Transparent conductive oxides (TCOs)—materials that have the twin desirable features of high optical transmission and electrical conductivity—play an increasingly significant role in the fields of photovoltaics and information technology. As an excellent TCO, Ta-doped anatase TiO2 shows great promise for a wide range of applications. Here, terahertz time-domain spectroscopy is used to study the complex optical conductivity σ̃ω of the TCO—heavily Ta-doped TiO2 thin films with different Ta-doping concentrations, in the frequency range of 0.3–2.7 THz and the temperature range of 10–300 K. Fitting the complex optical conductivity to a Drude-like behavior allows us to extract the temperature dependence of the effective mass, which suggests the existence of many-body large polarons. Moreover, the carrier scattering rate of Ta-doped TiO2 with different carrier concentrations agrees with the interacting polaron gas theory. Our results suggest that with increasing electron density in TiO2, the interaction between polarons is larger and electron–phonon coupling is smaller, which is beneficial for achieving high mobility and conductivity in TiO2.

In the past decade, the rapid development of information technology and consumer electronics has created huge demand for transparent conductive oxides (TCOs).1 The most commonly used TCO in large-scale application, indium tin oxide (ITO) thin film, suffers increasingly from the rarity of the post-transition metallic element indium.2,3 Also, one needs a variety of TCOs with different work functions and refractive indices for matching with other materials such as doped GaN in devices.3 Recently, Ta-doped anatase TiO2 has been found to be a good TCO with carrier density and mobility values similar to ITO, thus showing great promise for various applications.4,5

Due to the polar nature of TiO2, polarons—phonon-dressed electrons—rather than bare electrons, are the carriers that determine the conductivity in Ta-doped TiO2.5–7 The Fröhlich electron–phonon coupling constant (α) is used to classify polar materials: in materials with relatively large α (>6), the quasiparticles are small polarons and the conduction mechanism is hopping, which leads to a relatively low mobility, while in the case of weaker electron–phonon coupling (α<6), the quasiparticles are large polarons, which act like free electrons but with an effective mass still larger than the free electron mass.6 Large polarons (α<6) have been observed by angle-resolved photoemission spectroscopy (ARPES) on pure anatase TiO2 and by optical-pump THz-probe spectroscopy on pure rutile TiO2, while a polaronic liquid has been observed in pure anatase TiO2 using transport measurements.8–10 In doped TiO2, however, first-principles calculations predict the existence of small polarons (α>6) in Ta- and Nb-doped rutile and anatase TiO2, while transport and transient absorption measurements suggest the presence of large and small polarons, respectively, in both Nb- and Ta-doped anatase TiO2.5,11 In traditional theories, polarons are treated as quasiparticles, which do not interact with one another.5–7,11,12 This assumption is valid in the case of low carrier densities. However, in heavily doped semiconductors, such as Nb-doped SrTiO3 and Ta-doped TiO2, where the carrier density is very large (1021 cm−3), the interaction between carriers cannot be ignored.13 In this paper, we present terahertz time-domain spectroscopy (THz-TDS) studies of Ta-doped TiO2 thin films at different temperatures to study their frequency-dependent far-infrared conductivity. Our results reveal the existence of an interacting polaronic gas (IPG) in these TCOs and suggest that their large conductivity is caused by the combined effects of (1) a large carrier density and (2) a small electron–phonon coupling constant, α, due to Ta doping.

Three Ta-doped TiO2 thin films (labeled as sample A, sample B, and sample C) are grown by pulsed laser deposition (PLD) on a double-side polished SrLaAlO4 (SLAO) substrate with dimensions of 5 mm × 5 mm × 1 mm at a temperature of 880 K and at different oxygen partial pressures for different Ta-doping concentrations. The thickness and Ta-doping concentration of each thin film are measured by Rutherford backscattering spectrometry (RBS). Ion channeling measurements indicate that the Ta and Ti minimum yields are of the order of 10%, indicating an extremely high crystalline quality and substitutionality of Ta at the Ti site. Hall measurements are performed to obtain the carrier density of each sample, and the temperature-independent results are consistent with previous research.5 XRD measurements are performed to confirm the anatase phase of our samples, and the results suggest that only the component of the optical conductivity perpendicular to the optic axis of anatase (c-axis) is measured in this work. The details of sample characterization are given in the supplementary material. The carrier density, Ta-doping concentration, and thin film thickness of samples A, B, and C are given in Table I.

TABLE I.

Carrier density (from Hall measurements), Ta doping concentration (from RBS measurements), thin film thickness (from RBS measurements), and effective Fröhlich electron–phonon coupling constant αeff (from IPG model), with different Ta doping concentrations.

SampleCarrier density (1021 cm−3)Doping (at. %)Thickness (nm)αeff
Pure anatase TiO2a ∼10–2 … – ∼2.5 
0.5 2.4 160.8 2.49 
1.2 6.3 279.5 2.23 
1.5 6.1 216.4 1.92 
SampleCarrier density (1021 cm−3)Doping (at. %)Thickness (nm)αeff
Pure anatase TiO2a ∼10–2 … – ∼2.5 
0.5 2.4 160.8 2.49 
1.2 6.3 279.5 2.23 
1.5 6.1 216.4 1.92 
a

From Ref. 8.

THz-TDS is a noncontact far-infrared optical technique, which has been used to study the low-energy excitations of materials such as superconductors, topological insulators, and semiconductors.14 From the amplitude and phase of the transmitted complex THz electric field, we are able to extract the complex optical conductivity, σ̃ω, without the need for Kramers–Kronig transformation.15 In particular, THz-TDS has been used to study polaronic effects in semiconductors.9,16 The transmission THz spectra of our samples are measured using a commercial THz-TDS system (TeraView TPS-3000), coupled with a Janis ST-100 FTIR cryostat and a motorized vertical stage to move from the sample to reference (see Ref. 15 for details). The THz wave is generated and detected by photoconductive antennas fabricated on low temperature-grown GaAs films. The setup is well calibrated as shown in Fig. S1 in the supplementary material, where the vac–vac curve is the measurement of the sample holder without the sample and reference. Its transmission coefficient [T̃ω=ẼSampω/ẼRefω] is within 100%±1% in 0.3–2.5 THz, which represents the trustable region of our measurements.

FIG. 1.

Experimental data of sample A at selected temperatures. (a) THz waveform that passes through sample A and the pure substrate (reference); (b) transmission coefficient of sample A in the frequency domain, which shows an obvious temperature dependence. The data of samples B and C are similar.

FIG. 1.

Experimental data of sample A at selected temperatures. (a) THz waveform that passes through sample A and the pure substrate (reference); (b) transmission coefficient of sample A in the frequency domain, which shows an obvious temperature dependence. The data of samples B and C are similar.

Close modal

From the THz-TDS measurements, the time domain electric field signal of the sample and reference is obtained [Fig. 1(a)]. Figure 1(b) shows the transmission coefficient (T̃) of sample A (data of other samples are similar) at selected temperatures in the frequency domain. T̃ is obviously temperature dependent for temperature >100 K, while it changes only a little at low temperature, which is consistent with DC measurements (Fig. 4). The smaller transmission coefficient at lower temperatures indicates stronger THz absorption and higher conductivity.

The complex refractive index [ñω=nω+ikω] of the three samples at various temperatures can be extracted by numerically solving Eq. (1), which is a standard THz data analysis method,15 

T̃ω=2ñ1+ñsubexpiωdñ1cexpiωΔLñsub1c1+ññ+ñsub+ñ1ñsubñexp2iωdñc.
(1)

Here, ñ and ñsub are the complex refractive indices of the Ta-doped TiO2 thin film and substrate (SLAO), respectively, d is the thin film thickness, ΔL is the thickness difference between the sample and reference substrates, and c is the speed of light in vacuum. This equation takes into account multiple reflections in the Ta-doped TiO2 thin film. However, because of the relatively large refractive index and thickness of SLAO [Re ñsub>4.1, thickness 1 mm], the multiple reflections in the substrate are far from the main peak and excluded in our data analysis.

The extracted complex refractive index ñ is then used to calculate the complex conductivity [σ̃ω=σ1ω+iσ2ω] according to the relationship σ1=2ωϵ0nk and σ2=ωϵ0ϵn2+k2. Here, ϵ0 is the permittivity of free space and ϵ is the high-frequency dielectric constant, which is set to 1 initially, and will be a fitting parameter in the conductivity model discussed later.14 The obtained complex conductivity of all the samples as functions of frequency at different temperatures is shown in Fig. 2. The decrease in σ1 and σ2 with increasing temperature is consistent with the trend seen in the transmission coefficient [Fig. 2(b)]. Compared to other TCOs such as ITO, our three samples display a stronger temperature dependence of the conductivity, suggesting a stronger electron–phonon interaction, which is consistent with a polaronic nature of the carriers in TiO2.17 

FIG. 2.

Real and imaginary conductivity of samples A, B, and C. Solid lines are fits to the Drude model [Eq. (2)].

FIG. 2.

Real and imaginary conductivity of samples A, B, and C. Solid lines are fits to the Drude model [Eq. (2)].

Close modal

From Fig. 2, we see that σ̃ω exhibits a typical Drude response, where σ1ω has a maximum at zero frequency and decreases with increasing frequency, while σ2ω is zero at low frequency and increases with increasing frequency. Therefore, in order to understand the transport properties of our samples, the frequency-dependent complex conductivity is fitted by the Drude model,

σ̃ω=e2Nm*1γiωiϵ0ωϵ1,
(2)

where N, m*, and γ are the density, effective mass, and scattering rate of carriers in the thin film, respectively. The fits of Eq. (2) to the complex optical conductivity obtained from THz data are very good at all temperatures, with selected fitting curves shown in Fig. 2(a). In the fits, the values of the carrier density (N) of samples A, B, and C are set to 0.5, 1.2, and 1.5 × 1021 cm−3 according to Hall measurements and are temperature independent shown in Fig. S4, which is consistent with the degenerated doping nature of our samples.18 

From the fits, the parameters, scattering rate (γ) and effective mass (m*), of the three samples as functions of temperature are shown in Fig. 3. At low temperatures (<100 K), the temperature independence of γ [Fig. 3(a)] suggests defect scattering to be the dominant scattering process.19 However, the strong increase in γ at higher temperatures suggests a very different mechanism. This temperature dependence can also be seen in Fig. 2(a), where the Drude peak narrows with decreasing temperature. The narrowing of the Drude peak at low temperatures was also observed in Nb-doped SrTiO3 in Refs. 13 and 20, which the authors attributed to many-body polaronic effects. The effective mass m* also shows a strong temperature dependence [Fig. (3b)], in contrast to that in normal metals such as silver and gold, where m* is temperature independent.21 

FIG. 3.

(a) Scattering rate obtained from experimental data, and the curve comes from the fitting to Eq. (8). (b) Temperature-dependent effective mass of polarons in samples A, B, and C.

FIG. 3.

(a) Scattering rate obtained from experimental data, and the curve comes from the fitting to Eq. (8). (b) Temperature-dependent effective mass of polarons in samples A, B, and C.

Close modal

The above standard Drude model yielded good simultaneous fits to σ1ω and σ2ω and revealed a strong temperature dependence of γ (and m*). However, the Drude model does not provide for a microscopic origin for this strong temperature dependence, and for this, we present the interacting polaronic gas (IPG) model below.13 We offer the three justifications that Ta-doped TiO2 is an IPG. First, angle-resolved photoemission data showed that with doping, anatase-TiO2 can become a polaronic gas where the quasiparticles are large polarons;8 Second, due to the large carrier density (1021 cm−3 from Hall measurements, see Sec. I in the supplementary material), the average electron–electron distance is smaller than 1 nm, which is on the order of the polaron radius ( 1 nm, see Ref. 8) and the lattice constant of anatase TiO2 (a=b=0.38 nm and c=0.95 nm).22 This increases the correlation between adjacent polarons and makes the notion of an interacting polaronic gas in this material plausible. The carrier density in our samples is greater than that in another doped perovskite system—Nb-doped SrTiO3—where optical (THz to IR) spectroscopy data yielded a carrier density of n 1020 cm−3, which the authors attributed to the presence of an interacting polaronic gas.20 Third, in the conventional perturbation theory for large polarons, the effective mass m*=mb/1α/6 is taken to be temperature independent.7 Even in modified Feynman's polaron theory, m* should not change by 10% from 10 K to 300 K.23 Therefore, the strong temperature dependence of m* seen in our data suggests the presence of many-polaron effects. The same Nb-doped SrTiO3 paper mentioned above (Ref. 20) showed a similar temperature-dependent effective mass, where the ratio of the effective mass at high to low temperatures is ∼2.4, which the authors attributed to the presence of a polaronic liquid.20 In comparison, our data show an increasing m*300K/m*10K ratio of 1.7, 1.9, and 2.3 in samples with increasing carrier density (samples A, B, and C, respectively), suggesting a positive correlation between the strength of many-body effects and carrier density (doping concentration). Hence, correlations between polarons are not insignificant in our samples.

We now turn to the mathematical formulation of the IPG model. Considering both electron–phonon and electron–electron interactions as well as the Fermi statistics of the polaron gas, the model yields a complex optical conductivity that is mathematically similar to that of the Drude model in Eq. (2), but with the real and frequency-independent scattering rate γ replaced by a complex and frequency-dependent scattering rate γ̃IPG(ω),13 

σ̃IPGω=e2Nm*1γ̃IPG(ω)iωiϵ0ωϵ1,
(3)

where

γ̃IPGω=iχ̃ωω
(4)

and χ̃ω is the weighted sum of the contributions from different phonon modes,

χ̃ω=jαjχ̃j(ω);
(5)

the individual χ̃jω given by

χ̃jω=ωL,j26π2Nmb2mbωL,j12dq{gq,ωωL,j+g*q,ωωL,jgq,ωL,jg*q,ωL,j+1eβωL,j1GRq,ωωL,j+GRq,ωωL,j*GRq,ωL,jGRq,ωL,j*}}.
(6)

Here, ωL,j and αj are the phonon mode frequency and Fröhlich electron–phonon coupling constant, respectively, of the jth LO phonon, which is engaged in the polaron. g and GR are the casual and retarded Green's functions (see the details in Sec. II of the supplementary material).

The total scattering rate γ̃total(ω), which we fit to our data, is the sum of the real and temperature-independent contribution from defect scattering γ̃defect19 and the temperature-dependent and complex γ̃IPG,

γ̃totalω=γdefect+γ̃IPGω=γdefect+iχ̃ωω=γdefect+ijαjχ̃j(ω)ω.
(7)

By performing the DFT calculation (see Sec. III in the supplementary material), the LO phonon modes in Ta-doped TiO2 at frequencies of 6.0 THz (Eg), 7.1 THz (Eu), 13.0 THz (Eu), 15.3 THz (A1g), and 18.6 THz (Eg) should participate in polaron formation.

We use Eq. (5) (see Sec. II of the supplementary material for details) to compute χ̃j(ω) for each phonon mode (Fig. S6). We notice that, at high temperatures and in our THz frequency range, Imχ̃jωReχ̃jω and Imχ̃jωω. Therefore, the quantity iχ̃jω/ω becomes a real and frequency-independent quantity, which we call γIPG,j. On the other hand, at low temperatures, both Reχ̃jω and Imχ̃jω are very small in magnitude, and so the main contribution to the experimental γ̃totalT comes from the real and temperature-independent γdefect. Hence, whether at low or high temperatures, the complex and frequency-dependent scattering rate γ̃IPGω in the IPG model [Eq. (2)] can, therefore, be replaced by a real and frequency-independent quantity γIPG,j. This effectively reduces the IPG model to the Drude model and explains why both the IPG and Drude models fit the data equally well. However, we want to stress that the Drude model cannot give a microscopic explanation of the strong dependence of the scattering rate seen in our data, while the IPG model can. Our subsequent analysis using IPG, thus, assumes a real and frequency-independent γIPG,j, with the total scattering rate simplified from Eq. (6) to

γtotalT=γdefect+jαjγIPG,jT.
(8)

The Fröhlich electron–phonon coupling constant corresponding to each phonon mode (αj) can then be obtained by fitting the experimental scattering rate to Eq. (7), and finally, the value of effective Fröhlich coupling constant αeff=jαj, defined in Ref. 13, is obtained. The fitting results are given in Table I, and the values of αeff for all the samples are smaller than six, which is consistent with the large-polaron scenario. From the fitting, we can see that at higher carrier density, the electron–phonon coupling (αeff) is smaller, which is the result of screening effects arising from the higher electron density. This result is consistent with the observation that, as we go from sample A (smallest carrier concentration) to sample C (largest carrier concentration), the temperature dependence of γT is strongest for sample A and weakest for sample C.

Figure 4(a) shows the DC resistivity ρDC1/σ10.3THz of the three samples as a function of temperature. Combined with transport measurements on more samples (more data are given in Fig. S5 in the supplementary material), the fitting to a power law ρDCTβ on the data (both THz and transport measurements) shows the exponent evolving from β3 for low Ta-doping to β2 for high Ta-doping. This suggests that the system evolves from a e-ph scattering-dominated5,24 regime at low Ta doping to an e–e/carrier–carrier scattering-dominated regime. These results are consistent with our interacting polaronic gas analysis, where correlations between polarons are more significant at higher Ta doping.

FIG. 4.

(a) Temperature-dependent DC resistivity of each sample (at 0.3 THz). The curves represent for the power law fitting (ρ=ρ0+ρ1Tc). (b) Exponent c from power law fitting as a function of carrier density.

FIG. 4.

(a) Temperature-dependent DC resistivity of each sample (at 0.3 THz). The curves represent for the power law fitting (ρ=ρ0+ρ1Tc). (b) Exponent c from power law fitting as a function of carrier density.

Close modal

In conclusion, the THz-TDS technique is applied to study a Ta-doped TiO2 thin film. Utilizing the interacting polaron gas theory, the temperature dependence of the scattering rate is explained by suggesting that the carriers in those samples are interacting polarons rather than bare electrons. Also, the Fröhlich coupling constant α is obtained, which is highly dependent on the carrier density. Therefore, the exceptional electronic performance of Ta-doped TiO2 with higher doping concentrations is from not only its high carrier density but also the reduced electron–phonon coupling (smaller α).

See the supplementary material for (I) experimental details and sample characterization, (II) derivation of many-body polaron theory, and (III) DFT calculation.

We thank the Singapore Ministry of Education (MOE) Tier 1 (No. MOE2018-T1-1-097) and Tier 2 (No. MOE2019-T2-1-097) grants, NUSNNI-NanoCore at the National University of Singapore (NUS) and the Singapore National Research Foundation (NRF) under the Competitive Research Programs (CRP No. NRF-CRP15-2015-01), NUS cross-faculty grant, and FRC (ARF Grant No. R-144-000-278-112) for financial support. The work at Los Alamos was carried out under the auspices of the U.S. Department of Energy (DOE) National Nuclear Security Administration under Contract No. 89233218CNA000001 and was supported by the U.S. DOE BES under the No. LANL E3B5 and LANL LDRD Program. This research used resources of NERSC (DOE Contract No. DE-AC02-05CH11231) and the Institutional Computing Program at LANL with the support of the Center for Integrated Nanotechnologies, a DOE BES user facility. The x-ray diffraction work was performed at the Facility for Analysis, Characterization, Testing and Simulation (FACTS), Nanyang Technological University, Singapore.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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