# Optical Properties of Al- and Zr-Doped Rutile Single Crystals Grown by Tilting-Mirror-Type Floating Zone Method and Study of Structure-Property Relationships by First Principle Calculations

Authors: Md. Abdur Razzaque Sarker
Journal: Journal of Inorganic Chemistry
 (2014)

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## Abstract

High quality and transparent single crystals of undoped rutile TiO2, Al-doped rutile (Al : TiO2), and Zr-doped rutile (Zr : TiO2) have been grown successfully by tilting-mirror-type floating zone (TMFZ) using travelling solvent floating (TSFZ) technique. The effect of doping on the electronic and optical properties of rutile has been studied experimentally as well as by simulation calculations. The effect of doping on the quality of crystals was also investigated by observing optical micrograph and measuring etch pits density that reveals the presence of defects. Undoped rutile crystals were dark blue and comprised many low-angle grain boundaries. Al+3 and Zr+4 ions pin down the migration of dislocations during the cooling and create oxygen vacancies. Doping of the impurities would improve the electronic and optical properties of rutile. The elastic properties might be changed for doping of the impurities in the rutile crystals.


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## Body

## 1. Introduction

Rutile (TiO2) has excellent optical, mechanical, and chemical properties, and the demand for rutile single crystals has therefore been increasing to use as polarizer in a variety of optical devices because of their large refractive indices and birefringence [1]. It is of urgent necessity to grow rutile single crystals of high optical quality. The floating zone (FZ) method is one of the most promising techniques, which replace the Verneuil method, to grow rutile single crystals. Rutile single crystals were conventionally manufactured by the Verneuil method for gemstones [2]. However, the quality of Verneuil-grown crystals is usually poor, since the crystals undergo quenching, which induces stress birefringence [3]. The optical quality of the FZ-grown crystals was superior to that of commercially available Verneuil-grown rutile crystals [4]. Floating zone (FZ) method is a powerful technique for growth of single crystals without contaminations because it is a crucible free zone melting method. The segregation control of the dopant is very easy in this method by applying the travelling solvent floating zone technique. Hence, high quality single crystals with required dopant concentration are possible to grow by this technique from both congruently and incongruently melting compounds. The manufacturing use of the FZ method, however, is limited because the diameter of the crystals grown by this method is usually smaller than that by the other melt growth techniques such as the Czochralski, the Bridgman, and the Verneuil methods.The shapes of both the melt-feed and the melt-crystal interfaces are convex in the FZ method and this convexity increases with increase in diameters of grown crystal and feed. This makes the instability of molten zone for the unexpected contacts between the feed rod and the grown crystal. The instability of the molten zone limits the diameter of the grown crystals in the conventional FZ method. It is reported that the interface shape can be controlled by tilting the mirror in the tilting-mirror-type floating zone (TMFZ) method as shown in Figure1. The stability of the molten zone increases with increase in tilting angle that is favorable for growth of large diameter single crystals [9, 10]. It was revealed that the quality of the TMFZ-grown crystals increases with increase in tilting angle [11] and growth diameter [10].Figure 1
Schematic illustration of the tilting-mirror-type image furnace. The definition of tilting angle (θ) is given.In this study, I try to grow high quality undoped and doped rutile single crystal by TMFZ method for optical device applications. The electrical and optical properties of this crystal are dominated by oxygen vacancies or incorporation of impurities without changing the crystallographic structure [12]. Many impurities have been added into rutile (TiO2) to improve its electrical and optical properties depending on its applications. Higuchi and Kodaira reported that the number of low-angle grain boundaries in the FZ-grown rutile was drastically reduced with the addition of ZrO2 or Al2O3 during growth [4]. They obtained that the grain boundary free rutile single crystals contain 1.0 mol% ZrO2 and Al2O3 [13]. Here, we will investigate the effect of Al and Zr doping on the quality of rutile single crystals grown by the TMFZ method. I expect that the proper dopant affects the density of crystal defects such as etch pits, which indicate the presence of dislocations and low-angle grain boundaries.The electronic and optical properties of TiO2 have been investigated experimentally using various methods. Nowadays, first principleab initio and density functionally theory (DFT) electronic structure calculations have been extensively used for interpretation of various properties of undoped and doped rutile. The defect structure varies with the oxygen vacancy concentration and metal impurities, which may alter electronic and optical properties of rutile and play an important role in technological applications of TiO2. Commercial TiO2 rutile pigments are always doped with Al2O3 to enhance photochemical stability. It is assumed that Al doping creates defects in the rutile lattice that act as traps for the photogenerated charges [8]. The addition of small quantity of ZrO2 was very effective to improve the quality of rutile crystals [4] although theoretical studies are rare.In this study, I have successfully grown the undoped and doped (Al, Zr) rutile TiO2 single crystals by infrared heating TMFZ method using travelling solvent floating zone technique. We also investigate the effects of aluminum and zirconium doping on optical and electronic properties of rutile experimentally as well as by first principle calculations.


## 2. Experiment

A new tilting-mirror-type image furnace (Crystal Systems Corporation, model TLFZ-4000-H-VPO) was developed for my growth experiment. The tilting angle (θ), as shown in Figure 1, can be changed to up to 30° by a motor drive control. High-purity rutile powder (>99.99%) was put into rubber tubes to form a rod shape and pressed at a pressure of up to 3.0 × 108 Pa using a cold isostatic pressing machine (Nikkiso Co., Ltd., model CL3-22-60). The rods were sintered at 1600°C for 12 h in an oxygen flow. The sintered rods were zone-passed in a CO2 flow at a growth rate of 25 mm/h using a standard type four-mirror-type-image furnace (Crystal Systems Corporation, model FZ-4000H). The zone-passed feed rods were typically 10–12 mm in diameter and 80–100 mm in length.For solvent preparation, high purity Al2O3 (99.99%) or ZrO2 (99.90%) was used with TiO2. Appropriate amounts of TiO2 and Al2O3 or ZrO2 were weighted out and mixed well. The concentration of Al or Zr in the solvent was 1.0 at %. The applied conditions for every growth experiment were a growth rate of 5 mm/h, a feeding rate of 5 mm/h, upper shaft rotation of 3 rpm, lower shaft rotation of 50 rpm, a growth direction of 〈001〉 using a rutile seed crystal, and an atmospheric-pressure oxygen flow. In case of undoped rutile, the growth atmosphere was CO2 flow because CO2 flow is used for the commercial production of the rutile crystals. The tilting angle θ was 20°. It was reported that this tilting condition was best for growth of good quality large diameter single crystals with the most stabilized molten zone. The grown crystals were cut perpendicular to the growth direction in the range 23–25 mm from the seeding point. Then the crystals were cut parallel to (110) and the surface was polished like mirror. The polished surfaces were observed by optical microscope (Olympus U-MSSPG Japan) to investigate the phase purity. The polished samples were soaked in a mixture of (NH4)2SO4 and H2SO4 solutions (1 : 1 weight ratio) for 3 h at 300°C to etch their surfaces. The polished samples of ~1.5 mm in thickness were characterized by UV-Vis spectrometer (JASCO, V-660). The Agilent 4294A precision impedance analyzer was used for FTIR absorption spectrum.


## 3. Computational Methods

The calculations presented in this work were carried out by employing Cambridge serial total energy package (CASTEP) code [14] which utilizes the plane-wave pseudopotential based on the framework of density functional theory (DFT) method. The electronic exchange-correlation energy is treated under the generalized gradient approximation (GGA) in the scheme of Perdew-Bruke-Ernzerhof [15]. The interactions between the ions and electron are represented by ultrasoft Vanderbilt-type pseudopotentials for Ti, Al, Zr, and O atoms [16]. The basis set of valence electronic states was set to be 3d24s2 for Ti, 3s23p1 for Al, 4d25s2 for Zr, and 2s22p4 for O. The calculations use plane-wave cut-off energy of 1100 eV for all cases. For sampling of the Brillouin zone, a Monkhorst-Pack grid [17] of 7 × 7 × 11 k-points was employed for all calculations. All the structures were relaxed by Broyden-Fletcher-Goldfarb-Shenno (BFGS) methods [18]. For the geometry optimization, the convergence tolerances were set as follows: 5 × 10−6eV/atom for the total energy, 0.001 eV/Å for the maximum force on atoms, 0.02 GPa for the maximum stress, and 5 × 10−4  Å for the maximum atomic displacement. The total energy is conversed to within 0.01 m Ry/unit cell during the self-consistency cycle.


## 4. Results and Discussions

### 4.1. Bulk Properties of Rutile

The single crystals of pure Al-doped and Zr-doped rutile TiO2 were stably grown by infrared heating tilting-mirror-type floating zone (TMFZ) method using travelling solvent floating zone (TSFZ) technique. Figure 2 shows the photographs of conventional FZ grown undoped rutile crystals and TMFZ grown undoped rutile, Al-doped rutile and Zr-doped rutile crystals. From the view of the grown crystals, TMFZ grown crystals look nice in comparison to FZ grown crystals. The transparency of the FZ grown crystal along growth direction is not homogeneous and seems to be some inclusions inside. Figure 2 shows that the FZ grown undoped rutile TiO2 crystal is blue and TMFZ grown undoped rutile TiO2 crystal is dark blue; on the other hand, Al-doped rutile crystal is pale-yellow and Zr-doped rutile crystal is light blue color. The color of undoped and Zr-doped rutile is uniform along the growth directions. However, the color of Al-doped rutile crystal is slightly changed along the growth direction. This means that the segregation of Zr is uniform and that of Al is slightly changed during TSFZ growth. It was revealed that maximum solubility of Al and Zr in rutile is 1 at % because it was difficult to maintain the molten zone as stable for solvent of more than 1 at % Al or Zr dopant.Photographs of the crystals of (a) pure rutile TiO2 grown by FZ method and (b) pure rutile TiO2, (c) Al : TiO2, and (d) Zr : TiO2 grown by TMFZ method.
(a)
(b)
(c)
(d)


### 4.2. Structural and Electronic Properties

The BFGS algorithm is used to perform geometry optimization for tetragonal rutile TiO2 at experimental lattice constant of a=b=4.594 Å and c=2.958 Å [5]. The equilibrium lattice parameters and elastic constants are calculated at the optimized structure that contains the lattice constants of a = b = 4.642  Å and c = 2.965 Å. The optimized structure parameters, a, c, and u, the cohesive energy per TiO2 unit Eu, the ground state energy E0, the bulk modulus B0, the shear modulus G0, and the band gap Eg obtained with Perdew-Wang exchange-correlation functional PWGGA [8] and plane-wave pseudopotential DFT based CASTEP are given in Table 1 together with the corresponding experimental values [6, 7, 19]. The calculated structural parameters obtained with theoretical calculations are close to the experimental values. The value of a increases by 0.21% for 1 at % Zr doping in Ti site. The values of c increase by 0.34% for 1 at % Zr doping and by 0.68% for 1 at % Al doping in Ti site. The calculated values of Eu obtained with theories are in good agreement with experimental value (−1915 kJ/mole). The largest deviation, 7.26%, is obtained from the PWGGA calculations. The values of ground state energy, E0, increase by 7 eV for 1 at % Zr doping and by 26 eV for 1 at % Al doping in Ti site.Table 1
Comparison of calculated structural parametersa, c, and u, binding energy Eu, ground state energy E0, bulk modulus B0, shear modulus G0, and band gap Eg with experiment.
Expt.
PWGGA
CASTEP[this work]
TiO2
TiO2
TiO2
Al : TiO2
Zr : TiO2
a (Å)
4.59a
4.63d
4.64
4.64
4.65
c (Å)
2.96a
2.98d
2.96
2.98
2.97
u
0.305
0.305d
0.305
0.305
0.305
E
u (kJ/mol.)
−1915a
−2054d
−2048
−2022
−2042
E
0 (eV)
⋯
⋯
−4963
−4937
−4956
B
0 (GPa)
216b
223d
201
204
214
G
0 (GPa)
⋯
⋯
123
118
123
E
g (eV)
3.03c
1.90d
1.88
1.94
1.85
aReference [5].
bReference [6].
cReference [7].
dReference [8].The calculated value of bulk modulus for undoped rutile TiO2 is 201 GPa, for Al : TiO2 is 204 GPa, and for Zr : TiO2 is 214 GPa which is in good agreement with experimental value of 216 GPa [7]. The shear modulus of pure rutile TiO2 and Zr : TiO2 is 123 GPa, whereas that of Al : TiO2 is 118 GPa. It is well known that a material is being ductile or brittle depending on the ratio of shear modulus and bulk modulus (G0/B0). If G0/B0 > 0.5, the material is ductile; if G0/B0 < 0.5, the material is brittle. Here, all three samples are ductile and the ductility decreases slightly for Al or Zr doping. The ductility is one of the important properties of rutile in the device applications.Rutile is a semiconductor with a band gap (Eg) of 3.03 eV [19]. In the present study, the band structure was computed along the direction that contains the highest number of symmetry points within the Brillouin zone, namely, Γ–X–R–Z–Γ–M–A–Z. The calculated band structures of the pure rutile TiO2, Al : TiO2, and Zr : TiO2 are presented in Figure 3. Here, the top of the valence band is chosen to be the Fermi level (Ef = 0 eV). The valence band maximum and conduction band minimum occur at Γ point, making this material a direct band gap material (Γ–Γ). The calculated Eg values are given in Table 1. The calculated band gaps of the pure rutile TiO2, Al : TiO2, and Zr : TiO2 are 1.88 eV, 1.94 eV, and 1.85 eV that are the same with band gap value of other theoretical calculations [8, 20, 21]. The Γ point conduction band minimum in rutile is almost degenerate with minima M and R. The band gap slightly narrowed (0.03 eV) for Zr doping and increased (0.06 eV) for Al doping in pure rutile TiO2. The valence bands and conduction bands are clearly split up at Z–Γ–M path. It is observed that there are some impurity bands in case of Al : TiO2 and Zr : TiO2.Band structure of (a) pure rutile TiO2, (b) Al : TiO2, and (c) Zr : TiO2. The Ef represents the Fermi level.
(a)
(b)
(c)Some important features of the band gaps are presented in Table2. The minimal vertical transition (MVT) and minimal indirect transition (MIT) energies are given in this table. The MVT is along Γ–Γ direction. Experimental investigations using absorption and photoluminescence spectroscopy [7] and wavelength modulated transmission spectroscopy [22] have also determined that rutile is a direct forbidden gap semiconductor. In agreement with some theoretical investigations [23, 24], our calculations also predict that rutile has the smallest indirect gap along the Γ–M or Γ–R direction. The MIT energy along Γ–M or Γ–R is 1.97 eV which is very close to the MVT energy (1.88 eV). The second MVT is along M–M with energy of 2.85 eV and the second MIT is along Γ–Z with energy of 2.86 eV which are also very close.Table 2
Comparison of direct and indirect band gaps energies (eV) and symmetry of the valance to conduction band transition for rutile TiO2, Al-doped rutile, and Zr-doped rutile.
Valance to conduction band transition
Band gap energies (eV)
TiO2
Al : TiO2
Zr : TiO2
Γ
–
Γ
1.8805
1.9411
1.8529
Γ–Z
2.8656
3.0882
2.9117
Γ–R
1.9701
2.1176
1.9411
Γ–M
1.9701
2.1176
1.9411
X–X
3.2238
3.3529
3.1764
R–R
2.8656
3.0882
2.9117
Z–Z
4.2985
4.4117
4.4117
M–M
2.8546
3.0882
2.9117
A–A
3.2238
3.3529
3.2647In order to get more physical insight of the electronic band structure, we calculated the total density of states (TDOS) and partial density of states (PDOS) of pure rutile TiO2, Al : TiO2, and Zr : TiO2. A comparison of TDOS is shown in Figure 4(a). The TDOS is extended from about −5.92 eV to 7.78 eV where the Fermi-level position is set to 0 eV. Figure 4(a) shows that TDOS of Zr : TiO2 shifted slightly and that of Al : TiO2 shifted effectively from TDOS of TiO2. The density of states at the Fermi level is 0.441, 0.664, and 0.446 states/eV for pure rutile TiO2, Al : TiO2, and Zr : TiO2, respectively. Every DOS value at Fermi level reveals that rutile is a perfect semiconductor.(a) Total DOS of the pure and doped rutile. Partial DOS of (b) pure rutile TiO2, (c) Al :  TiO2, and (d) Zr : TiO2 crystals.
(a)
(b)
(c)
(d)Figure4(b) shows that the valance band (VB) and conduction band (CB) are composed of both Ti 3d and O 2p orbitals. The VB is mainly composed of O 2p states with some hybridization with Ti 3d orbitals and the top of the valence band is dominated by O 2p states. On the other hand, the conduction band has Ti 3d states with weak hybridization of O 2p states; and Ti 3d states are dominant here over O 2p states. It reveals that the transition between VB and CB is due to the O 2p and Ti 3d states, because these states have maximum contribution to VB and CB. The valance-band width of 6.29 eV is in good agreement with experimental results [25, 26]. On the other hand, the conduction bands have a width of 5.92 eV. The two peaks in the conduction band are a result of the ligand field splitting of the Ti 3dorbitals into two sets, t2g and eg orbitals, where eg orbitals are associated with upper conduction band [27]. The t2g and eg states represent the π and σ Ti 3d orbitals hybridized with O 2p states. Figure 4(c) shows the partial DOS of Al : TiO2 and 4(d) shows that of Zr : TiO2. In the Al : TiO2, the contribution of Al 3p is larger in the valance band than in the conduction band and the contribution of Al 2s is insignificant. In the Zr : TiO2, contribution of Zr 4p is significant in the valance band and that of Zr 4dis significant in the conduction band.Figure5 show the optical micrographs of conventional IR-FZ grown undoped TiO2 crystal and TMFZ grown undoped TiO2, Al : TiO2 and Zr : TiO2 crystals, respectively. The samples were polished mechanically like mirror on (110) surface and observed by optical microscopy. In a recent work, we showed that growth of large diameter single crystal is possible by TMFZ method [10] and quality of the grown crystals improved here [11]. In this work, I tried to compare the quality of the single crystals grown by the conventional FZ and TMFZ methods. In Figure 5(a), there are some grain boundaries in the FZ grown crystals, while Figures 5(b)-5(a) show no grain boundaries or secondary phases. Although little impurity spots were observed in Figure 5(b) for undoped rutile crystals, Figures 5(c) and 5(d) show that Al : TiO2 and Zr : TiO2 crystals grown by TMFZ method were almost defect-free. These results reveal that TMFZ is a powerful method for growth of high quality single crystals. To investigate the quality of the grown crystals in more details, I analyze the defects by measuring etch pits density (EPD). Figure 6 shows optical microphotographs of the samples on (110) surface after chemical etching. The observed etch pits are clearly more in the crystal grown by conventional FZ method compared with that in the crystal grown by TMFZ method as shown in Figures 6(a) and 6(b), respectively. The EPD in the crystals grown by conventional FZ method is 3.313 × 104 cm−2 and that in the crystals grown by the TMFZ (θ = 20°) method is 1.157 × 104 cm−2. So EPD was reduced by 50% with TMFZ growth. The EPDs also reduced for doping effects. The TMFZ (θ = 20°) grown Al : TiO2 and Zr : TiO2 have EPDs as 0.57 × 104 cm−2 as shown in Figures 6(c)–6(d), that is, 50% less compared to the EPDs in undoped TiO2 as shown in Figure 6(b). As the etch pits are associated with dislocations [28] and grain boundaries [29], the quality of the rutile crystals can be improved by tilting the mirror in TMFZ method as well as by doping the proper dopants. Hence, the doping of suitable foreign particles is effective to decrease the defects such as dislocations and low-angle grain boundaries in the rutile crystals.Optical micrographs of polished cross-section of (a) undoped rutile crystal by IR-FZ method and (b) undoped TiO2, (c) Al : TiO2, and(d) Zr : TiO2 crystals by TMFZ method.
(a)
(b)
(c)
(d)Optical microphotographs of etch pits on (100) of rutile single crystals grown by (a) conventional FZ method, (b) TMFZ method, (c) Al : TiO2, and (d) Zr : TiO2.
(a)
(b)
(c)
(d)The ac conductance of the samples was measured by impedance analyzer, where the signal frequency was varied from 100 Hz to 10 M Hz and oscillating voltage 300 mV was applied. Figure7(a) shows the frequency dependence ac conductance of doped and undoped TiO2 samples where the ac conductance increased with frequency up to 5 MHz and then decreased. The ac conductance is higher in undoped TiO2 sample than Al : TiO2 and Zr : TiO2 samples. Hatta et al. reported that the conductivity of Al : TiO2 crystal was larger by one order of magnitude than that of the pure crystal, which indicates that the diffusion rate of oxygen ions is much higher in the Al : TiO2 crystal than in the pure crystal [30]. In this work, it is shown that the frequency response ac conductance strongly depends on the doping effects of rutile. The impedance was also measured at frequency range of 100 Hz to 10 MHz by impedance analyzer. Figure 7(b) shows the frequency dependence impedance of doped and undoped TiO2 samples. The ac conductance is lower in undoped TiO2 sample than in Al : TiO2 and Zr : TiO2 samples.The frequency dependence of electrical conductance and impedance of undoped and doped rutile TiO2.
(a)
(b)As the impedance is opposite to ac conductance, the impedance also strongly depends on the doping effects of rutile.


### 4.3. Optical Properties of Rutile

Dielectric properties of the doped and undoped rutile TiO2 samples were studied by precision impedance analyzer. Silver paste was coated on both surfaces of each sample before measurement. Dielectric constant is a measure of materials ability to store electric charge. The frequency dependence capacitance of samples was measured within the frequency range of 100 Hz to 10 MHz at room temperature as shown in Figure 8(a). The capacitance of all samples is high at low frequency region and decreases with increase in frequency. The capacitance is higher in undoped TiO2 sample than in Al : TiO2 and Zr : TiO2 samples. Figure 8(b) shows the frequency dependence dielectric constant of the doped and undoped rutile TiO2 samples. The capacitance of the samples is high at low frequency region due to contribution of all kinds of polarization at low frequency and then decreases with increase in frequency. This is due to the change of space charge and ionic and orientation polarization at higher frequencies [31]. The dielectric constant is higher in undoped TiO2 sample than in Al : TiO2 and Zr : TiO2 samples. Frequency response capacitance and dielectric constant strongly depend on the doping effects on the TiO2 rutile crystals.The frequency dependence of capacitance and dielectric constant of undoped and doped rutile TiO2.
(a)
(b)The transmittance data of the disc-like samples with 1.5 mm in thickness is shown in Figure9. The intensity increased for the doping effect of Al and Zr on the pure rutile TiO2. This result is consistent with the color change of grown crystals as shown in Figure 2. This change corresponds to the change of oxygen deficiencies. The band edge is 420 nm for all samples, which is independent of the Al or Zr dopant. Although impurity bands are observed in band structure of Al : TiO2 and Zr : TiO2 as shown in Figures 3(b) and 3(c), it was insignificant and cannot influence the band edge. The transmission peak of pure rutile TiO2 is at 468 nm and that of Zr : TiO2 is at 549 nm, but the peak is wide. However, the transmission peak of Al : TiO2 is at 850 nm which is out of the visible region.Figure 9
Transmittance as a function of wavelength of light for undoped and doped rutile TiO2.Figure10 shows the Fourier transform infrared (FTIR) absorption spectra of pure and aluminum doped rutile powder samples. It shows that the spectra for pure rutile and Al : TiO2 are not identical because the dopant influences the chemical bonding and bond length and structure. At ~499 cm−1 of TiO2 absorption spectrum, there is a sharp peak, but at 704 cm−1, there is a broad peak due to the formation of Ti–O bonds. At ~664 cm−1 of Al : TiO2 absorption spectrum, there is a broad peak due to formation of Ti–O bonds as well as Al–O bonds since Al was doped in the Ti site. TiO2 sample shows that there are three sharp medium absorption peaks at 1084 cm−1, 1124 cm−1, and 1197 cm−1 which are all due to the vibrations of the rutile crystal lattice. The Al : TiO2 sample shows that there is a broad weak absorption peak at 1125 cm−1. Two medium peaks for both samples overlap at 1650 cm−1. There is a sharp strong peak at 3448 cm−1 for TiO2 sample and broad strong peak at 3479 cm−1 for Al : TiO2 sample; both absorption peaks are due to O–O bonds. The average absorption intensity of TiO2 sample is higher than that of Al : TiO2.Figure 10
FTIR absorption spectra of the (a) pure rutile TiO2 and (b) Al : TiO2.


## 4.1. Bulk Properties of Rutile

The single crystals of pure Al-doped and Zr-doped rutile TiO2 were stably grown by infrared heating tilting-mirror-type floating zone (TMFZ) method using travelling solvent floating zone (TSFZ) technique. Figure 2 shows the photographs of conventional FZ grown undoped rutile crystals and TMFZ grown undoped rutile, Al-doped rutile and Zr-doped rutile crystals. From the view of the grown crystals, TMFZ grown crystals look nice in comparison to FZ grown crystals. The transparency of the FZ grown crystal along growth direction is not homogeneous and seems to be some inclusions inside. Figure 2 shows that the FZ grown undoped rutile TiO2 crystal is blue and TMFZ grown undoped rutile TiO2 crystal is dark blue; on the other hand, Al-doped rutile crystal is pale-yellow and Zr-doped rutile crystal is light blue color. The color of undoped and Zr-doped rutile is uniform along the growth directions. However, the color of Al-doped rutile crystal is slightly changed along the growth direction. This means that the segregation of Zr is uniform and that of Al is slightly changed during TSFZ growth. It was revealed that maximum solubility of Al and Zr in rutile is 1 at % because it was difficult to maintain the molten zone as stable for solvent of more than 1 at % Al or Zr dopant.Photographs of the crystals of (a) pure rutile TiO2 grown by FZ method and (b) pure rutile TiO2, (c) Al : TiO2, and (d) Zr : TiO2 grown by TMFZ method.
(a)
(b)
(c)
(d)


## 4.2. Structural and Electronic Properties

The BFGS algorithm is used to perform geometry optimization for tetragonal rutile TiO2 at experimental lattice constant of a=b=4.594 Å and c=2.958 Å [5]. The equilibrium lattice parameters and elastic constants are calculated at the optimized structure that contains the lattice constants of a = b = 4.642  Å and c = 2.965 Å. The optimized structure parameters, a, c, and u, the cohesive energy per TiO2 unit Eu, the ground state energy E0, the bulk modulus B0, the shear modulus G0, and the band gap Eg obtained with Perdew-Wang exchange-correlation functional PWGGA [8] and plane-wave pseudopotential DFT based CASTEP are given in Table 1 together with the corresponding experimental values [6, 7, 19]. The calculated structural parameters obtained with theoretical calculations are close to the experimental values. The value of a increases by 0.21% for 1 at % Zr doping in Ti site. The values of c increase by 0.34% for 1 at % Zr doping and by 0.68% for 1 at % Al doping in Ti site. The calculated values of Eu obtained with theories are in good agreement with experimental value (−1915 kJ/mole). The largest deviation, 7.26%, is obtained from the PWGGA calculations. The values of ground state energy, E0, increase by 7 eV for 1 at % Zr doping and by 26 eV for 1 at % Al doping in Ti site.Table 1
Comparison of calculated structural parametersa, c, and u, binding energy Eu, ground state energy E0, bulk modulus B0, shear modulus G0, and band gap Eg with experiment.
Expt.
PWGGA
CASTEP[this work]
TiO2
TiO2
TiO2
Al : TiO2
Zr : TiO2
a (Å)
4.59a
4.63d
4.64
4.64
4.65
c (Å)
2.96a
2.98d
2.96
2.98
2.97
u
0.305
0.305d
0.305
0.305
0.305
E
u (kJ/mol.)
−1915a
−2054d
−2048
−2022
−2042
E
0 (eV)
⋯
⋯
−4963
−4937
−4956
B
0 (GPa)
216b
223d
201
204
214
G
0 (GPa)
⋯
⋯
123
118
123
E
g (eV)
3.03c
1.90d
1.88
1.94
1.85
aReference [5].
bReference [6].
cReference [7].
dReference [8].The calculated value of bulk modulus for undoped rutile TiO2 is 201 GPa, for Al : TiO2 is 204 GPa, and for Zr : TiO2 is 214 GPa which is in good agreement with experimental value of 216 GPa [7]. The shear modulus of pure rutile TiO2 and Zr : TiO2 is 123 GPa, whereas that of Al : TiO2 is 118 GPa. It is well known that a material is being ductile or brittle depending on the ratio of shear modulus and bulk modulus (G0/B0). If G0/B0 > 0.5, the material is ductile; if G0/B0 < 0.5, the material is brittle. Here, all three samples are ductile and the ductility decreases slightly for Al or Zr doping. The ductility is one of the important properties of rutile in the device applications.Rutile is a semiconductor with a band gap (Eg) of 3.03 eV [19]. In the present study, the band structure was computed along the direction that contains the highest number of symmetry points within the Brillouin zone, namely, Γ–X–R–Z–Γ–M–A–Z. The calculated band structures of the pure rutile TiO2, Al : TiO2, and Zr : TiO2 are presented in Figure 3. Here, the top of the valence band is chosen to be the Fermi level (Ef = 0 eV). The valence band maximum and conduction band minimum occur at Γ point, making this material a direct band gap material (Γ–Γ). The calculated Eg values are given in Table 1. The calculated band gaps of the pure rutile TiO2, Al : TiO2, and Zr : TiO2 are 1.88 eV, 1.94 eV, and 1.85 eV that are the same with band gap value of other theoretical calculations [8, 20, 21]. The Γ point conduction band minimum in rutile is almost degenerate with minima M and R. The band gap slightly narrowed (0.03 eV) for Zr doping and increased (0.06 eV) for Al doping in pure rutile TiO2. The valence bands and conduction bands are clearly split up at Z–Γ–M path. It is observed that there are some impurity bands in case of Al : TiO2 and Zr : TiO2.Band structure of (a) pure rutile TiO2, (b) Al : TiO2, and (c) Zr : TiO2. The Ef represents the Fermi level.
(a)
(b)
(c)Some important features of the band gaps are presented in Table2. The minimal vertical transition (MVT) and minimal indirect transition (MIT) energies are given in this table. The MVT is along Γ–Γ direction. Experimental investigations using absorption and photoluminescence spectroscopy [7] and wavelength modulated transmission spectroscopy [22] have also determined that rutile is a direct forbidden gap semiconductor. In agreement with some theoretical investigations [23, 24], our calculations also predict that rutile has the smallest indirect gap along the Γ–M or Γ–R direction. The MIT energy along Γ–M or Γ–R is 1.97 eV which is very close to the MVT energy (1.88 eV). The second MVT is along M–M with energy of 2.85 eV and the second MIT is along Γ–Z with energy of 2.86 eV which are also very close.Table 2
Comparison of direct and indirect band gaps energies (eV) and symmetry of the valance to conduction band transition for rutile TiO2, Al-doped rutile, and Zr-doped rutile.
Valance to conduction band transition
Band gap energies (eV)
TiO2
Al : TiO2
Zr : TiO2
Γ
–
Γ
1.8805
1.9411
1.8529
Γ–Z
2.8656
3.0882
2.9117
Γ–R
1.9701
2.1176
1.9411
Γ–M
1.9701
2.1176
1.9411
X–X
3.2238
3.3529
3.1764
R–R
2.8656
3.0882
2.9117
Z–Z
4.2985
4.4117
4.4117
M–M
2.8546
3.0882
2.9117
A–A
3.2238
3.3529
3.2647In order to get more physical insight of the electronic band structure, we calculated the total density of states (TDOS) and partial density of states (PDOS) of pure rutile TiO2, Al : TiO2, and Zr : TiO2. A comparison of TDOS is shown in Figure 4(a). The TDOS is extended from about −5.92 eV to 7.78 eV where the Fermi-level position is set to 0 eV. Figure 4(a) shows that TDOS of Zr : TiO2 shifted slightly and that of Al : TiO2 shifted effectively from TDOS of TiO2. The density of states at the Fermi level is 0.441, 0.664, and 0.446 states/eV for pure rutile TiO2, Al : TiO2, and Zr : TiO2, respectively. Every DOS value at Fermi level reveals that rutile is a perfect semiconductor.(a) Total DOS of the pure and doped rutile. Partial DOS of (b) pure rutile TiO2, (c) Al :  TiO2, and (d) Zr : TiO2 crystals.
(a)
(b)
(c)
(d)Figure4(b) shows that the valance band (VB) and conduction band (CB) are composed of both Ti 3d and O 2p orbitals. The VB is mainly composed of O 2p states with some hybridization with Ti 3d orbitals and the top of the valence band is dominated by O 2p states. On the other hand, the conduction band has Ti 3d states with weak hybridization of O 2p states; and Ti 3d states are dominant here over O 2p states. It reveals that the transition between VB and CB is due to the O 2p and Ti 3d states, because these states have maximum contribution to VB and CB. The valance-band width of 6.29 eV is in good agreement with experimental results [25, 26]. On the other hand, the conduction bands have a width of 5.92 eV. The two peaks in the conduction band are a result of the ligand field splitting of the Ti 3dorbitals into two sets, t2g and eg orbitals, where eg orbitals are associated with upper conduction band [27]. The t2g and eg states represent the π and σ Ti 3d orbitals hybridized with O 2p states. Figure 4(c) shows the partial DOS of Al : TiO2 and 4(d) shows that of Zr : TiO2. In the Al : TiO2, the contribution of Al 3p is larger in the valance band than in the conduction band and the contribution of Al 2s is insignificant. In the Zr : TiO2, contribution of Zr 4p is significant in the valance band and that of Zr 4dis significant in the conduction band.Figure5 show the optical micrographs of conventional IR-FZ grown undoped TiO2 crystal and TMFZ grown undoped TiO2, Al : TiO2 and Zr : TiO2 crystals, respectively. The samples were polished mechanically like mirror on (110) surface and observed by optical microscopy. In a recent work, we showed that growth of large diameter single crystal is possible by TMFZ method [10] and quality of the grown crystals improved here [11]. In this work, I tried to compare the quality of the single crystals grown by the conventional FZ and TMFZ methods. In Figure 5(a), there are some grain boundaries in the FZ grown crystals, while Figures 5(b)-5(a) show no grain boundaries or secondary phases. Although little impurity spots were observed in Figure 5(b) for undoped rutile crystals, Figures 5(c) and 5(d) show that Al : TiO2 and Zr : TiO2 crystals grown by TMFZ method were almost defect-free. These results reveal that TMFZ is a powerful method for growth of high quality single crystals. To investigate the quality of the grown crystals in more details, I analyze the defects by measuring etch pits density (EPD). Figure 6 shows optical microphotographs of the samples on (110) surface after chemical etching. The observed etch pits are clearly more in the crystal grown by conventional FZ method compared with that in the crystal grown by TMFZ method as shown in Figures 6(a) and 6(b), respectively. The EPD in the crystals grown by conventional FZ method is 3.313 × 104 cm−2 and that in the crystals grown by the TMFZ (θ = 20°) method is 1.157 × 104 cm−2. So EPD was reduced by 50% with TMFZ growth. The EPDs also reduced for doping effects. The TMFZ (θ = 20°) grown Al : TiO2 and Zr : TiO2 have EPDs as 0.57 × 104 cm−2 as shown in Figures 6(c)–6(d), that is, 50% less compared to the EPDs in undoped TiO2 as shown in Figure 6(b). As the etch pits are associated with dislocations [28] and grain boundaries [29], the quality of the rutile crystals can be improved by tilting the mirror in TMFZ method as well as by doping the proper dopants. Hence, the doping of suitable foreign particles is effective to decrease the defects such as dislocations and low-angle grain boundaries in the rutile crystals.Optical micrographs of polished cross-section of (a) undoped rutile crystal by IR-FZ method and (b) undoped TiO2, (c) Al : TiO2, and(d) Zr : TiO2 crystals by TMFZ method.
(a)
(b)
(c)
(d)Optical microphotographs of etch pits on (100) of rutile single crystals grown by (a) conventional FZ method, (b) TMFZ method, (c) Al : TiO2, and (d) Zr : TiO2.
(a)
(b)
(c)
(d)The ac conductance of the samples was measured by impedance analyzer, where the signal frequency was varied from 100 Hz to 10 M Hz and oscillating voltage 300 mV was applied. Figure7(a) shows the frequency dependence ac conductance of doped and undoped TiO2 samples where the ac conductance increased with frequency up to 5 MHz and then decreased. The ac conductance is higher in undoped TiO2 sample than Al : TiO2 and Zr : TiO2 samples. Hatta et al. reported that the conductivity of Al : TiO2 crystal was larger by one order of magnitude than that of the pure crystal, which indicates that the diffusion rate of oxygen ions is much higher in the Al : TiO2 crystal than in the pure crystal [30]. In this work, it is shown that the frequency response ac conductance strongly depends on the doping effects of rutile. The impedance was also measured at frequency range of 100 Hz to 10 MHz by impedance analyzer. Figure 7(b) shows the frequency dependence impedance of doped and undoped TiO2 samples. The ac conductance is lower in undoped TiO2 sample than in Al : TiO2 and Zr : TiO2 samples.The frequency dependence of electrical conductance and impedance of undoped and doped rutile TiO2.
(a)
(b)As the impedance is opposite to ac conductance, the impedance also strongly depends on the doping effects of rutile.


## 4.3. Optical Properties of Rutile

Dielectric properties of the doped and undoped rutile TiO2 samples were studied by precision impedance analyzer. Silver paste was coated on both surfaces of each sample before measurement. Dielectric constant is a measure of materials ability to store electric charge. The frequency dependence capacitance of samples was measured within the frequency range of 100 Hz to 10 MHz at room temperature as shown in Figure 8(a). The capacitance of all samples is high at low frequency region and decreases with increase in frequency. The capacitance is higher in undoped TiO2 sample than in Al : TiO2 and Zr : TiO2 samples. Figure 8(b) shows the frequency dependence dielectric constant of the doped and undoped rutile TiO2 samples. The capacitance of the samples is high at low frequency region due to contribution of all kinds of polarization at low frequency and then decreases with increase in frequency. This is due to the change of space charge and ionic and orientation polarization at higher frequencies [31]. The dielectric constant is higher in undoped TiO2 sample than in Al : TiO2 and Zr : TiO2 samples. Frequency response capacitance and dielectric constant strongly depend on the doping effects on the TiO2 rutile crystals.The frequency dependence of capacitance and dielectric constant of undoped and doped rutile TiO2.
(a)
(b)The transmittance data of the disc-like samples with 1.5 mm in thickness is shown in Figure9. The intensity increased for the doping effect of Al and Zr on the pure rutile TiO2. This result is consistent with the color change of grown crystals as shown in Figure 2. This change corresponds to the change of oxygen deficiencies. The band edge is 420 nm for all samples, which is independent of the Al or Zr dopant. Although impurity bands are observed in band structure of Al : TiO2 and Zr : TiO2 as shown in Figures 3(b) and 3(c), it was insignificant and cannot influence the band edge. The transmission peak of pure rutile TiO2 is at 468 nm and that of Zr : TiO2 is at 549 nm, but the peak is wide. However, the transmission peak of Al : TiO2 is at 850 nm which is out of the visible region.Figure 9
Transmittance as a function of wavelength of light for undoped and doped rutile TiO2.Figure10 shows the Fourier transform infrared (FTIR) absorption spectra of pure and aluminum doped rutile powder samples. It shows that the spectra for pure rutile and Al : TiO2 are not identical because the dopant influences the chemical bonding and bond length and structure. At ~499 cm−1 of TiO2 absorption spectrum, there is a sharp peak, but at 704 cm−1, there is a broad peak due to the formation of Ti–O bonds. At ~664 cm−1 of Al : TiO2 absorption spectrum, there is a broad peak due to formation of Ti–O bonds as well as Al–O bonds since Al was doped in the Ti site. TiO2 sample shows that there are three sharp medium absorption peaks at 1084 cm−1, 1124 cm−1, and 1197 cm−1 which are all due to the vibrations of the rutile crystal lattice. The Al : TiO2 sample shows that there is a broad weak absorption peak at 1125 cm−1. Two medium peaks for both samples overlap at 1650 cm−1. There is a sharp strong peak at 3448 cm−1 for TiO2 sample and broad strong peak at 3479 cm−1 for Al : TiO2 sample; both absorption peaks are due to O–O bonds. The average absorption intensity of TiO2 sample is higher than that of Al : TiO2.Figure 10
FTIR absorption spectra of the (a) pure rutile TiO2 and (b) Al : TiO2.


## 5. Conclusions

High quality rutile single crystals free from low-angle grain boundaries were successfully grown with addition of 1 at % Al2O3 and ZrO2 by infrared heating tilting-mirror-type floating zone (TMFZ) method using travelling solvent floating zone (TSFZ) technique. The structural and electronic properties of pure and doped rutile have been analyzed by first principle calculations based on DFT theory using CASTEP code. The calculated structural and electronic parameters are in good agreement with experimental values. The ductility of the rutile single crystals decreases slightly for Al or Zr doping. The conventional FZ grown rutile crystal contains defects such as dislocations and low-angle grain boundaries. However, Al- or Zr-doped rutile single crystals grown by TMFZ method are free from defects. Therefore, the addition of Al or Zr in the rutile TiO2 is the most effective to suppress the defects in order to improve the quality of the grown crystals. The transmission peak of the Al : TiO2 is far from that of pure rutile. It was revealed that, in case of Al doping, substitution of two Ti4+ by Al3+ introduces an oxygen vacancy in the rutile TiO2 that would change the color.



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