Tailoring thermal conduction in anatase TiO2

Thermal conductivity (κ) plays an essential role in functional devices. It is advantageous to design materials where one can tune κ in a wide range according to its function: single-crystals and nanowires of anatase polymorph of titanium dioxide, broadly used in applications ranging from photovoltaics, reflective coatings to memristors, have been synthesized in large quantities. Here we identify a new, strong diffusion mechanism of heat by polaronic structures due to oxygen vacancies, which considerably influences both the absolute value and the temperature dependence of κ. The additional decrease of κ is achieved in anatase nanowires organized into foam, where porosity and the quasi-one-dimensional size-effect dramatically hinder the propagation of heat, resulting in an extremely low κ = 0.014 W/Km at room-temperature. Doping this anatase foam could herald promising applications, in particular in thermoelectricity. The thermal properties of a material often determine its suitability for application and use in devices. Here, the thermal conductivity of anatase TiO2 is tuned over three orders of magnitude from bulk crystals to foam samples, by controlling polaronic effects and texturing.

A ctive and passive thermal management play a critical role in nearly every sector of engineering sciences. Alongside developing strategies to optimise heat transfer in various devices, understanding and tuning the intrinsic ability of a material to propagate heat is a major issue. Materials with high thermal conductivities (κ) are sought for their capacity to transfer heat swiftly and evenly. Electronic packaging is one of the most important examples where the temperature of a circuit has to be limited to avoid overheating [1][2][3] . Mediums with high κ are also required in cars, rocket engines, or nuclear reactors, where heat has to be evacuated efficiently to prevent hazardous failure of the mechanical parts. Conversely, other applications demand thermally insulating materials. Compounds with low values of κ (<0.1 WK −1 m −1 ) are employed to prevent severe heat losses in buildings, heat pumps, car engines and power plants 1 . Moreover, thermal conductivity is one of the key parameters ruling the conversion of energy in thermoelectric materials. In the latter case, κ has to be the lowest to enhance the thermoelectric figure of merit 4 . Over the last decade, an ever-intensive scientific effort has been witnessed to develop new technology concepts for producing ultra-low-κ materials. One strategy focuses on structuring a media at meso/nano-scales, thus truncating the phonon-meanfree path 5 . A hundred-fold reduction of the thermal conductivity in silicon nanowires in respect of bulk was reported, and comparable results were obtained in other nanostructured alloys 6 . In a similar approach, compounds with complex cage structures (clathrates, skutterudites) and mixed-lattice atoms (half-Heusler alloys) exhibit low-thermal conductivities 7,8 . Here, the random vibrations of groups of atoms in the lattice hinder the propagation of heat by phonons. Corresponding phenomena have been described in organometallic perovskites, where crystal-liquid duality, together with the formation of large polarons, account for the very low κ 9-11 . Nielsen et al. were reasoning that compounds with anharmonic phonon modes stand amongst the best candidates to moderate heat transport 12 . Although this progress demonstrates the ability to lower κ significantly, fundamental insight into the mechanisms of thermal transport is still needed for designing more efficient materials for thermal management. In addition, many of the proposed low-κ compounds are difficult to scale up for applications, owing to the scarcity of their constitutive elements or to intricate synthesis (e.g. by high-pressure or nano-lithographic processes). Developing materials with few elements and up-scalability is the optimal strategy.
In this context, TiO 2 emerges as a prominent candidate, since titanium is the ninth most abundant element on Earth, and TiO 2 is non-toxic and stable under atmospheric conditions. It provides a wide spectrum of functionalities for attractive applications, allowed by a variety of different architectures at nanoscales 13 . In particular, photocatalytic activity of thin TiO 2 coatings proves antibacterial action and self-cleaning properties [14][15][16] . These features can also be achieved when titanate nanowires are arranged into a fibrous network, thus providing an enhanced filtering with significant potential for water and air purification 17,18 . Moreover, nanoparticles of TiO 2 contribute to good performances of solarcells regarding the collection of photo-induced charges 19 . For niobium-doped TiO 2 a high power-factor was reported, placing this material among promising-thermoelectric converters 20 . Besides energy-and environment-oriented applications, TiO 2 nanostructures achieved humidity sensing 21 , they were employed to reinforce polystyrene composites 22 and are foreseen as central elements of non-volatile memories (memristors) 23 . Most of the beneficial characteristics of TiO 2 stem from the anatase phase of the material. Although thermal conductivity of rutile, another polymorph of TiO 2 commonly found in its bulk form, was measured decades ago (κ = 8 WK −1 m −1 at room-temperature) 24 , the same quantity for bulk-anatase crystals or nanowires remains largely unexplored, mainly due to difficulties in synthesis. In the present work their measurements are exposed, and we identify a new, strong diffusion mechanism of heat by polaronic structures due to oxygen vacancies, which considerably influences both the absolute value and the temperature dependence of κ. A further step in decreasing κ is achieved in anatase nanowires organised into foam. In this case, porosity and the quasi-one-dimensional size-effect of thin nanowires dramatically hinder the propagation of heat, resulting in an extremely low κ = 0.014 W/Km at roomtemperature. Figure 1 shows the broad range of anatase samples for which thermal conductivity is reported here. The large, millimetres-long single-crystals (SC) on Fig. 1a are unique for anatase in which growth our laboratory is specialised. One can easily vary their oxygen stoichiometry, from the pristine one (transparent) on the left, towards the increased concentration of oxygen vacancies (brown and black) achieved by heat treatment in a hydrogen-rich atmosphere. The black SC on the right, from the pristine, 3.2 eV gapped semiconductor, evolved towards a semi-metallic sample due to the shallow-donor levels introduced by the offstoichiometry 25,26 . Complementarily, electron paramagnetic resonance (EPR) provides a measure of the spin-1/2 defect concentration originated by oxygen vacancies: 0.34, 21 and 51 ppm for the transparent, brown and black crystals respectively (see Supplementary Note 1, Supplementary Figs. 1 and 2). The concentration of defects for another "pristine"-anatase crystal was reported earlier by Chauvet et al., and its value~15 ppm is an order of magnitude higher than our SC, but far below 10,000 ppm (1%) 27 . In Fig. 1b, the egg-white-like, airy sample (referred to in the following text as "foam") was prepared from the quantity of TiO 2 powder exposed in the right Petri-dish, by first transforming it into nanowires. The breakthrough in its preparation is the scaling-up of the growth of nanowires from milligrams (hydrothermal synthesis) to kilograms by a different synthesis route 19 . Figure 1c displays a zoom on the foam by scanning electron microscopy (SEM), revealing the architecture of the nanowires, and Fig. 1d exhibits it after its compression applying 100 MPa of uniaxial pressure. The density and the inter-wire connections are clearly increased. For the sake of completeness, an assembly of anatase nanoparticles (mean diameter 50 nm), already used in many applications, was also prepared and measured (Fig. 1e).

Results
Thermal conductivity of anatase single crystals. Thermal conductivity of anatase SC was measured in the temperature range of 10-300 K; a steady-state, four-point method was applied, described previously by Jaćimović et al. 28 . The three curves in Fig. 2a correspond to the crystals displayed in Fig. 1a, from the top κ of the transparent crystal and below with increasing amounts of oxygen vacancies. A significantly higher value of κ for the more electrically insulating crystals is the first indication that itinerant charge carriers are not the main heat transporters. Indeed, using the Wiedemann-Franz law, even for the most electrically conducting crystal (the black SC), the electronic part κ el is at the level of 10 −3 WK −1 m −1 , a negligible fraction of the total κ at roomtemperature (5 WK −1 m −1 ) 29 . Consequently, in all our anatase SCs, heat is propagated almost entirely by the phonon subsystem (sketched in Fig. 2b, for the pristine, transparent SC). Using the experimental data in Fig. 2a measured in a broad temperature range, our primary interest is to explain the dramatic drop of κ with increasing concentration of oxygen vacancies. This could greatly help the control of thermal properties of anatase, ranging from SCs to nanostructured materials.
Recent ab initio calculations relying on non-perturbative microscopic treatment of static defects characterised by latticeelastic-constant and mass variances around these defects, demonstrated that standard perturbative approaches can considerably underestimate phonon-scattering rates. Namely, highorder scattering processes may result in large corrections of these rates 30,31 , which, if combined with a concentration of defects that surpass 1%, can account for a reduction 32-34 of the thermal conductivity that is comparable to the reduction of κ(T) observed in Fig. 2a. However, the concentrations of defects in our SCs (10 16 cm −3 , 6.3 × 10 17 and 1.5 × 10 18 cm −3 , for the transparent, brown and black SC, respectively) are far below 3 × 10 20 cm −3 (2-4 orders of magnitude), the threshold corresponding to 1% of defects. This clearly indicates the existence of an additional mechanism for the phonon-scattering that has not been considered so far. In the quest for identifying this extra contribution, we have to recall that Ti-O are dominantly ionic bonds 35,36 , which is underlined by a large difference between static (ε 0 ) and optical (ε ∞ ) dielectric constants, ε 0 >> ε ∞ 37 . Therefore, the presence of an excess charge in such medium can significantly affect the surrounding-lattice polarisation (deformation), particularly in the limit of low concentration of Thermal conductivity of anatase single-crystals. a Temperature dependence of the thermal conductivity of insulating-transparent, brown and conducting-black anatase TiO 2 single-crystals, respectively from higher to lower κ. The solid lines are fits of κ including the resonant-dynamical-scattering term. Sketches of the difference between heat propagation in b pristine single-crystal and in c crystal with oxygen vacancies. Only one vacancy is displayed with an electron localised in its surrounding. Such defects exhibit strong polaronic effects, which heavily scatter phonons and reduce κ. The colour gradients from red to blue illustrate temperature differences across the material. Experimental error at 300 K is 15%, and decreases with lowering temperature as the black-body-radiation contribution diminishes a b e d c itinerant charge carriers, when the screening is weak, as it is the case in anatase (even for the electrically conducting SC) 38 . Density functional theory (DFT) calculations found a diversity of band-gap states and resonances in anatase TiO 2 (in bulk and at surfaces), for various impurities and intrinsic defects 39,40 . In particular, polaronic defects related to charged-oxygen vacancies in anatase have been reported both theoretically 41,42 and experimentally 26,43 . Atoms in the vicinity of such defects relax to a new equilibrium position due to their strong coupling to the charge in excess, a process that can be compared to the formation of a small-localised polaron 44 . Such a localised-polaronic defect is characterised by a large binding energy E b ≳ 300 meV, corresponding to a band-gap state that is, to a great extent, spectrally and spatially decoupled from the rest of the system 45 . A polaronic defect develops its own dynamics, characterised by typical frequencies of its internal excitations ω p,i . Since E b >> ħω p,i in the temperature range considered herein (where k B T~ħω p,i ), the surplus of electric charge remains well-confined around the vacancy. Moreover, a charged-polaronic defect, involving internal degrees of freedom and their corresponding excitations, interacts with crystalline phonons in its surrounding, contributing importantly to their scattering (sketched in Fig. 2c). This scattering is dynamical, and as such, fundamentally different from static situations described earlier with variances of the lattice parameters. Furthermore, without any screening, the forces are long-ranged, enhancing particularly low-momentum scattering rates. In order to validate this new, polaronic scenario we turn to a standard model giving very reasonable approximations of thermal properties, yet involving a small set of parameters. In this context, we start from the Boltzmann transport equation, using the relaxation time approximation (see Supplementary Notes 2 and 3, Supplementary Eq. (S14)) 46 . The total scattering time τ is obtained from Matthiessen's rule, τ −1 = ∑ i τ i −1 , where τ i represents different frequency-(ω) and temperature-(T) dependent contributions. In the particular situation of semiconducting materials, three sources of scattering are commonly invoked 47,48 : (i) frequency-independent-Casimir scattering rate at the boundaries of the sample τ À1 where v s is the average speed of sound and L a characteristic dimension of the SC; (ii) frequency-dependent-Rayleigh scattering at static-point-like impurities, τ À1 R ¼ a R ω 4 ; and (iii) anharmonic phonon-phonon (Umklapp) scattering incorporating both ω and T dependencies, τ À1 U $ a U Tω 2 exp ÀT U =T ð Þ (with T U a characteristic energy for the Umklapp process). Using these three scattering mechanisms only, the best fit of the experimental data for the brown and the black SCs give Casimir-scattering times a c ( Table 1) that are orders of magnitude lower than realistic values. The latter may be estimated from a C ¼ v s = ffiffi ffi S p , with the average speed of sound, v s ≈ 7 × 10 3 ms −1 , and the characteristic length scale given by the sample cross-section S, similar for all SC samples in Fig. 1, ffiffi ffi S p % 10 À3 m. This is a further evidence calling for an additional source of scattering for phonons. As shown in Supplementary Note 2 and 3, the polaronic-scattering mechanism leads to a resonant form of the phonon-relaxation time: with N d the concentration of polaronic defects, and b i constants with dimensions of frequencies, defined by details of their coupling to crystalline phonons. Fitting κ across the whole temperature range with τ p included, one does not only obtain a qualitative, but also a remarkable quantitative agreement with the experimental data. Striving to retain a minimal number of parameters in our modelling, τ p in Eq. (1) relies on two resonant frequencies only, yielding ω p,1 = 7.8 meV and ω p,2 = 16.4 meV. As discussed in Supplementary Notes 2, these frequencies, characterising internal excitations of polaronic defects, correspond to renormalized-local-phonon modes in the neighbourhood of a vacancy site, adiabatically softened due to the strong polaronic coupling with the charge in excess. Simultaneously, the values of the parameters related to the Casimir-, Rayleigh-and Umklapp-scattering are kept the same for all three SCs, foregrounding the influence of polaronic defects on κ. In other words, the parameter proportional to the concentration of defects N d is the only one altered, allowing fitting curves to fairly align with the experimental data of the SCs-data in Fig. 2a. From these fits, one finds that N d in the conducting SC (black) is about one order of magnitude higher than in the insulating SC (brown) (see Supplementary Note 4), in good agreement with the measured concentration of defects. Therefore, we infer that N d is closely related to the amount of oxygen vacancies in our specimens 26 . To our knowledge, the dynamical phonon-scattering rate τ À1 p ω ð Þ was derived for the first time, successfully explaining the reduction of κ in anatase TiO 2 by oxygen vacancies. This mechanism should act in the rutile phase of TiO 2 as well, which has shown a strong dependence on the oxygen stoichiometry 24 . Complementarily to it, a further reduction of κ in anatase was thought to be achievable by texturing the material at micro-and nano-scales. This is precisely the course of action taken and presented hereafter for the anatase nanowires (foams) and nanoparticles displayed in Fig. 1c-e, respectively.
Thermal conductivity of the anatase foams and nanoparticles. The thermal conductivities of the anatase foams and nanoparticles are shown in Fig. 3a, b, mirrored in Fig. 3c-e by sketches of the sample's textures. Compared to the most thermally insulating anatase single-crystal, κ of the as-grown foam is reduced by a factor of 600 at room-temperature, attaining the ultra-low value of κ = 0.014 WK −1 m −1 . This number competes with that of organic and silica aerogels at room-temperature and makes anatase foam a (thermal) superinsulator 49 . Yet, structural inhomogeneities and the high porosity of this foam alone (φ = 95%) cannot account for such a marked reduction of κ. Indeed, an estimation of the thermal conductivity of the foam κ f as a sole consequence of the porosity can be obtained within the percolation theory: where κ b is the thermal conductivity of the pristine anatase SC. For t = 1.5 50 , this estimation gives κ f = 0.1 WK −1 m −1 at room-temperature, an order of magnitude above the experimental value. Hence, reasons for the particularly low value of κ f and its temperature dependence have to be sought in the peculiar morphology of the foam at nanoscale: properties of individual nanowires composing the foam and the number of contacts between them drastically affect the thermal transport. On average, anatase nanowires can be approximated as crystalline rods of~200 nm length and 10 nm diameter 51 , with lattice fringes in the [101] plane 19 . As for the contacts between nanowires, transmission-electron microscopy verifies their crystalline nature and their small cross-section (~10 nm), with the [001] crystal orientation across two nanowires 19 . Further analysis of the scanning electron microscopy (SEM) images of the asgrown foam (Figs. 1c and 3c) indicate that each nanowire connects to only few (~3 on average) neighbouring wires, whereas upon compression this number rises to~25 (Figs. 1d and 3d). Considering a single nanowire, only phonons with momenta aligned along its axis contribute significantly to the heat transport. Namely, the rod's rough boundaries diffusively scatter phonons whose momenta are not aligned along the rod's axis. This effect is enhanced at low temperatures by the small number of unit cells along the rod's diameter, which makes the phonon spectrum strongly gapped for excitations in the perpendicular directions. These effects of reduced dimensionality can be probed at low temperatures, where κ scales as T β , and β is the effectivelow-temperature dimension of the phonon subsystem (see Supplementary Note 5). For the as-grown foam, according to a simple fit, β = 1.0 (Fig. 3a), which permits the Callaway formula (Supplementary Eq. (S14)) to be implemented with a onedimensional density of states 52 . Owing to the scarce amount of defects (2.5 ppm), we deliberately omit the effects of polaronic and static defects. As a result, the fit of the whole temperature dependence of κ in Fig. 3a can hardly be distinguished from the experimental data. At low temperatures, the mean free path l ph (corresponding to phonons with momenta aligned along the rod's axis) is comparable to the average length of nanowires, pointing towards a partially ballistic, quasi-one-dimensional transport between contact areas. However, temperatures higher than one fifth of the Debye temperature (ϴ D /5) prompt Umklapp scattering, hence notably decreasing l ph . We benefit from compaction as a simple and powerful way to continuously tune thermal (and mechanical) properties of the foam. As the porosity drops by~70%, κ of the compressed foam increases by two orders of magnitude. A larger low-temperature exponent β = 1.5 (Fig. 3b) supports the drastic change in the nanowiresconnectivity scheme after compression. Interestingly, the assembly of TiO 2 nanoparticles (Figs. 1e and 3e) exhibit analogous thermal transport properties as the compressed foam (blue dots in Fig. 3b), especially at low temperatures. This suggests that both nanostructured materials are characterised by a similar connectivity scheme in the three-dimensional volume, with a small active portion of nanoparticles surface actually contributing to heat transport (as for nanowires) 53 .

Discussion
A summary of our measurements in Fig. 4 shows the outstanding flexibility of anatase TiO 2 -based materials in tuning thermal conductivity over three orders of magnitudes at 300 K by polaronic and nano-structuring effects. The dynamic-polaronicscattering mechanism, prompted by charged-oxygen vacancies, has not been discussed before. It has been uncovered by theoretical modelling and found to dominantly account for the reduction of κ in bulk anatase. Yet, beyond oxygen vacancies in anatase, this model encompasses wider classes of defects and impurities, opening the door to a more accurate understanding of polaronic effects on phonon-thermal transport. This also brings attention to the multiple consequences of doping efforts in anatase, many of Compressed foam Nanoparticles Fig. 3 Thermal conductivity of the anatase foams and nanoparticles. a Temperature dependencies of the thermal conductivity of as-grown and b the compressed anatase TiO 2 foams (green triangles). The solid lines correspond to fitting curves. The blue circles represent the data for the assembly of nanoparticles. Experimental error at 300 K is 15%, and decreases with lowering temperature as the black-body-radiation contribution diminishes. c-e Sketches of the three nanostructures based on their scanning electron micrograph images which aspire to reduce the energy gap and to collect a broader spectral range in photovoltaics. Our study shows that one does not only change the electronic, but substantially the thermal properties as well, through the resonant scattering of phonons on polaronic defects. This fundamentally new finding gains relevance in many other functional oxides, such as SrTiO 3 , ZnO and CaMnO 3 , where polarons are present.
Next to that, it is remarkable that in assemblies of nanowires produced in large scales, and not only in lithographically fabricated micron-square surfaces, one can engineer structures with ultra-low-thermal conductivity. The value of 0.014 WK −1 m −1 is important and beneficial by itself because it calls for an excellent thermal insulator, but especially also by the way it was reached: by the discreet-phonon dispersion due to the one-dimensionality of the nanowires. Since the same mechanism does not limit the electrical conductivity, this foam would be an excellent material for thermoelectric applications if one doped them to metallicity 54 . Recalling that in anatase thin films doped with 6% of Nb, the high power factor (S 2 σ, where S is the Seebeck coefficient and σ the electrical conductivity) of 14 µWK −2 cm −1 gave only a figure of merit (ZT) of 0.1 because of the high thermal conductivity (8.5 WK −1 m −1 ) of the thin film, but not attractive enough for practical purposes. If one could achieve a high σ via doping the nanowires, the foam could deliver a high ZT, and, together with its chemical and high temperatures stability, would be suitable for scalable applications.

Methods
Synthesis of anatase TiO 2 single crystals. Single-crystals of anatase TiO 2 were grown by chemical-vapour transport as described elsewhere 26,29 . Anatase powder of high purity (<300 ppm of Al detected) and powder of NH 4 Cl (transport agent) were sealed in a quartz tube in argon gas. The 3-mm-thick, 2-cm-wide, and 20-cmlong ampoule was placed in a horizontal tubular furnace and the temperature at its ends were set to 740 and 610°C. After 2 weeks, crystals of a few cubic millimetres were grown on the cooler side of the ampoule. The prepared samples are brown, (Fig. 1a), signalling the existence of impurities, while other transparentstoichiometric samples are colourless with a band gap of 3.2 eV. Earlier, it was demonstrated by Sekiya et al. that this colour indicates the presence of the oxygen vacancies with electrons remaining at their positions 55 . From an energy point of view, these electrons are localised deep in the band gap and do not contribute to the electrical conductivity. This is confirmed by the measured electrical resistivity of 10 12 Ω cm at room-temperature. Several specimens of these crystals were placed in a silica glass tube, inserted into an electric tubular furnace and heated under hydrogen at 650°C for several hours. Such treatment is known to introduce additional oxygen vacancies whose energetic states are circa 100 meV below the conduction band 18,55 . Two effects were observed: the crystals change their colour to dark blue, almost black (Fig. 1b) and electrical resistivity drops to 1 Ω cm at room-temperature. The electrical resistivity measurements showed that such material has metallic character down to 60 K when it starts to behave as insulator, in agreement with earlier findings 25,29 .
Synthesis of anatase TiO 2 foams. The preparation of TiO 2 fibrous network (foam) is reported elsewhere 19 . Some of the foam was compressed under a uniaxial pressure of 100 MPa. Pellets were then shaped into parallelepipeds suitable for measurements of κ.
Anatase TiO 2 nanoparticles were bought from Sigma-Aldrich Inc. (powder, −325 mesh, >99% purity) and mixed with polyvinil alcohol (PVA) and poly ethylene glycol (PEG). Secondly the mixture was compressed (380 MPa) to form pellets. In order to completely remove the organic molecules (PVA and PEG), 10 h sintering at 600°C was performed. As for the foams, evenly shaped samples were obtained by dicing the pellets.
Measurements of thermal conductivity. Thermal conductivity measurements of all specimens were performed using the steady-state method described in the work of Jaćimović et al. 28 . Experimental error at 300 K is 15%, and decreases with lowering temperature as the black-body-radiation contribution diminishes. While, in some cases, anatase is known to host anisotropic physical properties, measurements of κ in different crystallographic directions for the same crystal did not lead to a noticeable anisotropy of this quantity within experimental error.
Measurements of the defect concentrations. Electron paramagnetic resonance (EPR) at 9.4 GHz microwave frequency was performed on a Bruker X-band spectrometer. A conventional field modulation technique was employed with lockin detection, providing the first derivative of the EPR-absorption spectra. The EPR intensity was calibrated with CuSO 4 · 5H 2 O reference samples. The EPR intensity of each signal component is determined both by double integration of the EPR signal and by fitting the (derivative) Lorentzian curves, as is customary in the EPRrelated literature 56 .  Fig. 4 Overview of the thermal conductivity of single-crystals and textured anatase polymorph of TiO 2 . The thermal conductivity of the transparent (orange triangles), brown (pink circles) and conducting-black (red diamonds) single-crystals, compressed (dark green triangles) and as-grown (light green diamonds) foams made of nanowires and of nanoparticles (blue circles), span four orders of magnitude. For comparison, the solid line shows the thermal conductivity of rutile TiO 2 single-crystal, reproduced from the work of Thurber et al. 24