On the dielectric constant of titanium dioxide

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Introduction
The static and low-frequency dielectric constant, , of rutile TiO2 has been surrounded with controversy since as early as 1952, when Nicolini [1] reported an extremely high value of around 10,000 for the dielectric constant of ceramic rutile TiO2.Similar values of (1 − 3) × 10 4 were later observed by Parker and Wasilik [2] for single crystalline rutile TiO2.It was immediately realized, however, that such high values may be a result of an incorrectly designed experiment or an incorrect interpretation.Indeed, the method used in Ref. [2] is based on capacitance measurements over the full thickness of the crystal, where metallic contacts are deposited on opposite sides.Parker and Wasilik [2] suggested that, in case of non-negligible free carrier concentration, i.e., non-negligible effective net doping, Schottky contacts can be unintentionally formed.This is particularly relevant in the case of reduced TiO2, where oxygen vacancies give rise to ntype doping.In such samples, the total capacitance of the crystal is determined predominantly by the depletion region associated with the Schottky contact, or "by an electron-deficient barrier layer at the electrode-crystal interface" as formulated by Parker and Wasilik [2].
Based on these findings, Parker [3,4] performed new theoretical and experimental studies of  in rutile TiO2 crystals.Special care was taken to prepare highly resistive TiO2 by "heavy oxidation" as phrased in Ref. [3].Highly resistive, oxygen-rich rutile TiO2 crystals were then investigated by measuring the capacitance between a parallel plate capacitor with the crystal inside.Dielectric constants of 170 and 86 were measured at 300 K along the c-and a-axes, respectively.At 1.6 K, the dielectric constants along the c-and a-axis were deduced to be 257 and 111, respectively.No frequency dependence was observed in the frequency range 10 2 -3x10 6 Hz.
Later, Samara and Peercy [5] measured pressure and temperature dependencies of .Similar to the previous investigations,  was determined from capacitance measurements.No frequency dependence was assumed, based on the findings by Parker [3], and the measurements were performed at 100 kHz.It has been shown that data for the dielectric constant, ε, can be fitted over the whole temperature range by the modified Curie-Weiss law derived first by Barrett [6] for perovskite-type crystals: where T is temperature, and A0, C0, T1 and T0 are fitting parameters.At 296 K, the dielectric constant along the c-and a-axes was determined to be 166.7 and 89.8, respectively.As the temperature was decreased to 4 K, the dielectric constants increased to 251 along the c-axis and to 114.9 along the a-axis.These values are close to those determined by Parker [3].
Reports on extremely high  of reduced rutile TiO2 continue to appear in the literature.For example, Chu [7] has reported values in a range of 100 − 10000.The dielectric constants were deduced from impedance measurements over the full crystal thickness, with gold contacts deposited on opposite sides of the sample.
This concept is similar to that used in the earlier studies [2].Recently, Li et al. [8] reported a colossal dielectric permittivity in hydrogen-reduced rutile TiO2 crystals.Similar to earlier studies,  was deduced from impedance measurements of the crystal with silver contacts.One can notice, however, that neither Chu [7] nor Li et al. [8] have considered formation of Schottky barriers at the metal-TiO2 interface and the corresponding depletion regions, which was considered by Parker and Wasilik [2].

Results and discussion
Annealing of TiO2 in reducing and hydrogen-rich atmosphere has long been known to result in conductive, n-type material (see, for instance, Ref. [9]).Two main mechanisms are believed to be responsible: (i) formation of donors assigned to oxygen vacancies (VO) and titanium interstitials (Tii) in the reducing atmosphere [10] and (ii) introduction of interstitial hydrogen (Hi) donors during hydrogenation [11,12].
Besides, Hi can interact with acceptors and passivate them, increasing the net n-type conductivity.We have demonstrated previously [13,14] that annealing of TiO2 in N2 at 1100-1200 o C or in forming gas (FG), 10%at.H2 and 90%at.N2, at 600 o C leads to increase in conductivity.In the case of heat treatment in FG, the increase can be correlated with concentration of Hi.For annealing in N2, the increase in conductivity occurs without a corresponding increase in hydrogen concentration.These observations support the feasibility of the two mechanisms that involve VO/Tii and Hi.The effect of hydrogen-induced donors was used to form a pronounced donor profile at a well-defined depth by ion implantation.
Capacitance-voltage (CV) measurements are a well-established technique for probing the depth distribution of donors and acceptors.Fig. 1 depicts CV measurements at different temperatures (Tmeas) for a sample annealed in N2 at 1100 • C for 60 min (TiO2-N2).The red, dotted curve represents measurements performed at 300 K on the as-prepared, conductive sample, i.e., after the N2 heat treatment but prior to H + implantation.
Blue, solid curves represent measurements recorded after the sample was implanted with 200-keV H + to a dose of 3 × 10 13 cm −2 .The different blue curves represent measurements recorded at different Tmeas.Fig. 1(a) shows that capacitance of the Schottky diode decreases with increasing reverse voltage, V, as expected for a diode.Within the depletion approximation [15], one can derive the following expression: where N(V) is the doping concentration at the depth of the depletion region for a given V, ε0 is the vacuum permittivity, ε is the relative permittivity, or dielectric constant, C is the capacitance at the given V, q is the electron charge and A is the area of the diode.Since ε0 and ε do not depend on V, the product N(V)ε0ε will maintain the shape of N(V).These voltages correspond to the capacitance values indicated in Fig. 1(a) with filled circles.For example, for measurements at 100 K, the reverse bias Vpeak=−3.8V (voltage when the depletion region reaches the donor peak concentration) corresponds to Cpeak= 160 pF.Similarly, for measurements at 250 K, the reverse bias Vpeak=−6.2V (donor peak concentration) results in Cpeak= 131 pF.On the other hand, the depth of the concentration peak for hydrogen-induced donors is known from secondary ion mass spectrometry (SIMS) measurements: dpeak = 0.97 µm (see Supplementary Material).
Within the depletion approximation, the capacitance, C, and the depletion depth, d, are related as C=0A/d.
One can thus find ε from: The results of the analysis from Fig. 1 and Eq. ( 3) are summarized in Table I.No frequency dependence has been observed within the range between 1 kHz and 1 MHz (see Supplementary Material).
It should be noted, however, that in some cases we could not perform this analysis at the given experimental conditions.This is illustrated in Fig. 1(b), where the curve for Tmeas = 300 K does not reveal a well-defined donor concentration peak.The analysis described by Fig. 1 and Eqs. ( 2) and (3) has been applied for more detailed studies of ε in a number of samples annealed in FG (TiO2-FG) and the TiO2-N2 samples.Fig. 2 demonstrates that ε determined in the present study is very close to that determined for insulating, oxygen-rich rutile TiO2 by Samara and Peercy [5].For reduced TiO2, ε along the c-axis is in the range 160 − 240 for temperatures 50 -300 K.It decreases as temperature increases and can be described by the modified Curie-Weiss law.We do not observe a significant difference in ε between reduced TiO2 (annealed in N2 at 1100 • C) and reducedhydrogenated TiO2 (annealed in FG at 600 • C).

Conclusion
In conclusion, measurements of the static dielectric constant, ε, in conductive TiO2 are challenging and causing controversy.We propose a method for deducing ε from capacitance measurements.The method involves formation of Schottky barrier diodes and hydrogen implantation.The implantation results in a well-pronounced donor concentration profile, corresponding to the implanted hydrogen profile.The donor profile is then characterized using capacitance-voltage measurements, and ε can be deduced.We observe that ε of reduced, conductive rutile TiO2 is similar to that of oxygen-rich, insulating rutile TiO2 established previously.We can not confirm claims of colossal dielectric permittivity in hydrogenated and reduced rutile
Conductive n-type TiO2 samples of bluish colour were obtained by heat treatments in forming gas (FG) flow (N2 + H2 with [H2]/[N2] ≈ 1/9) at 600 • C for 90 min (hydrogenating and reducing heat treatment) or in N2 flow at 1100 • C for 60 min (reducing heat treatment).After the heat treatments, circular 150-nm thick Pd contacts with a diameter of around 400 μm were deposited through a shadow mask, resulting in Schottky barrier diodes with a rectification of up to eight orders of magnitude [ 17 ].After initial electrical measurements, the samples were implanted at room temperature with 200-keV H + ions to different doses in the range 6×10 12 -3×10 14 cm −2 .

B. Experimental set-up
Secondary ion mass spectrometry (SIMS) measurements were performed using a Cameca IMS 7f spectrometer with a primary beam of 15-keV Cs + ions.A constant erosion rate was assumed for depthcalibration, where the crater depths were measured using a DekTak Stylus Profilometer.
After the electrical measurements, each Schottky diode was measured by the profilometer for accurate determination of the diode area.
Capacitance-Voltage (CV) measurements were carried out under dark conditions at temperatures in the range between 20 K and 300 K using an Agilent 4284A LCR Meter at six different probing frequencies between 1 kHz and 1 MHz and with a probing AC amplitude of 30 mV.The LCR-meter was used in two modes: (1) so-called parallel mode (Cp-Gp) and (2) so-called series mode (Cs-Rs).

Figures
Figure 1 Capacitance

Supplementary Files
This is a list of supplementary les associated with this preprint.Click to download. epsrTiO2SciRepSupplementary.pdf

Fig. 1 .
Fig. 1.Capacitance (a) of a TiO2-N2 sample (probing frequency 60 kHz) and the N(V)ε0ε product (b) as functions of applied voltage.The red, dotted curve shows the as-prepared sample, i.e., after the N2 heat treatment but prior to H + implantation.The blue, solid curves are for the H-implanted sample measured at different temperatures (Tmeas).The peaks in N(V)ε0ε are indicated by the drop-down lines.The corresponding capacitance values, obtained at the same applied voltages, are also marked in (a).

Fig. 1 (
Fig.1(b) displays N(V)ε0ε as a function of V.For the as-prepared sample, the data reveal a somewhat nonuniform, but monotonous N(V)ε0ε as a function of V. Hydrogen implantation leads to formation of a pronounced peak in N(V)ε0ε.One can thus identify the voltages at which the edge of the depletion region reaches the peak of N(V)ε0ε, as indicated in Fig.1(b) with filled circles.For example, at 100 K, the depletion region edge reaches the donor concentration peak at Vpeak=−3.8V, and at 250 K it occurs at Vpeak=−6.2V.

Fig. 2 .
Fig. 2.Temperature dependence of the c-axis dielectric constant () of conductive, ntype TiO2 obtained after heat treatments in FG and N2; and a modified Curie-Weiss dependence with the parameters determined by Samara and Peercy [5] (solid curve).
of Norway is acknowledged for the support to the Norwegian Micro-and Nano-Fabrication Facility, NorFab, project number 295864.Financial support by the Research Council of Norway via the EEA-JRPRO-NO-2013-1 European Project (PERPHECT), the Research Center for Sustainable Solar Cell Technology (FME SUSOLTECH, project number 257639), and the Norwegian PhD Network on Nanotechnology for Microsystems (project number 221860/F60), is gratefully acknowledged.Financial support by the Faculty of Mathematics and Natural Sciences at the University of Oslo via the strategic research initiative FOXHOUND is gratefully acknowledged.
(a) of a TiO2-N2 sample (probing frequency 60 kHz) and the N(V)ε0ε product (b) as functions of applied voltage.The red, dotted curve shows the as-prepared sample, i.e., after the N2 heat treatment but prior to H+ implantation.The blue, solid curves are for the H-implanted sample measured at different temperatures (Tmeas).The peaks in N(V)ε0ε are indicated by the drop-down lines.The corresponding capacitance values, obtained at the same applied voltages, are also marked in (a).

Figure 2 Temperature
Figure 2

Table I .
Data used for analysis of temperature dependence of ε for the TiO2-N2 sample with CV data plotted in Fig.1.The implantation depth dpeak = 0.97 µm.The diode area