02 1/3 3 103

3.6. Impurities in SnO2: role played by 3d TMs

In this section, we present the magnetic properties with focuson the MM and SCO characteristics seen in DMO systemssuch as Sn1−xTMxO2 and Sn1−xTMxO2−y(VO)y , in which weconsider low impurity/defect concentrations with x = 0.04and y = 0.02. The 3d series with TM = V, Cr, Mn, Fe,Co and Ni have been investigated. The theoretical analysispresented here takes into consideration the influence of anoxygen vacancy in the electronic and magnetic properties ofthis series of TM doped systems, emphasizing the role playedby the neighbouring oxygen atoms (plus the O vacancy) on theobtained magnetic moment trends for the different centres.

The equilibrium structures of the DMA configurationswere obtained by relaxing 72 atom supercells with respectto their internal coordinates. These calculations wereperformed within the GGA+U approach, as implemented inthe VASP code [23], since in the case of SnO2, GGA aloneunderestimates the binding energy of the semicore Sn (d) statesso severely that the SnO2 GGA band gap is only 0.60 eV (seefigure 1), as compared to the experimental value of 3.60 eV. Inorder to correct the position of the Sn (d) states, we includedan on-site Coulomb correlation interaction U = 3.9 eV to theSn (d) states, as done previously by Singh et al [60]. Thisvalue of U was obtained by calculating Uatom for the atomand screening the value by dividing it by the high frequencydielectric constant of SnO2. Within this approach we obtainedthe following values for the lattice constants and band gapa = 4.730 Å, c = 3.164 Å, u = 0.306 and Egap = 1.65 eV.

In figures 7(a) and 7(b) we show the supercells used.For the Sn1−xTMxO2 (TM=V, Cr, Mn, Fe, Co and Ni)DMA systems, a single tin atom was replaced with a TMatom, simulating x = 0.04 or 4% impurity concentration.In the Sn1−xTMxO2−y(VO)y DMAs, an additional oxygenvacancy (VO) is made by removing a single oxygen atom andthus forming a TM–VO complex pair, with x = 0.04 andy = 0.02 (or 2% oxygen vacancy concentration). For bothconfigurations the lattice parameters were fixed at a = b =9.67 Å and c = 9.73 Å.

The magnetic systems were simulated by imposing certainvalues for the total magnetic moment, m, in the intervalbetween 0.0 and 7.0 µB/cell. Figure 8 shows the calculated

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J. Phys. D: Appl. Phys. 47 (2014) 413001 Topical Review

Table 4. Theoretical (NT ) and experimental (N exp) charge states, state (GS is the ground state), magnetic moment per cell m (µB/cell) andmagnetic moment per TM mTM (µB), spin configurations of the TM, eTM, followed by that of the system, eS, and volume reduction, Vol.Red.(%) around the TM for each DMA [7, 9, 13, 61–70].

System NT N exp State m mTM Spin conf. Vol. Red.

Sn0.96V0.04O2 V4+ (3d1) V4+ GS 1 1 e(↑)

V 17Sn0.96V0.04O1.98(VO)0.02 V4+ (3d1) GS 1 1 e(↑)

V 10Sn0.96Cr0.04O2 Cr4+ (3d2) Cr3+ — 0 0 e(↑)

Cr e(↓)

Cr 11Cr4+ (3d2) GS 2 2 2e(↑)

Cr 9.7Sn0.96Cr0.04O1.98(VO)0.02 Cr4+ (3d2) — 0 0 e(↑)

Cr e(↓)

Cr 15Cr3+ (3d3) GS 2 3 3e(↑)

Cr e(↓)

S 7.4Cr3+ (3d3) GS 4 3 3e(↑)

Cr e(↑)

S 5.7Sn0.96Mn0.04O2 Mn4+ (3d3) Mn4+ — 1 1 2e(↑)

Mne(↓)

Mn 5.0Mn4+ (3d3) Mn3+ GS 3 3 3e(↑)

Mn 4.0Sn0.96Mn0.04O1.98(VO)0.02 Mn3+ (3d4) GS 3 4 4e(↑)

Mne(↓)

S 7.7Sn0.96Mn0.04O1.98(VO)0.02 Mn3+ (3d4) — 5 4 4e(↑)

Mne(↑)

S 7.4Sn0.96Fe0.04O2 Fe4+ (3d4) Fe3+ — 0 0 2e(↑)

Fe 2e(↓)

Fe 14Fe4+ (3d4) Fe2+ GS 2 2 3e(↑)

Fe e(↓)

Fe 13Fe4+ (3d4) — 4 4 4e(↑)

Fe 4.6Sn0.96Fe0.04O1.98(VO)0.02 Fe3+ (3d5) — 2 3 4e(↑)

Fe e(↓)

Fe e(↓)

S 11Fe2+ (3d6) GS 4 4 5e(↑)

Fe e(↓)

Fe 6.3Fe2+ (3d6) - 6 4 5e(↑)

Fe e(↓)

Fe 2e(↑)

S 7.7Sn0.96Co0.04O2 Co4+ (3d5) Co3+ GS 1 1 3e(↑)

Co 2e(↓)

Co 13Co3+ (3d6) Co2+ — 3 2 4e(↑)

Co 2e(↓)

Co e(↑)

S 10Sn0.96Co0.04O1.98(VO)0.02 Co3+ (3d6) — 1 2 4e(↑)

Co 2e(↓)

Co e(↓)

S 13Co2+ (3d7) GS 3 3 5e(↑)

Co 2e(↓)

Co 6.3Sn0.96Ni0.04O2 Ni4+ (3d6) Ni2+ GS 0 0 3e(↑)

Ni 3e(↓)

Ni 13

total energy as a function of m for the Sn0.96TM0.04O2 DMAs.In each plot of figure 8, the energy of the ground state wasset to zero. For V, Cr and Mn the ground state correspondsto high-spin configurations, whereas it is represented by low-spin configurations for Ni and Co. For Fe, two high-spinconfigurations were found, with m = 2 and 4µB/cell, with theformer being the ground state. We now define a spin-crossoverenergy, SE, associated with the change in the spin state, fromHS to LS and vice versa. The calculated values for the spin-crossover energies are SE0/2

Cr = 144 meV, SE1/3Mn = 80.0 meV,

SE0/2Fe = 18.0 meV, SE4/2

Fe = 9.00 meV and SE3/1Co = 17.0 meV.

All of these values are found in table 3. A trend in the valuesof m observed for the HS and LS states can be understood byanalysing the charge states of the TM and the first neighbouratoms around it. For the Sn0.96TM0.04O2 DMAs, the TM4+

impurity replacing Sn4+ leads to the magnetic states. Inmost cases, the neighbourhood does not contribute to themagnetization of the system. For example, in the case ofchromium, the Cr4+ (3d2) impurity replacing Sn4+ allows themagnetic statesm = 0 and 2µB/cell. The two spin-up electronsfrom the Cr impurity (2e(↑)

Cr ) give rise to the metastable statewith m = 2µB/cell, while one spin-up electron and one spin-down electron from the Cr impurity, e(↑)

Cr e(↓)

Cr , leads to themetastable state m = 0µB/cell. Cobalt is an exception, wherethe existence of the magnetic metastable state m = 3µB/cellinvolves a contribution from the neighbouring atoms. Table 4shows theoretical (NT ) and experimental (Nexp) charge states,state (GS is the ground state), magnetic moment per cell m

(µB/cell), and magnetic moment per TM mTM (µB), spinconfigurations of the TM, eTM, followed by that of the system,

eS, and volume reduction, Vol.Red. (%) around the TM foreach DMA.

Considering the Sn0.96TM0.04O1.98(VO)0.02 DMAs wherean O vacancy was created close to the TM site, the metastablestates and SEs change for most cases. For chromium, anew ground state configuration with magnetic moment m =4µB/cell, appears with SE4/2

Cr = 171 meV and the SE0/2Cr being

reduced to about 78% of that value. For manganese, themetastable state of 1µB/cell disappears and a new high-spinstate arises (m = 5µB/cell). For iron, a new high-spin statearises (m = 6µB/cell) and the metastable state of close tozero magnetic moment vanishes. The ground state becomesthat with m = 4µB/cell instead of 2µB/cell, and the obtainedenergy for the cross-over SE2/4

Fe is 124 meV. For cobalt, theground state is found with 3µB/cell and the metastable statewith 1µB/cell. The spin-crossover energy SE1/3

Co = 103 meV.For vanadium the state of 1µB/cell remains, and for nickela HS state appears with magnetic moment m = 2µB/cell.Figure 9 presents the results for the magnetization dependenceof the total energy when an oxygen vacancy is taken intoconsideration in the DMAs.

As shown in table 4, with the presence of the oxygenvacancy, the impurity state changes for most of the TMimpurities, and the neighbouring atoms are found to contributeto the magnetic moment of the centre. Chromium changesfrom the Cr4+ (3d2) state to Cr3+ (3d3). Three spin-upelectrons from Cr, 3e(↑)

Cr , plus one spin-down electron, e(↓)

S ,arising from the neighbouring atoms allow for the metastablestate with m = 2µB/cell, while three spin-up electrons fromCr, 3e(↑)

Cr , plus one spin-up electron, e(↑)

S , arising from the

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J. Phys. D: Appl. Phys. 47 (2014) 413001 Topical Review

Figure 9. Total energy, per cell, versus magnetic moment for the complex TM + VO in SnO2 diluted alloys considering TM = (a) vanadium,(b) chromium, (c) manganese, (d) iron, (e) cobalt and (f ) nickel doping. The total energy of the ground state for each system is set to zero.

neighbouring atoms, give rise to the metastable state with m =4µB/cell. These results are in agreement with the experimentaldata. Misra et al [13] have shown that the observed electronparamagnetic resonance spectrum (EPR) of Cr3+ doping ionspresent in samples of SnO2 nanoparticles exhibits a FM orderedcomponent attributed to oxygen vacancies.

We also observed that MM and structural modificationaround the TM atoms are correlated. The obtained value for themagnetic moment depends on the volume around the impurity,which affects intra-atomic exchange interactions and inter-atomic electronic motion due to the crystalline field. Moruzzi

[71] has already observed the same effect for other systemsincluding TMs. The atomic d-level in the octahedral symmetryof the Sn site of the rutile structure splits into two levels (t2g andeg) when replaced by Cr, where the energy difference betweenthese levels is called crystal field splitting. Each level, t2g

and eg , can be split still further due to intra-atomic exchangeinteractions. Therefore, the occurrence of the LS and HS statesdepends on the effective balance between these two interactionfields. In other words, the low-spin state occurs when thecrystal field splitting is larger than the intra-atomic exchangesplitting (lowest volume), and otherwise, the high-spin state is

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J. Phys. D: Appl. Phys. 47 (2014) 413001 Topical Review

the ground state (highest volume). Considering a sphericalvolume around the TM atom, we observe that after fullrelaxation the corresponding volume is reduced in all of thesystems studied. As discussed recently [18] a greater reductionin volume means lower values for the magnetic moment. Thecalculated reduced volumes obtained around each TM impurityare presented in table 4.

In the present section, we describe the study of the mag-netic properties of Sn1−xTMxO2 and Sn1−xTMxO2−y(VO)ywith x = 0.04 and y = 0.02 where TM=V, Cr, Mn, Fe, Coand Ni. The occurrence of MM was observed in mostsystems.Metastable states for Cr, Mn, Fe, Co and Ni were found. Thespin-crossover energies (SE) were calculated. In the case ofSn0.96TM0.04O1.98(VO)0.02 systems where an O vacancy wascreated close to the TM site, we show that the metastabilityand SE energy change and in all cases there is a strong in-fluence of these defects. For chromium, a new ground statearises with magnetic moment m = 4µB/cell and SE0/2

Cr re-duces to about 78%. For manganese, the metastable state of1µB/cell disappears and the state of 3µB/cell remains. Foriron, a new high-spin state appears, and the state close to zeromagnetic moment disappears. The ground state is 4µB/cellinstead of 2µB/cell. For cobalt, the ground state is found with3µB/cell and the metastable state with 1µB/cell. For vanadiumthe state of 1µB/cell remains and for nickel a high-spin stateappears with magnetic moment m = 2µB /cell. We shouldmention that recently we performed calculations for these sys-tems without taking into account the on-site Coulomb corre-lation interaction U [17–19], and different lattice parameters.Although some changes in the spin-crossover energy and mag-netic ground state are observed for Sn0.96Cr0.04O1.98(VO)0.02,similar behaviours for all cases were obtained.

4. Summary

We have reviewed our density functional theory results on thestructural, electronic and vibrational properties of the high-Ksemiconductor oxides SnO2, HfO2, ZrO2, TiO2 and SrTiO3

in their most stable structural phases, including dielectricproperties as derived from the electronic structure, and takinginto account the lattice contribution. It is shown that as far asthe static dielectric constant of these oxides is concerned, a verygood agreement between theory and experiment is reached,provided that, besides the electrons, the phonon contributionsare taken into account in the calculations. Moreover, thishigher lattice contribution is due to the great values of the Borneffective charges, together with the presence of transversaloptical phonons with frequencies smaller than 250 cm−1.

Finally, we presented our recent results for transitionmetal-doped SnO2, forming diluted magnetic alloys. Atheoretical prediction of spin-crossover in dilute magneticsemiconductors based on SnO2 was presented. Our resultssuggest that mainly Cr-, Fe- and Co-doped SnO2 in dilutedmagnetic alloy conditions, together with the presence of Ovacancies, may be used in devices for spintronic applicationsthat require different magnetization states.

Acknowledgments

We would like to thank A T Lino, V C Anjos, C C Silva, A RR Neto, T D Aguiar, M Marques, V N Freire, G A Farias andL V C Assali for the very fruitful collaboration over severalyears on the subject of this paper.

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