domain: chim phys

text:
T Sounalet
email: thomas.sounalet@subatech.in2p3.fr

C Alliot

N Audouin

N Bozovic

F Brelet

R Dureau

L Granger

A Guertin

F Haddad

K Kamalakannan

E Nigron

M Tarinas

Production of Pb-203 from target manufacturing to chemical separation Tl/Pb

Introduction

Why203 Pb? 203 Pb emits a strong 279 keV gamma line and has a half-life of 51.9h.

203

Pb is suited for SPECT imaging.

What else? 203 Pb forms a theranostic pair with 212 Pb which is used for targeted alphatherapy.

Which impurities are to be avoid for 203 Pb production? 201 Pb (t1/2 = 9.33h) is to be avoided and it must be limited during production. The only way to minimize this production is to respect the cooling time of a minimum of several hours for its value to be considered low enough: below 0.07%.

Objectives

The The Tl deposit is suited for the irradiation The calculation predicts that for 5h of irradiation with 50 µAe in the energy range of 32.5 MeV to 30 MeV, it is necessary to cool down for 61h after end of bombardment if the enrichment of 205 Tl is 99.75%. From these parameters of irradiation, 299 mCi can be produced at EOB leading to 133 mCi after 61h of EOB of 203 Pb.

How to separate Tl and Pb to obtain final product of 203

Pb with a high purity?

For routine production of 203 Pb large targets are mandatories. To mimic this and prepare chemistry procedure to separate Pb from Tl, a natural Tl deposit of 13.52 cm 2 and 40 µm of thickness has been manufactured by electrodeposition. The Tl deposit was irradiated at an energy of 32.5 MeV with an intensity of 30 µA for a duration of 1h. A Pb resin from Triskem was used to separate Tl and Pb. 1 mol/L of HNO3 was used to dissolve the deposit and filled firstly in the column with the volume of 150 mL to recover Tl. In a second step, 1 mol/L of ammonium acetate with the volume of 15 mL was filled in the column to recover 203 Pb. To obtain a final product of 203 Pb with a small volume, the solution is dried through evaporation and the residue is dissolved in 2 mL of 0.25 mol/L of ammonium acetate at pH 7.

Photograph of a Tl deposit after irradiation in a hot cell

Results of final product of 203 Pb and of recycle natural thallium

The final product of 203 Pb has been analysed by y-ray spectrometry and ICP-OES to determine the purity of 203 Pb. The yield rates are determined by using 2 radiotracers, The chemical yield of Pb and Tl are 85% and 82% respectively. The final amount of Tl is 27 µg and for stable Pb, the value is below the detection limit (<4.4 µg).

Conclusion and perspectives

The final product of 203 Pb and the yield rates are in accordance to expectations based on cross sections measurements. The next step is to use 205 Tl to produce very high purity of

  Photograph of a Tl deposit on gold backing

Pb, with less than 0.07% of 201 Pb after several hours of cooling and less than 0.05 µg/mCi of chemical purity of Tl by increasing the volume of Tl elution.


Alexandre Jaoui

Shan Jiang

Xiaokang Li

Yasuhide Tomioka

Isao H Inoue

Johannes Engelmayer

Rohit Sharma

Lara Pätzold

Thomas Lorenz

Benoît Fauqué

Kamran Behnia

Glass-like thermal conductivity and narrow insulating gap of EuTiO 3

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. Introduction

In most insulating crystals, the flow of heat can be understood by considering the response of a gas of phonons to a temperature gradient. This picture, first drawn by Peierls [START_REF] Peierls | Zur kinetischen Theorie der Wärmeleitung in Kristallen[END_REF], using a linearized Boltzmann equation, is remarkably successful in describing the thermal conductivity, κ, of most insulators [START_REF] Lindsay | Ab initio thermal transport in compound semiconductors[END_REF][START_REF] Subedi | First principles study of thermal conductivity of In2O3 in relation to Al2O3, Ga2O3[END_REF]. It explains why at an intermediate temperature, κ peaks [START_REF] Vandersande | The thermal conductivity of insulators and semiconductors[END_REF], separating a high-temperature decrease by warming (due to anharmonicity), and a low-temperature decrease by cooling (due to phonon depopulation). In amorphous solids, on the other hand, there is no such peak in κ(T ) and heat diffuses thanks to off-diagonal coupling across harmonic branches [START_REF] Allen | Diffusons, locons and propagons: Character of atomic vibrations in amorphous Si[END_REF][START_REF] Seyf | A method for distinguishing between propagons, diffusions, and locons[END_REF][START_REF] Simoncelli | Unified theory of thermal transport in crystals and glasses[END_REF][START_REF] Isaeva | Modeling heat transport in crystals and glasses from a unified lattice-dynamical approach[END_REF].

Some crystals, however, display glass-like thermal conductivity [START_REF] Visser | Thermal conductivity of manganite perovskites: Colossal magnetoresistance as a lattice-dynamics transition[END_REF][START_REF] Takahata | Low thermal conductivity of the layered oxide (Na,Ca)Co2O4: Another example of a phonon glass and an electron crystal[END_REF][START_REF] Li | Phonon-glass-like behavior of magnetic origin in single-crystal Tb2Ti2O7[END_REF][START_REF] Tachibana | Thermal conductivity of pyrochlore R2Ti2O7 (R=rare earth)[END_REF][START_REF] Sugii | Thermal hall effect in a phonon-glass Ba3CuSb2O9[END_REF][START_REF] Dutta | Ultralow thermal conductivity in chain-like TlSe due to inherent Tl + rattling[END_REF][START_REF] Hanus | Uncovering design principles for amorphous-like heat conduction using two-channel lattice dynamics[END_REF]. The thermal conductivity of these materials can be very low and/or feature a monotonic temperature dependence (lacking the T -1 decrease at high temperature). They are sought after, since a low lattice thermal conductivity would lead to a large thermoelectric figure of merit [START_REF] Snyder | Complex thermoelectric materials[END_REF].

EuTiO 3 (ETO) is a perovskite with a cubic structure at room temperature [START_REF] Bussmann-Holder | Hybrid paramagnon phonon modes at elevated temperatures in EuTiO3[END_REF]. Its electric permittivity increases upon cooling without giving rise to a ferroelectric instability. This quantum paraelectric behavior [START_REF] Müller | SrTiO3: An intrinsic quantum paraelectric below 4 k[END_REF] is akin to what has been seen in other ABO 3 compounds such as SrTiO 3 (STO) and KTaO 3 (KTO). In contrast to them, however, it magnetically orders at T N = 5.5 K, with an antiferromagnetic alignment of the nearest neighbor Eu 2+ spins [START_REF] Mcguire | Magnetic structure of EuTiO3[END_REF][START_REF] Katsufuji | Coupling between magnetism and dielectric properties in quantum paraelectric EuTiO3[END_REF]. Like STO, but at higher temperatures, it goes also through an antiferrodistortive transition where adjacent TiO 6 octahedra rotate in opposite directions [START_REF] Bussmann-Holder | Relation between structural instabilities in EuTiO3 and SrTiO3[END_REF]. Upon doping, either with oxygen vacancies [START_REF] Engelmayer | Charge transport in oxygen-deficient eutio3: The emerging picture of dilute metallicity in quantum-paraelectric perovskite oxides[END_REF] or La substitution of Ti [START_REF] Tomioka | Magnetotransport properties of Eu1-xLaxTiO3 (0 ≤ x ≤ 0.07) single crystals[END_REF], it becomes a dilute metal [START_REF] Engelmayer | Charge transport in oxygen-deficient eutio3: The emerging picture of dilute metallicity in quantum-paraelectric perovskite oxides[END_REF][START_REF] Tomioka | Magnetotransport properties of Eu1-xLaxTiO3 (0 ≤ x ≤ 0.07) single crystals[END_REF][START_REF] Maruhashi | Anisotropic quantum transport through a single spin channel in the magnetic semiconductor EuTiO3[END_REF], but without a superconducting ground state.

Here, we show that the temperature dependence of thermal conductivity in ETO is glass-like and its amplitude, over a broad temperature range starting from room temperature and extending to cryogenic temperatures, is lower than in STO and KTO. The drastic attenuation of thermal conductivity occurs well above the Néel temperature and when the magnetic entropy is saturated to its expected value. This points to an unusual version of spin-phonon coupling such as the phonon-paramagnon hybridization postulated by Bussmann-Holder et al. [START_REF] Bussmann-Holder | Hybrid paramagnon phonon modes at elevated temperatures in EuTiO3[END_REF].

Evidence for spin-lattice coupling in ETO has been around for more than two decades [START_REF] Katsufuji | Coupling between magnetism and dielectric properties in quantum paraelectric EuTiO3[END_REF]. However, it was restricted to temperatures comparable to the magnetic ordering. According to our findings, the phonon mean free path is affected by spins at Eu sites even at high temperature when there is neither a magnetic order nor a field-dependent entropy.

Monitoring the temperature dependence of electrical conductivity and thermopower in insulating EuTiO 3 , we find an activation gap of 0.11 eV, indicating that the intrinsic transport band gap in EuTiO 3 is as low as ∼0.2 eV, much smaller than the gaps found by optical studies, but close to what is expected by Ab Initio calculations [START_REF] Ranjan | Electronic structure and magnetism of EuTiO3: a first-principles study[END_REF][START_REF] Malyi | Local positional and spin symmetry breaking as a source of magnetism and insulation in paramagnetic EuTio3[END_REF].

The narrow gap between the chemical potential and Eu 2+ energy level and the random orientation of large magnetic moments in the paramagnetic state emerge as principal suspects shortening the lifetime of heat carrying phonons at elevated temperatures in the paramagnetic state.

In contrast to many other crystals displaying glass-like conductivity, ETO is not a spin liquid candidate, but a simple G-type antiferromagnet and there is no magnetic frustration. In this context, the glassy behavior could be framed in a picture of off-diagonal coupling across harmonic modes [START_REF] Simoncelli | Unified theory of thermal transport in crystals and glasses[END_REF][START_REF] Hanus | Uncovering design principles for amorphous-like heat conduction using two-channel lattice dynamics[END_REF], which in this case would be phonons and paramagnons.

II. Results

A. Activation and band gap

Fig. 1a shows the temperature dependence of electrical resistivity ρ in as-grown EuTiO 3 (ETO) single crystals. One can see that ρ increases by seven orders of magnitude upon cooling from 360 K to 50 K.

An Arrhenius activation behavior becomes visible by plotting lnρ as a function of the inverse of temperature (Fig. 1b). The activation gap is ∼ 0.11 eV. The Hall resistivity (see the supplement [27]) shows also an Arrhenius behavior with a comparable ∆. At low temperature, resistivity begins to saturate when the Hall carrier density is as low as 10 10 cm -3 (See the supplement [27]). This implies an extremely low level of extrinsic donors . An activated behavior in longitudinal and Hall resistivities of as-grown ETO crystals was previously reported by Engelmayer et al. [START_REF] Engelmayer | Charge transport in oxygen-deficient eutio3: The emerging picture of dilute metallicity in quantum-paraelectric perovskite oxides[END_REF], whose study focused on the emergence of metallicity in oxygen-deficient ETO.

This activated behavior is also confirmed by our measurements of the Seebeck coefficient (See Fig. 1(c)) and and the temperature dependence of the electric permittivity [START_REF] Engelmayer | Charge transport in oxygen-deficient eutio3: The emerging picture of dilute metallicity in quantum-paraelectric perovskite oxides[END_REF].

The thermoelectric power has a negative sign and an amplitude in the range of mV/K, typical of a narrow gap semiconductor. The Seebeck coefficient in an intrinsic semiconductor is expected to be ≈ k B e ∆ k B T [START_REF] Behnia | Fundamentals of Thermoelectricity[END_REF]. Thus, by plotting S as a function of T -1 (see the inset), one expects to see a straight line whose slope yields ∆. As seen in the inset of Fig. 1c, this is indeed what our data yields, with ∆ ≈ 0.1eV . Note that this temperature dependence is qualitatively distinct from what is expected in extrinsic semiconductors [START_REF] Johnson | Theory of thermoelectric power in semiconductors with applications to germanium[END_REF] as seen, for example, in the case of Nb-doped STO [START_REF] Collignon | Heavy nondegenerate electrons in doped strontium titanate[END_REF].

Thus, the temperature dependence of electric and thermoelectric conductivity both point to a similar energy gap between the chemical potential and the conduction band. The Hall and the Seebeck coefficients are both negative, indicating that carriers are electron-like and thermally excited to live in the conduction band originating from Ti-d orbitals.

Both ρ and S show an anomaly near 260 K, which we identify as the temperature of the structural transition in ETO [START_REF] Bussmann-Holder | Hybrid paramagnon phonon modes at elevated temperatures in EuTiO3[END_REF][START_REF] Rushchanskii | Firstprinciples prediction of oxygen octahedral rotations in perovskite-structure EuTiO3[END_REF]. Like STO [START_REF] Lytle | X-ray diffractometry of low-temperature phase transformations in strontium titanate[END_REF], this transition makes ETO tetragonal [START_REF] Kennedy | Studies of the antiferrodistortive transition in EuTiO3[END_REF]. We cannot rule out a very small difference in the amplitude of the activation gap between the cubic and the tetragonal phases (See the inset of Fig1b). We also detected an unexpected and reproducible hysteresis near this phase transition. Taken at its face value, this indicates that this structural phase transition is first order (see the supplement).

Assuming that the chemical potential is at halfway between the chemical potential and the valence band leads us to conclude that the band gap of ETO is ≈ 0.22 eV. This is remarkably smaller than the 3.2 eV gap of STO [START_REF] Collignon | Metallicity and superconductivity in doped strontium titanate[END_REF], but only slightly smaller than what a recent DFT calculation [START_REF] Malyi | Local positional and spin symmetry breaking as a source of magnetism and insulation in paramagnetic EuTio3[END_REF] found (0.27-0.33 eV). According to another and earlier theoretical study [START_REF] Ranjan | Electronic structure and magnetism of EuTiO3: a first-principles study[END_REF], the magnitude of the band gap in ETO depends on the amplitude of the Hubbard U , and a realistic U (5-6 eV) would lead to a band gap of 0.2-0.4 eV. Thus, the narrow gap detected by our transport studies is close to what is intrinsically expected in this solid. Optical probes, however, have detected a larger gap (0.8-0.9 eV) [START_REF] Lee | Optical band gap and magnetic properties of unstrained EuTiO3 films[END_REF][START_REF] Akamatsu | Crystal and electronic structure and magnetic properties of divalent europium perovskite oxides EuMO3 (m = ti, zr, and hf): Experimental and first-principles approaches[END_REF]. Such larger energy scales are possible indications that the Density of States (DOS) near the Eu-f level is not featureless [START_REF] Malyi | Local positional and spin symmetry breaking as a source of magnetism and insulation in paramagnetic EuTio3[END_REF].

Let us keep in mind the contrast between ETO on one hand and STO and KTO on the other hand. The first has a valence band originating from Eu-f orbitals close to the chemical potential, while the two others have a valence Band emanating from O-p orbitals and much further away from the chemical potential.

B. Thermal conductivity

Fig. 2 shows the temperature dependence of thermal conductivity, κ, of two ETO crystals from slightly above room temperature down to dilution refrigerator temperatures. We measured several crystals and found that the thermal conductivity of all lies somewhere between the two cases shown in this figure. Moreover, we could not detect a correlation between the amplitude of κ and a weak variation of the saturation magnetization observed across various ETO samples (see the supplement [27]).

In contrast to typical crystalline insulators [START_REF] Vandersande | The thermal conductivity of insulators and semiconductors[END_REF], κ does not show a prominent peak. As seen in the inset, there is a clear anomaly at T AF D and below this temperature κ rises by ten percent increase with cooling. However, there is no sign of a kinetic regime with κ ∝ T -1 , as seen in many other insulators [START_REF] Berman | Thermal conduction in Solids[END_REF][START_REF] Xu | Thermal conductivity of bulk In2O3 single crystals[END_REF].

C. Crystals, glasses and glass-like crystals

In order to put our observation in a proper context, we compare our data with what has been reported in the case of SiO 2 , which shows a spectacular difference in the thermal conductivity in its crystalline and amorphous structures [START_REF] Zeller | Thermal conductivity and specific heat of noncrystalline solids[END_REF]. As seen in Fig. 3, κ in amorphous silica monotonically decreases with cooling, in contrast to crystalline quartz, which has a prominent peak. At any given temperature, the crystal conducts heat at least an order of magnitude more than the glass [START_REF] Vandersande | The thermal conductivity of insulators and semiconductors[END_REF]. Not only the thermal conductivity of ETO is similar to silica in temperature dependence, but also in the cryogenic temperature range surrounding the Néel temperature (T N ≃ 5.5K), the ETO crystalline samples conduct heat less than amorphous silica. The order of magnitude of thermal conductivity and its temperature dependence in ETO is comparable with other crystalline solids displaying a glass-like thermal conductivity, such as Tb 2 Ti 2 O 7 [START_REF] Li | Phonon-glass-like behavior of magnetic origin in single-crystal Tb2Ti2O7[END_REF], Tb 3 Ga 5 O 12 [START_REF] Inyushkin | Low-temperature thermal conductivity of terbium-gallium garnet[END_REF], Na 4 Ir 3 O 8 [START_REF] Fauqué | Thermal conductivity across the metal-insulator transition in the single-crystalline hyperkagome antiferromagnet Na3+xIr3O8[END_REF], Pr 2 Ir 2 O 7 [START_REF] Uehara | Phonon thermal hall effect in a metallic spin ice[END_REF] and La 0.2 Nd 0.4 Pb 0.4 MnO 4 [START_REF] Visser | Thermal conductivity of manganite perovskites: Colossal magnetoresistance as a lattice-dynamics transition[END_REF]. In order to allow a direct comparison, Fig. 3 includes the data for Tb 2 Ti 2 O 7 [START_REF] Li | Phonon-glass-like behavior of magnetic origin in single-crystal Tb2Ti2O7[END_REF], the compound with the lowest thermal conductivity among these frustrated magnets. Note that in contrast to other members of this club of materials, EuTO 3 is a G-type antiferromagnet and is not frustrated.

As a further comparison, we also include κ of EuCoO 3 [START_REF] Berggold | Anomalous expansion and phonon damping due to the Co spin-state transition in RCoO3[END_REF], which displays a temperature dependence typical of a crystalline insulator. EuCoO 3 is a perovskite like ETO, but it has an orthorhombic symmetry, and the valence is Eu 3+ with only 6 electrons in the 4f shell. According to Hund's rules, this causes a vanishing magnetic moment (J = L -S = 0), in agreement with the experimentally observed van Vleck susceptibility [START_REF] Baier | Spin-state transition and metal-insulator transition in La1-xEuxCoO3[END_REF], drastically different from the large local moments of 7 µ B /Eu 2+ in ETO. Comparison with crystals, glasses and glasslike crystals: Thermal conductivity as a function of temperature in crystalline quartz, in vitreous silica [START_REF] Zeller | Thermal conductivity and specific heat of noncrystalline solids[END_REF], in EuTiO3 and in the frustrated magnet Tb2Ti2O7 [START_REF] Li | Phonon-glass-like behavior of magnetic origin in single-crystal Tb2Ti2O7[END_REF]. The latter two crystalline compounds conduct heat like a glass rather than like a crystal. Also shown is the crystal-like thermal conductivity of non-magnetic EuCoO3 [START_REF] Berggold | Anomalous expansion and phonon damping due to the Co spin-state transition in RCoO3[END_REF].

κ ( W K -1 m -1 ) T ( K ) FIG. 4.
Thermal conductivity of three quantum paraelectric solids: Thermal conductivity of the two EuTiO3 samples in a linear scale compared with thermal conductivity of SrTiO3 [START_REF] Martelli | Thermal transport and phonon hydrodynamics in strontium titanate[END_REF] and KTaO3 [START_REF] Tachibana | Thermal conductivity of pyrochlore R2Ti2O7 (R=rare earth)[END_REF][START_REF] Salce | Disorder and thermal transport in undoped KTaO3[END_REF]. Note the drastic reduction of thermal conductivity in the paramagnetic EuTiO3.

The visible sample dependence of the data for each material is much smaller than the differnce between the three compounds.

D. ETO compared to its non-magnetic sisters Fig. 4 compares the thermal conductivity of ETO and two other ABO 3 perovskites. SrTiO 3 (STO) and KTaO 3 (KTO) are also quantum paraelectric, but not magnetic, solids. Our new data on these two materials is in good agreement with previous studies of heat transport in STO [START_REF] Martelli | Thermal transport and phonon hydrodynamics in strontium titanate[END_REF][START_REF] Steigmeier | Field effect on the cochran modes in SrTiO3 and KTaO3[END_REF][START_REF] Li | Phonon thermal hall effect in strontium titanate[END_REF] and in KTO [START_REF] Salce | Disorder and thermal transport in undoped KTaO3[END_REF][START_REF] Steigmeier | Field effect on the cochran modes in SrTiO3 and KTaO3[END_REF][START_REF] Li | Phonon thermal hall effect in strontium titanate[END_REF]. In both cases, there is also a visible sample dependence, which is more pronounced near the peak. However, this sample dependence is much smaller than the difference between the three compounds. At room temperature, this difference is small, yet visible: κ(300K) is ≈ 9 W/(K.m) in ETO, ≈ 11 W/(K.m) in STO and ≈ 17 W/(K.m) in KTO. Much more drastic is the difference in the temperature dependence between the three sister compounds. The enhancement with cooling observed in the other perovskites is absent in ETO. This difference extends over the full temperature range above the magnetic ordering.

It is instructive to scrutinize the specific heat of the three sister compounds. Fig. 5(a) shows that C/T of STO, ETO and KTO approach each other towards room temperature and reach C ≈ 100 J/(mol.K). With five atoms, one expects the specific heat saturating to the Dulong-Petit value of 5 N A k B = 125 J/(mol.K), which is indeed what happens to STO, heated to 1800 K [START_REF] De Ligny | High-temperature heat capacity and thermal expansion of SrTiO3 and SrZrO3 perovskites[END_REF], a remarkably high temperature, broadly compatible with the highest energy scale in the phonon spectrum of these solids (≈ 100 meV) [START_REF] Rushchanskii | Firstprinciples prediction of oxygen octahedral rotations in perovskite-structure EuTiO3[END_REF][START_REF] Aschauer | Competition and cooperation between antiferrodistortive and ferroelectric instabilities in the model perovskite SrTiO3[END_REF]. A systematic difference in the specific heat evolves at lower temperatures. As is shown in the inset of Fig. 5(a), all three solids show a peak in the temperature dependence of C T 3 . In the case of STO, this peak is known to be caused by the presence of Einstein modes [START_REF] Burns | Comment on the low temperature specific heat of ferroelectrics, antiferroelectrics, and related materials[END_REF][START_REF] Mccalla | Low-temperature specific heat of doped SrTiO3: Doping dependence of the effective mass and Kadowaki-Woods scaling violation[END_REF]. Similar peaks are visible for KTO and for ETO, and the position of this peak shifts with the increase in the molar mass. In STO (184 g/mol), it occurs at 30 K, in ETO (248 g/mol) at 25 K, and in KTO (267 g/mol) at 12 K. In the case of ETO, a distinct additional contribution shoots up below 15 K, which signals strong magnetic fluctuations above the Néel ordering temperature of the Eu 2+ spins.

E. Field dependence

As reported previously [START_REF] Petrović | Electric field modulation of the tetragonal domain orientation revealed in the magnetic ground state of quantum paraelectric EuTiO3[END_REF], the specific heat in ETO displays a strong dependence on magnetic field. In order to separate the magnetic C mag and the phononic C phon contribution, we subtracted from the total specific heat of EuTiO 3 the measured specific heat of SrTiO 3 (with a temperature re-scaled by a factor of 0.83 such that the positions of the C/T 3 maxima of both materials match). Note that this scaling factor is close to the ratio of the molar masses M STO /M ETO = 0.86, the expected ratio of their Debye temperatures. zero-field anomaly signals the antiferromagnetic order at T N = 5.5 K that gets strongly broadened already in a field of 1 T. This reflects the magnetic-field-induced switching to the polarized magnetic state, which sets in around 1.2 T in ETO (see, e.g., the supplement [27] or [START_REF] Midya | Large adiabatic temperature and magnetic entropy changes in EuTiO3[END_REF]) and, consequently, the C mag /T data in larger fields no longer signal a spontaneous magnetic ordering transition, but a continuous evolution of magnetic entropy as it is the case for ferromagnets in an external magnetic field. In fact, the C mag /T data of EuTiO 3 strongly resembles the corresponding data obtained on the Eu 2+ -based ferromagnet EuC 2 , which orders at T C = 14 K [START_REF] Heyer | Thermodynamic properties and resistivity of the ferromagnetic semiconductor EuC2[END_REF]. The magnetic entropy obtained by integrating C mag /T is displayed in the inset of Fig. 5(b) and reveals that the full magnetic entropy N A k B ln(2S + 1) of a spin 7/2 system is reached above about 15 K for zero field and also for 1 T, whereas for fields above 2 T the entropy evolution drastically broadens and extends to much higher temperatures.

The field dependence of κ, shown in Fig. 6, further points to an intricate coupling between Eu 2+ spins and heat-carrying phonons. Thermal conductivity begins to depend on magnetic field below 15 K. Interestingly, as seen in the inset of Fig. 5, this is the temperature below which there is a significant magnetic entropy.

Above T N , magnetic field slightly amplifies κ, indicating a weakening of the spin-induced localisation of phonons. This field-induced amplification of κ in ETO is modest, in contrast with to the thirty-fold field-induced amplification of the ultra-low thermal conductivity in Tb 2 Ti 2 O 7 [START_REF] Li | Phonon-glass-like behavior of magnetic origin in single-crystal Tb2Ti2O7[END_REF].

Below 3 K, well below the Néel temperature, the fielddependence becomes non-monotonic (See Fig. 6. We leave the quantitative understanding of the field dependence of κ in EuTiO 3 as a subject of study for future investigations.

F. Thermal diffusivity and phonon mean-free-path

Replacing Sr by Eu reduces the thermal conductivity in a wide temperature range and enhances the specific heat below 15 K. Therefore, the ratio of thermal conductivity (in W/(K.m)) to specific heat per volume (in J/(K. m 3 )), i.e. the thermal diffusivity, D, (in m 2 /s) is drastically modified. It is plotted in Fig. 7. The most striking feature of D(T ) is its two-orders-of-magnitude drop at the Néel temperature. Within the entry to the antiferromagnetically ordered phase, at 5.5 K, it becomes exceptionally low. Its minimum, 0.03 mm 2 /s, is almost two orders of magnitude lower than the thermal diffusivity of a typical glass [START_REF] Yang | Heat capacity and thermal diffusivity of a glass sample[END_REF]. This low thermal diffusivity, a consequence of the combination of an unusually low lattice thermal conductivity and a very large magnetic entropy may find applications in heat management in a cryogenic context.

The thermal conductivity of an insulator can be written as:

κ = s,q C s (q)v 2 s (q)τ s (q) (1)
The index s refers to different modes and q is the wavevector. C s , v s and τ s are, specific heat, velocity and scattering time. There are modes contributing to the total specific heat (C = s C s ), but not to thermal conductivity, because of their negligible velocity. Theoretically [START_REF] Bussmann-Holder | Hybrid paramagnon phonon modes at elevated temperatures in EuTiO3[END_REF], paramagnons in the paramagnetic state have no dispersion and therefore do not carry heat. They can, however, reduce the phonon thermal transport. In ETO, phonons not only dominate thermal conductivity, but also, at least down to 15 K, the specific heat. Therefore, one can simply write: κ = 1/3Cv s ℓ ph . This neglects the q dependence of the scattering time and assumes that all modes have the same velocity. Taking v s = 6.8 km/s, the measured sound velocity in STO [START_REF] Okai | Pressure dependence of the structural phase transition temperature in SrTiO3 and KMnF3[END_REF], and in reasonable agreement with the dispersion of acoustic branches in ETO [START_REF] Rushchanskii | Firstprinciples prediction of oxygen octahedral rotations in perovskite-structure EuTiO3[END_REF], one can estimate ℓ ph , shown in the inset of Fig. 7. Comparing it to the inverse of the wave-vector of thermally excited phonons: q s = k B T hvs , one finds that below 20 K, q s ℓ ph ∼ = 1 , reminiscent of the Anderson localization. There is a drop at 15K, below which specific heat is no more phonon dominated.

III. Discussion

Evidence for a coupling between spin and lattice degrees of freedom in this compound was first reported by Katsufuji and Takagi [START_REF] Katsufuji | Coupling between magnetism and dielectric properties in quantum paraelectric EuTiO3[END_REF], who found that when the spins order at 5.5 K, the electric permittivity of ETO drops by 7 percent and this drop is suppressed by the application of a magnetic field of the order of 3 T. This magneto-electric effect implies coupling between Eu 2+ spins and the soft mode governing the electric permittivity. Reuvekamp et al. [START_REF] Reuvekamp | Tiny cause with huge impact: polar instability through strong magneto-electric-elastic coupling in bulk EuTiO3[END_REF] have found a quantitative agreement between the amplitude of the magneto-electric effect and the low-temperature magnetostriction of the system.

Our main finding is that lattice-spin coupling drastically attenuates the propagation of heat in ETO, even at temperatures where magnetic ordering is absent and magnetic entropy is practically saturated at its maximum value k B ln(2S + 1)/spin. This implies that even when the spins are randomly oriented, heat-carrying phonons couple to Eu 2+ states and their large magnetic moments (6.9-7 µ B ). According to ref. [START_REF] Malyi | Local positional and spin symmetry breaking as a source of magnetism and insulation in paramagnetic EuTio3[END_REF], without incorporating the loss of spin symmetry, the DFT calculations cannot FIG. 8. Crystal structure of EuTiO3 in its paramagnetic phase: Random magnetic moments at Eu sites are located between TiO6 octahedra, which are tilted off each other. Superexchange interaction between these spins involves Ti and oxygen atoms. This can impede phonons to have a well-defined wave-vector over long distances.

explain the absence of metallicity and the finite band gap of the paramagnetic phase.

The random orientation of magnetic moments at Eu sites (See Fig. 8) is the most plausible source of phonon localization. The superexchange interaction between Eu spins occurs through Ti ions [START_REF] Akamatsu | Antiferromagnetic superexchange via 3d states of titanium in EuTiO3 as seen from hybrid hartree-fock density functional calculations[END_REF]. The inter-atomic force constant can depend on the relative orientation of spins. The calculated phonon frequencies for parallel or antiparallel alignment of adjacent spins are not the same [START_REF] Fennie | Magnetic and electric phase control in epitaxial EuTiO3 from first principles[END_REF], which implies that phonons cannot keep a well-defined dispersion in presence of random spin orientation.

The narrow energy separation between the Eu 2+ energy level and the chemical potential may also play a role. In some Eu-based metals, the thermoelectric response has been linked to the temperature dependence of Eu valence [START_REF] Stockert | Valence effect on the thermopower of Eu systems[END_REF]. Remarkably, a theoretical study [START_REF] Feng | Phonon transport in perovskite SrTiO3 from first principles[END_REF] has concluded that the contribution of optical phonons to the overall lattice thermal conductivity is unusually large in this lattice structure. A coupling between heat-carrying optical phonons and Eu valence may be the source of observed κ attenuation.

Inelastic neutron scattering is a promising probe of this physics. In the case of Tb 3 Ga 5 O 12 , for example, it documented the coupling between spin and lattice [START_REF] Petit | Spin and lattice dynamics in the two-singlet system Tb3Ga5O12[END_REF]). A recent study on STO [START_REF] Fauqué | Mesoscopic fluctuating domains in strontium titanate[END_REF] has found evidence for an unusual hybridization between acoustic and optic phonon branches. No neutron scattering data is presently available to compare ETO with STO.

Finally, let us note that the glass-like thermal conductivity of EuTiO 3 , in contrast to spin-liquid crystals, occurs in a solid with a simple G-type anti-ferromagnetic ground state [START_REF] Mcguire | Magnetic structure of EuTiO3[END_REF]. A formal theoretical treatment may be achieved by complementing the picture drawn by Eq.1 with an 'off-diagonal' coupling between different vibrational states [START_REF] Simoncelli | Unified theory of thermal transport in crystals and glasses[END_REF][START_REF] Isaeva | Modeling heat transport in crystals and glasses from a unified lattice-dynamical approach[END_REF][START_REF] Hanus | Uncovering design principles for amorphous-like heat conduction using two-channel lattice dynamics[END_REF]:

κ od = s̸ =s ′ ss ′ ,q C ss ′ (q)v 2 ss ′ (q)τ ss ′ (q) (2)
This equation was conceived for non-magnetic crystals, in which harmonic coupling occurs across phonon branches [START_REF] Simoncelli | Unified theory of thermal transport in crystals and glasses[END_REF][START_REF] Hanus | Uncovering design principles for amorphous-like heat conduction using two-channel lattice dynamics[END_REF]. It can be extended to magnetic modes.

IV. Acknowledgements

Supplementary Materials for Glass-like thermal conductivity and narrow insulating gap of EuTiO 3 S1. Crystal growth EuTiO 3 single crystals were grown in two different labs. In Cologne, the floating-zone technique was used as described in Ref. [START_REF] Engelmayer | Charge transport in oxygen-deficient eutio3: The emerging picture of dilute metallicity in quantum-paraelectric perovskite oxides[END_REF]. In order to prevent the formation of the Eu 2 Ti 2 O 7 pyrochlore phase, the growth was performed in Argon atmosphere and without any preliminary powder reaction, similar to Ref. [START_REF] Katsufuji | Transport and magnetic properties of a ferromagnetic metal: Eu1-xRxTiO3[END_REF]. Powders of Eu 2 O 3 , TiO, and TiO 2 were mixed in a ratio to reach EuTiO 3-δ , pressed to a rod, and installed into the floating-zone furnace. In a series of growth attempts, it turned out that it was necessary to mix the above materials with a nominal oxygen deficiency of δ = 0.05. This allowed to compensate possible off-stoichiometries of the starting materials or an oxygen capture during the growth process. Single crystallinity of the grown crystal was confirmed by Laue images and phase purity was verified by X-ray powder diffraction.

In Tsukuba, similarly to the above description, EuTiO 3 crystals were prepared by the floating zone method [START_REF] Tomioka | Magnetotransport properties of Eu1-xLaxTiO3 (0 ≤ x ≤ 0.07) single crystals[END_REF].

S2. Magnetization

Figure S1(a) displays magnetization data M(T) of a EuTiO 3 single crystal for different magnetic fields between 10 and 500 mT applied along a [100] c direction of the cubic room temperature phase. At the lowest fields, the antiferromagnetic ordering causes a maximum in M (T ). This peak becomes a kink at an intermediate field of 500 mT. The corresponding inverse susceptibility follows a straight line over the entire temperature range from 300 K down to about 7 K (See panels (b) and (c)). The Curie-Weiss fit yields an effective magnet moment of µ ef f = 7.8 µ B /f.u. and a positive Weiss temperature of θ W = 3.55 K, which signals a net ferromagnetic coupling. These values agree with the literature. The positive θ W has been explained by the fact that in EuTiO 3 the antiferromagnetic coupling of the Eu 2+ spins to 6 nearest neighbors (NN) is overcompensated by a larger ferromagnetic coupling to 12 next nearest neighbors [START_REF] Mcguire | Magnetic structure of EuTiO3[END_REF][START_REF] Akamatsu | Crystal and electronic structure and magnetic properties of divalent europium perovskite oxides EuMO3 (m = ti, zr, and hf): Experimental and first-principles approaches[END_REF][START_REF] Petrović | Electric field modulation of the tetragonal domain orientation revealed in the magnetic ground state of quantum paraelectric EuTiO3[END_REF][START_REF] Midya | Large adiabatic temperature and magnetic entropy changes in EuTiO3[END_REF][START_REF] Chien | Magnetic properties of EuTiO3, Eu2TiO4, and Eu3Ti2O7[END_REF]. almost reaches the saturation magnetization expected for a spin 7/2 system (dashed line) (e) Magnetization curves measured on 2 different pristine EuTiO3 single crystals (JE93, JE111), which both were insulating, in comparison to 2 metallic EuTiO3 samples, which were weakly electron doped by vacuum annealing [START_REF] Engelmayer | Charge transport in oxygen-deficient eutio3: The emerging picture of dilute metallicity in quantum-paraelectric perovskite oxides[END_REF]. Insets show enlarged views of the same data in the region of (f) the saturation magnetization and of (g) a spin-flop transition occurring at ≈ 0.2 T when the field is applied along a cubic [100]c direction (3 samples), but remains absent for H[110]c (doped sample JE93).

𝜇 0 𝐻 (mT) 𝑯 ∥ [100] c (a) (b) (c) (d)
Note that there is no magnetic frustration resulting from the ferromagnetic and antiferromagnetic couplings because both couplings support the G-type antiferromagnetic order. Nevertheless, a comparatively weak magnetic field of about 1.2 T is sufficient to induce a polarized magnetic state. This is because only the weaker NN antiferromagnetic coupling needs to be overcome. As shown in Fig. S1(d), the saturation magnetization M sat ≃ 6.73 µ B /f.u. reaches about 96 % of the expected spin-only value of M sat ≃ gµ B S z = 7µ B for Eu 2+ with half-filled 4f shell. An analogous 4 % reduction is obtained from the Curie-Weiss fit in the paramagnetic phase for the Curie constant, which results in a 2 % reduction of the effective magnetic moment µ ef f = 7.8 µ B /f.u. in comparison to the expected µ ef f = g S(S + 1)µ B = 7.94 µ B for S = 7/2.

As is shown in Fig S1(e) this weakly reduced saturation magnetization has been observed in various pristine insulating samples that were obtained from different growth attempts. In contrast, however, other EuTiO 3 samples that were cut from the same mother crystals, but were made conducting via vacuum annealing at high temperature, see e.g. [START_REF] Engelmayer | Charge transport in oxygen-deficient eutio3: The emerging picture of dilute metallicity in quantum-paraelectric perovskite oxides[END_REF], reached the expected saturation magnetization of 7µ B /Eu 2+ . As it is known that the vacuum annealing of STO and ETO causes electron doping which is traced back to a reduction of the oxygen content, the different saturation magnetizations of pristine versus doped ETO can be interpreted as follows: appearently, the pristine ETO samples have a weak oxygen surplus that results in a corresponding partial oxidization of Eu 2+ to Eu 3+ which has a vanishing magnetic moment thereby reducing the overall magnetization. Vice versa, with decreasing oxygen content, the magnetization of pristine ETO should first increase but remain insulating until all Eu 3+ ions are reduced to Eu 2+ and only with further oxygen reduction a metal-insulator transition sets in via electron doping into the 3d band of Ti [START_REF] Engelmayer | Charge transport in oxygen-deficient eutio3: The emerging picture of dilute metallicity in quantum-paraelectric perovskite oxides[END_REF].

S3. Methods for measuring thermal conductivity

We measured thermal conductivity with a standard one-heater-two-thermometers method. Above 2K, the measurements were carried out with a home-built setup in a Quantum Design PPMS. Down to 100mK, the measurements were realized in a Dilution refrigerator system. The sensors used for the measurement of ETO were Cernox chips Cx-1030 and RuO 2 thermometers. Above 20 K and up to room temperature, type E thermocouples were used. Close attention was paid to the overlap between different sets of experimental data. The thermometers were directly glued to the samples with Dupont 4922N silver paste. These sensors were also thermally isolated from the copper sample holder by long manganin wires with very low thermal conductance. The ETO/STO samples were connected to a heat sink (copper finger) with Dupont 4922N silver paste on one side and to a RuO 2 -resistor (used as a heater) on the other side. A heating current was applied by flowing a current I through the heater from a DC current source (Keithley 6220). We evaluated the heating power from the relation I × V with V : the electric voltage measured across the heater by a digital multimeter (Keithley 2000). κ was controlled to be independent of the applied thermal gradient ∆T by changing ∆T /T . (∆T /T ) max was kept under 10%. Finally, calibration of the sensors was realized during each experiment and upon verification.

S4. Sample dependence

As discussed above, the magnetization data of pristine ETO crystals indicate the presence of a certain mixture of Eu 2+ and Eu 3+ valencies, and it is known from many intermetallic materials that Eu (or other rare-earth) valence instabilities can drastically influence the magnetic and/or transport properties [START_REF] Stockert | Valence effect on the thermopower of Eu systems[END_REF]. Therefore, we studied several EuTiO 3 crystals in order to check whether there is some correlation between the surprisingly low, glass-like thermal conductivity and the reduced saturation magnetization. As can be seen from Fig. S2(a), the overall variation of the κ(T ) curves obtained on the different samples is covered by the data obtained on samples C1 and T1, which are discussed in the main article. The magnetization data of the pristine samples C2 and T1 are essentially identical to each other and agree with the data of the other pristine samples shown in Fig. S1. While this is not surprising for C1, which is another piece from the same mother crystal JE111, this was not necessarily to be expected for T1, which was grown independently under somewhat different conditions (see above) and it is evident from panel (a) that overall the κ(T ) curve of T1 is lying above those measured on the samples C1, C2, and C3. On the other hand, the magnetization of C3 is the only one which essentially reaches 7µ B . This can be traced back to the fact this sample was vacuum annealed for 1 h at 700 • C. Resistivity measurements before and after annealing did not show a measurable difference, whereas the enhanced magnetization indicates a significant variation of Eu valence towards almost pure Eu 2+ . Nevertheless, the thermal conductivity of this particular C3 sample remains as low and glassy as those of all the ETO samples. Thus, we conclude that the weak Eu 2+/3+ off-stoichiometry is not the main mechanism that induces the unusual glass-like phonon heat transport in the ETO crystals.

We also confirmed the activated behavior of the resistivity in several samples measured in three different laboratories. Figure S3 indicates temperature profiles of resistivity, ρ of two EuTiO 3 crystals prepared by the floating zone method [23]. Tiny anomalies can be discerned at near 260 K, which is the transition between cubic and tetragonal structure, consistent with the discussion in Sec. II. A of the main paper. The inset shows lnρ vs. T -1 of these crystals. The estimated value of the activation gap, ∆ is comparable to that of samples T1 and T2 in Figure 1(b).

S5. The Hall coefficient

As shown in Fig. S4, we found that the Hall carrier density displays an Arrhenius behavior from 180 K down to 70 K. A deviation starts below the latter temperature, when the resistance was in the range of 10 8 ohms rendering the data unreliable. The corresponding carrier density was 10 10 cm -3 . At very low temperature, the saturated Hall number corresponds to a carrier density of ≈ 10 9 cm -3 .

S6. Hysteresis near the structural transition

The structural transition in ETO is believed to be similar to the structural transition in STO an antiferrodistortive transition in which adjacent octahedra rotate in opposite orientations [START_REF] Bussmann-Holder | Hybrid paramagnon phonon modes at elevated temperatures in EuTiO3[END_REF]. In the case of STO, this transition is known as a rare case of second order structural phase transition. However, in our study of resistivity, we found that the anomaly at ≈ 260 K presents a hysteresis, which was found to be reproducible. The data is shown in Fig S5 . This may point to the first-order nature of this structural transition in ETO.

FIG. 1 .

 1 FIG. 1. Electrical resistivity, thermopower and activation gap of EuTiO3: a) Resistivity, ρ, as a function of temperature in two single crystals of EuTiO3 in a semilogarithmic plot. The inset shows the unit cell (in the cubic phase). b) Arrhenius plot of the same data: lnρ vs. T -1 . The solid blue line corresponds to ∆ = 0.11eV . The inset shows the temperature variation of ∆ = -kBT 2 ∂lnρ ∂T , with an arrow pointing to TAF D , the temperature of antiferrodistortive transition. c) Seebeck coefficient, S as a function of temperature. The data is restricted to T > 120K, below which measurement becomes problematic. The inset shows S as a function of T -1 . The solid blue lines corresponds to ∆ = 0.1eV . The arrow points to TAF D .

FIG. 2 .

 2 FIG. 2. Thermal conductivity of two EuTiO3 crystals: a) Thermal conductivity, κ, as a function of temperature in two EuTiO3 crystals in a log-log plot. The arrow points to a minimum near the Néel temperature. The inset shows a linear plot, with an arrow pointing to TAF D .

  FIG. 3.Comparison with crystals, glasses and glasslike crystals: Thermal conductivity as a function of temperature in crystalline quartz, in vitreous silica[START_REF] Zeller | Thermal conductivity and specific heat of noncrystalline solids[END_REF], in EuTiO3 and in the frustrated magnet Tb2Ti2O7[START_REF] Li | Phonon-glass-like behavior of magnetic origin in single-crystal Tb2Ti2O7[END_REF]. The latter two crystalline compounds conduct heat like a glass rather than like a crystal. Also shown is the crystal-like thermal conductivity of non-magnetic EuCoO3[START_REF] Berggold | Anomalous expansion and phonon damping due to the Co spin-state transition in RCoO3[END_REF].

  ( T a c h i b a n a ) K T O ( S a l c e ) K T O ( t h i s w o r k ) S T O ( M a r t e l l i ) S T O ( t h i s w o r k ) E T O ( S a m p l e C 1 ) E T O ( S a m p l e T 1 )

FIG. 5 .

 5 FIG. 5. Specific heat in three quantum paraelectric solids: a) Temperature dependence of the total specific heat in KTaO3, SrTiO3 and EuTiO3 plotted as C/T vs. T . The inset is an enlarged view of the same data shown as Cp T 3 vs. T . Note the presence of Einstein modes in all three solids and the additional magnetic specific heat in ETO emerging below 15 K. b) Evolution of Cmag/T of ETO for different magnetic fields up to 10 T. The inset shows the corresponding magnetic entropy Smag = Cmag/T dT of ETO and the maximum magnetic entropy NAkB ln(2S + 1) of a spin 7/2 system is indicated by the dashed line.

FIG. 6 .

 6 FIG. 6. Field dependence of thermal conductivity in EuTiO3 a) κ vs. T from 2 K to 10 K for B = 0 T to 9 T. b) Normalised κ vs. B for T = 2.4 K to 15.6 K for B//[001].

FIG. 7 .

 7 FIG. 7.Thermal Diffusivity of EuTiO3 and the meanfree-path of phonons: Temperature dependence of thermal diffusivity, D, by taking the ratio of thermal conductivity and the specific heat per volume in sample C1. Note the drastic drop across the magnetic transition. The insets shows the Temperature dependence of the mean free path in the paramagnetic phase, assuming that all phonons have the same scattering time and velocity for the same sample. The red solid line represents the inverse of the wavevector of the thermally excited acoustic phonons.

  FIG. S1. (a) Magnetization of EuTiO3 as a function of temperature for different magnetic fields applied along the cubic [100]c direction. (b) Inverse susceptibility χ -1 = µ0H/M at 500 mT with a Curie-Weiss fit (line) for T > 20 K (c) Detailed view of the low-temperature regime, revealing the Weiss temperature θW = 3.55 K. (d) Magnetization curve measured at 2 K, whichalmost reaches the saturation magnetization expected for a spin 7/2 system (dashed line) (e) Magnetization curves measured on 2 different pristine EuTiO3 single crystals (JE93, JE111), which both were insulating, in comparison to 2 metallic EuTiO3 samples, which were weakly electron doped by vacuum annealing[START_REF] Engelmayer | Charge transport in oxygen-deficient eutio3: The emerging picture of dilute metallicity in quantum-paraelectric perovskite oxides[END_REF]. Insets show enlarged views of the same data in the region of (f) the saturation magnetization and of (g) a spin-flop transition occurring at ≈ 0.2 T when the field is applied along a cubic [100]c direction (3 samples), but remains absent for H[110]c (doped sample JE93).

  FIG. S2. (a)Comparison of thermal conductivity data κ(T ) of EuTiO3, which were measured on different samples, which stem from different sources (T1 versus C1, C2, C3) (b) Magnetization data measured on the same samples apart from C1, that got broken in the meanwhile. Note that the field axis was corrected for demagnetization effects to account for the rather different sample geometries.

FIG

  FIG. S3. Temperature profiles of resistivity in two EuTiO3 crystals (samples A1 and A2). Inset displays ln ρ vs. T -1 . Dotted line is calculated using D = 101 meV.

  FIG. S4. The Hall carrier density as a function of the inverse of a temperature in an Arrhenius plot.

  FIG. S5. a) Temperature dependence of resistivity near the structural transition for upward and downward sweeps either with a continuous ramp or by steps. b) The temperature derivative of resistivity for an upward and a downward sweeps with discrete steps.

We thank Annette Bussman-Holder, Yo Machida, Igor Mazin and Nicola Spaldin for stimulating discussions. This work was supported in France by JEIP-Collège de France, and by the Agence Nationale de la Recherche (ANR-18-CE92-0020-01; ANR-19-CE30-0014-04) and a grant by the Ile de France regional council. In Germany, it was supported by the DFG (German Research Foundation) via Project No. LO 818/6-1. I.H.I.In Japna, it was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant No. 18H03686 and the Japan Science and Technology Agency (JST) CREST Grant No. JPMJCR19K2.


Shan Jiang

Benoît Fauqué

Kamran Behnia

T-square dependence of the electronic thermal resistivity in metallic strontium titanate

The temperature dependence of the phase space for electron-electron (e-e) collisions leads to a T-square contribution to electrical resistivity of metals. Umklapp scattering are identified as the origin of momentum loss due to e-e scattering in dense metals. However, in dilute metals like lightly doped strontium titanate, the origin of T-square electrical resistivity in absence of Umklapp events is yet to be pinned down. Here, by separating electron and phonon contributions to heat transport, we extract the electronic thermal resistivity in niobium-doped strontium titanate and show that it also displays a T-square temperature dependence. Its amplitude correlates with the T-square electrical resistivity. The Wiedemann-Franz law strictly holds in the zero-temperature limit, but not at finite-temperature, because the two T-square prefactors are different by a factor of ≈ 3, like in other Fermi liquids. Recalling the case of 3 He, we argue that T-square thermal resistivity does not require Umklapp events. The approximate recovery of the Wiedemann-Franz law in presence of disorder would account for a T-square electrical resistivity without Umklapp.

Landau and Pomeranchuk [1], and contemporaneously Baber [2] postulated that electron-electron collisions cause a quadratic temperature dependence in electrical resistivity of metals. Subsequent experiments found that this is prominent in metals hosting strongly-correlated electrons (Such as UPt 3 [3] or strontium titanate [4]), but also those with a small carrier concentration (like bismuth [5] and graphite [6]). In these cases, at sufficiently low temperature, resistivity, ρ, can be expressed as :

ρ = ρ 0 + AT 2 (1)
Here, ρ 0 is the residual resistivity, which depends on disorder. The prefactor, A, on the other hand is intrinsic to each metal. The ubiquity of equation 1 across various families of Fermi liquids raises two questions: 1) What makes the exchange of momentum between two colliding electrons detrimental to the electrical conduction? 2) What sets the amplitude of A?

The two identified answers to the first question are Umklapp and the Baber mechanism. An Umklapp event occurs when the momentum vector sum of the colliding electrons gets out of the Brillouin zone, leading to a loss of momentum equivalent to one reciprocal unit vector [7,8]. The Baber mechanism [2] refers to the existence of two distinct electron reservoirs whose momentum exchange is a bottleneck in the path of momentum leak from the electron bath to the phonon bath. The second question was addressed first by Rice [9] and then by Kadowaki and Woods [10] (See also [11,12]) who argued that the amplitude of A scales with the square of the T -linear specific heat, γ 2 , because both depend on the density of states.

The persistence of T-square electric resistivity in metallic strontium titanate (STO) [13,14] to the extreme dilute limit [15] raised new questions about these answers. The Fermi surface in this dilute metal is too small to allow Umklapp scattering and it consists of a single pocket in the extreme dilute limit [16][17][18]. Thus, none of the two mechanisms can generate T-square resistivity. Moreover, the proper scaling relation was found to be between A and the Fermi energy, A ∝ E -2 F , instead of the standard Kadowaki-Woods scaling, A ∝ γ 2 , which fails in STO [19].

Following this observation, transport properties of dilute metallic STO were studied up to temperatures well above room temperature [20] and the effective mass was found to increase with warming. Two theoretical studies [21,22] showed that the temperature dependence of electrical resistivity in STO one can be explained with a scenario based on the scattering of electrons by two soft transverse optical (TO) phonons. This would account for the persistence of T-square resistivity above the degeneracy temperature. On the other hand, low-temperature T-square resistivity (well below the minimum energy of TO phonons) remained a mystery. Another development was the discovery of T-square resistivity in dilute metallic Bi 2 O 2 Se [23], another solid with a small Fermi surface, and without any soft phonon mode. The T-square prefactor was found to scale with the Fermi energy. This demonstrated that STO is not an isolated case and called for an e-e scattering scenario in absence of Umklapp.

The T-square thermal resistivity of electrons in Fermi liquids is less known and even much less explored. Defining the electronic thermal resistivity as W T = T /κ e , one expects:

W T = (W T ) 0 + BT 2 (2)
Here, (W T ) 0 is the residual thermal resistivity, expected to obey the Wiedemann Franz (WF) law:

L 0 (W T ) 0 = ρ 0 with L 0 = π 2 3 k 2 B e 2 = 2.44 × 10 -8 V 2 K -2 .
On the other hand, L 0 B > A, because, compared to energy flow, momentum flow is less affected by small-angle arXiv:2304.04841v2 [cond-mat.str-el] 11 Jul 2023 scattering [START_REF] Ziman | Principles of the Theory of Solids[END_REF][START_REF] Li | [END_REF]. Experiments on various metals, including Ni [26], Al [27], W [28], Sb [29], CeRhIn 5 [30], WP 2 [31], UPt 3 [32]) have confirmed both these expectations.

A quantitative connection between this physics and heat transport in the normal liquid 3 He was recently highlighted [33]. In normal liquid 3 He , thermal conductivity becomes proportional to the inverse of temperature [34] at very low temperatures, which means that thermal resistivity W T = T /κ, is proportional to T 2 . The evolution of this T-square resistivity with pressure follows the scaling seen for A and B with E F in metals [33]. There is no Umklapp in normal liquid 3 He and the Fermi surface is a single sphere. Thus, T-square thermal resistivity can occur in a Fermi liquid without Umklapp and the amplitude of B is directly linked to its Landau parameters, which also set the Fermi temperature.

Here, we present a study of electric and thermal conductivity in SrTi 1-x Nb x O 3 at two different carrier concentrations (n = 3.1×10 20 cm -3 and n = 1.8×10 20 cm -3 ). Despite the dominance of the lattice contribution to the heat transport in strontium titanate [35][36][37], we succeeded in extracting the electronic contribution to heat transport by exploiting the differentiating effect of the magnetic field on phonons and electrons. Such a method was employed previously in the case of bismuth and antimony [29,[START_REF] White | The Philosophical Magazine[END_REF][39][40]. We found that W T follows Eq. 2 and L 0 B > A. Thus T-square resistivity in SrTi 1-x Nb x O 3 cannot be distinguished from other metals in which the e-e origin of the T-square resistivity is uncontested. This leads us to conclude that T-square (electric and thermal) resistivity can be caused without Umklapp as a consequence of the T-square decrease in the amplitude of the (momentum and energy) diffusivity in a Fermi liquid caused by fermion-fermion scattering. A comprehensive theory of this phenomenon is yet to be elaborated.

Fig 1 shows the temperature dependence of the total thermal conductivity in two samples of SrTi 1-x Nb x O 3 . As discussed in the supplementary material [41], the temperature dependence of electrical resistivity in our samples is comparable to what has been reported previously for this doping level and their residual resitivity is close to the lower end of the spectrum. In order to see the relative share of the electronic and the phononic contributions to the total heat transport, κ xx /T is compared with L 0 σ xx , which represents the upper boundary of electronic thermal conductivity according to the WF law. With decreasing temperature, κ xx /T approaches L 0 σ xx . As the temperature tends to zero (see inset), they tend to join each other. κ xx /T and L 0 σ xx are both modified by the presence of a perpendicular 12 T magnetic field . To quantify longitudinal conductivity in presence of magnetic field, we measured both the electrical and the thermal Hall resistivities and inverted the resistivity tensor.

The field-induced decrease in electrical conductivity σ xx (i.e. the magnetoresistance) was the subject matter of a previous study [42], which found that both longi-tudinal and transverse conductivity follow the behavior expected in the semi-classical picture:

σ xx = neµ 1 + µ 2 B 2
(3)

σ xy = neµ 1 + µ 2 B 2 µB (4)
Here, n is the carrier density, and e is the electron charge. Mobility, µ(B, T ), is the only adjustable parameter depending on temperature and magnetic field. It monotonically decreases with increasing magnetic field and/or temperature. A remarkable (and poorly understood) fact about metallic STO is that the field dependence of mobility shows little dependence on the orientation of the magnetic field [42].

The thermal conductivity tensor κ, on the other hand has an electronic κ e and a lattice κ ph component in longitudinal:

κ xx (B) = κ e xx (B) + κ ph xx .
As in previous studies on semi-metals [29,[START_REF] White | The Philosophical Magazine[END_REF]39], one can separate the two components by assuming that the field dependence of the lattice thermal conductivity is negligible compared to the field dependence of electronic thermal conductivity. In insulating strontium titanate, thermal conductivity is purely phononic. There, at 24 K, a magnetic field of 12 T reduces κ xx at most by 7 × 10 -3 [43] and generates a finite thermal Hall conductivity of κ ph xy ≈ 0.09W/K.m. In our metallic samples, the effect of magnetic field on κ e xx and the amplitude of κ e xy (see below) are orders of magnitude larger.

Fig 2 shows the temperature dependence of the transverse thermal conductivity divided by temperature (-κ xy /T ). In the whole temperature range, it remains close (but smaller than L 0 σ xy ), which is what is expected for the electronic part. The measured signal is much larger than κ ph xy /T measured in insulating STO [43] [44]. Thus, we can safely identify the field-induced change in thermal conductivity ∆κ xx with the thermal magnetoresistance of electrons : In both samples, these two quantities converge at low temperature and their difference grows with increasing temperature. This implies the validation of the WF law at zero temperature, a departure from it at finite temperature. The finite-temperature departure from the WF law is more significant in the sample with lower carrier density.

∆κ xx = κ e xx (B = 0) -κ e xx (B) (5)
In order to extract the longitudinal electronic thermal conductivity at zero magnetic field (κ e xx (B = 0)) from thermal magneto-conductivity (∆κ xx ), we need an additional assumption: At any given temperature, the field dependence of κ e xx , is similar to the field dependence of the electrical conductivity (expressed by Eq. 3). Since the field-induced reduction in conductivity, in both thermal and electrical channels, is due to the same Lorentz T ( K )

κ x x / T ( W K -2 m -1 ) # 1 n = 3 . 1 ×1 0 2 0 c m -3 µ 0 H : 0 T 1 2 T κ x x / T : L 0 σ x x : ( a ) µ 0 H : 0 T 1 2 T κ x x / T : L 0 σ x x : # 2 n = 1 . 8 ×1 0 2 0 c m -3 T ( K ) ( b ) FIG. 1.
Thermal conductivity in Nb-doped SrTiO3: Thermal conductivity divided by temperature (κxx/T ) at B= 0 and B= 12 T compared with the electrical conductivity multiplied by the Sommerfeld value (L0σxx) in sample #1 (a) and in sample #2 (b). κxx/T increases with warming, because of the phonon contribution, which rises faster than T . L0σxx, which is a rough estimate of electronic contribution to κxx/T decreases with warming due to the reduction of electrical conductivity by inelastic scattering. Note the reduction induced by magnetic field in both. The inset is a zoom on the low-temperature data, showing that they tend to join in the zero-temperature limit. 0 . 0 0 4 0 . 0 0 8 0 . 0 0 0 [37,43].

0 1 0 2 0 3 0 0 1 0 2 0 3 0 0 . 0 1 0 . 0 2 # 1 ( a ) ∆κ x x / T L 0 ∆σ x x κ/ T , L 0 σ ( W K -2 m -1 ) # 1 -κ x y / T L 0 σ x y κ x y p h ( b ) # 2 -κ x y / T L 0 σ x y κ x y p h T ( K ) ( d ) ∆κ x x / T L 0 ∆σ x x # 2 ( c ) κ/ T , L 0 σ ( W K -2 m -1 ) T ( K )
1 0 2 4 0 0 . 1 1 1 0 1 0 2 4 0 0 . 1 1 1 0 1 1 0 0 . 4 4 0 1 0 -3 1 0 -2 1 0 -1 1 0 0 1 0 1 1 0 2 # 2 κ t o t a l κ p h κ e L 0 σ 0 T κ x x ( W K -1 m -1 ) T ( K ) ( a ) κ t o t a l κ p h κ e L 0 σ 0 T # 1 ( b ) κ x x ( W K -1 m -1 ) T ( K ) # 1 # 2 S r T i 1 -x N b x O 3 [ 1 8 ] S r T i O 3 [ 3 6 , 4 4 ] S r 1 -x C a x T i O 3 [ 3 7 ] κ ( W K -1 m -1 ) T ( K ) ( c )
κ P h FIG. 3. Electron and phonon contributions to the thermal conductivity (a) The total thermal conductivity (κ total xx ) and its electronic (κ e xx ) and phononic (κ ph xx ) components as a function of temperature in sample #1. Also shown is the electrical conductivity multiplied by L0. (b) same for sample #2. (c) Comparison of the phonon thermal conductivity in the two samples with total thermal conductivity of undoped STO [36,45] with κ ph xx in SrTi1-xNbxO3 n = 2.6 × 10 20 cm -3 just above the superconducting transition [18] and with Sr0.991Ca0.009TiO3 [37].

force, this is a reasonable assumption. It implies that the Lorenz ratio (L = κ e xx σxxT ) has a negligible field dependence. This assumption is consistent with our fielddependent data, which shows at a given temperature L L0 is less than unity, but its amplitude does not depend on magnetic field (See Fig. S2 in the supplement [41]). Thus, the magnetoresistance, which is not quadratic in magnetic field is set by the field dependence of residual resistivity and there is no detectable field-induced change in inetlastic scattering. This approach leads us to [41] :

κ e xx (B = 0) = ∆κ xx σ xx (B = 0) ∆σ xx (6)
Having quantified κ e xx from our data, we can deduce κ ph xx by subtracting the electronic component from the total conductivity. xx in undoped STO [36]. As one can see in the figure, κ ph of our metallic samples, with about 1% of Ti atoms replaced by Nb, is similar to the total κ of insulating samples of Sr 1-x Ca x TiO 3 , with about 1% of Sr atoms are replaced by Ca. In both cases, κ ph is reduced in comparison to pristine STO, because the substituting atoms are randomly distributed and their average distance is comparable with the order of magnitude of the wavelength of thermally excited phonons. The rough similarity between Nb doping (which brings mobile electrons) and Ca substitution (which does not), indicates that scattering by mobile electrons plays a minor role.

Let us now turn our attention to the electronic thermal resistivity, W T , obtained by inverting κ e xx /T . Fig 4 shows ρ and L 0 W T as a function of T 2 for the two samples. One can see that in both samples, Equations The two black solid lines show the lower and the upper limit to the slope of thermal resistivity in the two samples.

1 and 2 hold. ρ 0 and L 0 W T 0 are identical at low temperature confirming the validity of the WF Law in the zero-temperature limit. In both samples, the slope of L 0 W T (T 2 ) (B in equation 2) is larger than the slope of ρ(T 2 ) (A in equation 2). This behavior is similar to what has been observed in semimetals (like W, WP 2 , and Sb) and heavy-fermions (like UPt 3 , and CeRhIn 5 ) (See Tab.I) and corresponds to what is theoretically expected in the e-e scattering picture [START_REF] Li | [END_REF].

The fermiology of doped strontium titanate has been the subject of several recent studies [14,17,18,46,47]. Experiments have confirmed that, as expected by band calculations [14], three bands associated with Ti orbitals are successively filled, as the doping increases. In the two samples studied here, the carrier density is such that the Fermi surface consists of three concentric pockets all three centered at the gamma point. The average radius of the outer pocket is bounded by the carrier density: k max F < (3π 2 n) 1/3 . This yields 2.1 (1.7) nm -1 in sample #1 (#2). The width of the Brillouin zone is G = 2π a = 16.1nm -1 , where a = 0.3905 nm is the lattice parameter. Since k max F < G 4 , Umklapp events cannot occur. This distinguishes metallic strontium titanate from other metals displaying T-square ρ and W T with amplitudes linked to each other by the WF law.

The other Fermi liquid with a T-square thermal resistivity in absence of Umklapp is normal liquid 3 He [START_REF] Dobbs | Helium Three[END_REF][START_REF] Leggett | [END_REF]. The dominant contribution to thermal conductivity (in the zero-temperature limit) is proportional to the inverse of temperature. This κT term is strictly equivalent to the inverse of B, the slope of W T as a function T 2 , and, as first calculated by Abrikosov and Khalatnikov [50], is proportional to the fermion-fermion scattering time, which quadratically decreases with temperature. Extracted from thermal conductivity, this scattering time was dubbed τ κ , and τ κ T 2 was extensively measured by Greywall [34]. Theoretically, this quantity was computed by quantifying the Landau parameters of the Fermi liquid [51][52][53]. The agreement between the theoreticallycomputed and the experimentally-measured κT and τ κ T 2 is within experimental uncertainty at saturating vapor pressure and less than a factor of 2 near the melting pressure.

Let us now see how metallic strontium titanate fits in this picture. τ κ is given by [34]:

κ = 1 3 C V V m v 2 F τ κ (7)
Here, C V is the molar specific heat, V m is the molar volume and v F the Fermi velocity. This means that, in analogy with the case of normal liquid 3 He [34], B, the prefactor of T-square thermal resistivity, is inversely proportional to τ κ T 2 :

1 τ κ T 2 = v 2 F 3 γ V m B (8)
Using the reported values of γ (ranging from 1.55 to 1.9 mJ/mol.K 2 [19,35,54] at this doping level) and extracting the average Fermi wave-vector from carrier density, one can quantify the Fermi velocity and find τ κ T 2 . The results are listed in Tab. II. Unsurprisingly, τ κ T 2 is orders of magnitude larger in STO than in 3 He, which has a lower Fermi temperature and higher fermion-fermion collision cross section.

A more instructive basis for comparison is a dimensionless collision cross-section defined as [53,[55][START_REF] Vollhardt | The superfluid phases of helium 3[END_REF][START_REF] Behnia | [END_REF]:

ζ = ℏE F τ κ T 2 k 2 B (9)
The amplitude of this quantity in a Fermi liquid is set by a combination of its Landau parameters [55,58,59]. As seen in table II, ζ in 3 He, strongly correlated and close to both localisation and a magnetic instability [60], varies from 35 to 60. In contrast, ζ in STO, at this doping level (where the effective mass is close to four times the bare electron mass [16,35]), is ≈ 3.

Thus, not only the existence of the T-square thermal resistivity in metallic strontium titanate, but also its amplitude can be accounted for by considering it as a Fermi liquid with moderate correlations. As for T-square electrical resistivity (at low temperatures, that is below the degeneracy temperature of electrons and the minimum energy of the soft TO phonons), it could be accounted for, assuming a rough recovery of the Wiedemann-Franz law in presence of disorder. However, the theory for such a scenario is yet to be elaborated. It may require including the gradient of momentum flow caused by disorder and phonon scattering.

Material ρ0 (nΩ • cm) A (nΩ • cm • K -2 ) B (nΩ • cm • K -2 ) B/A reference W
(5.66 ± 0.04) × 10 -2 (8.7 ± 0.3) × 10 -4 (5.3 ± 0.15) × 10 -3 6. Supplemental Material for "T-square electron thermal resistivity in metallic strontium titanate" S1. Materials and methods [13,20,62] and Sr 1-x La x TiO 3 [13] at similar carrier doping levels. Above 100 K, the resistivity of samples with the same carrier concentration is very similar. Below 80 K, they show different residual resistivities. Our samples tend to display a comparatively lower residual resitivity. in absence of electrical current (J e =0 and J Q ̸ =0). In the measurement of W xx the temperature is the average temperature between T 1 and T 2 . In the measurement of W xy , the average temperature is taken between T 1 and T The variation of thermal and electric conductivity caused by the magnetic field are similar. This is true for both the longitudinal and transverse response. The Lorentz force is thus the main source for the variation of thermal conductivity. The field-dependent Lorenz ratio in longitudinal and transverse are shown in Fig S3 (c) and (f). The finite temperature departure of the Wiedemann Franz law does not change with the magnetic field. This confirms our assumption of a field-independent Lorenz number.

S3. Absence of phonon drag thermal Hall effect

Fig S4 shows the Seebeck effect of sample #2 compared with a SrTiO 3-x sample with a carrier density two orders of magnitude lower n = 1.6 × 10 18 cm -3 . While a phonon drag peak is found in the low-doped material it is absent in sample #2. Moreover samples #1 and #2 have a much smaller Hall angle compared with low doped SrTiO 3-σ (n = 1 × 10 18 cm -3 ). Therefore, no phonon drag contribution to κ xy is expected to arise in sample #2 and also #1 [37].

S4. Extracting electronic thermal conductivity from thermal and electrical magnetoresistance

Two assumptions have been done to extract the temperature dependence of the electronic thermal conductivity. First we assume that the phonon contribution is independent of the magnetic field. The change in the total conductivity (κ xx ) is thus equal to the change in field of the electronic thermal conductivity : ) and in a lightly doped SrTiO3-σ (n=1.6e10 18 cm -3 ). No phonon drag peak is observed in sample #2. The Hall angle in sample #1 is much lower than in lightly doped SrTiO3-σ.

Fig 2 (

 2 Fig 2 (a) and (c) compare ∆κ xx /T with L 0 ∆σ xx .In both samples, these two quantities converge at low temperature and their difference grows with increasing temperature. This implies the validation of the WF law at zero temperature, a departure from it at finite temperature. The finite-temperature departure from the WF law is more significant in the sample with lower carrier density.In order to extract the longitudinal electronic thermal conductivity at zero magnetic field (κ e xx (B = 0)) from thermal magneto-conductivity (∆κ xx ), we need an additional assumption: At any given temperature, the field dependence of κ e xx , is similar to the field dependence of the electrical conductivity (expressed by Eq. 3). Since the field-induced reduction in conductivity, in both thermal and electrical channels, is due to the same Lorentz

FIG. 2 .

 2 FIG. 2. Longitudinal and Hall conductivity: (a) The difference in longitudinal thermal conductivity divided by temperature between zero field and 12 T. (∆κxx = κxx(0T ) -κxx(12T )) in sample #1. Also shown is the difference in the longitudinal electric conductivity multiplied by the Sommerfeld value (∆σxx = σxx(0T ) -σxx(12T )). (c) Same for sample #2. (b) The transverse thermal conductivity κxy divided by temperature, compared with the transverse electric conductivity σxy multiplied by L0 (d) same for sample #2. Also shown in (b) and (d) is the κxy/T caused by phonons in undoped pure STO[37,43].

  Fig 3 shows the results. One can see in panels (a) and (b) that, above 20 K, κ ph xx becomes an order of magnitude larger than κ e . However, since κ ph xx decreases faster than κ e xx with cooling, the electron contribution becomes prominent below 10 K. It is almost equal to L 0 σ 0 T at low temperature, but becomes significantly lower at higher temperatures. The extracted κ ph xx , shown in Fig 3 (c), is significantly lower than κ ph

FIG. 4 .

 4 FIG.4. Electronic thermal resistivity: (a) Electric and thermal resistivity as a function of the square of the temperature for sample #1 (b) Same as in (a) for sample#2. In both cases, ρ and WT have the same intercept but different slopes. The two black solid lines show the lower and the upper limit to the slope of thermal resistivity in the two samples.

  FIG.S1. Electrical resistivity (ρ): ρ vs. temperature in log scale from 2 K to 300 K. Our samples (#1 and #2) are compared with those with similar carrier concentrations in previous studies of SrTi1-xNbxO3[20,62] and Sr1-xLaxTiO3[13].

SrTi 1 -

 1 x Nb x O 3 crystals were commercially provided by SurfaceNet GmbH. The nominal Nb content for sample #1 (#2) is 1 wt% (0.5 wt%). The expected carrier concentration for #1 and #2 is 3.3 × 10 20 cm -3 and 1.7 × 10 20 cm -3 , respectively, in good agreement with the carrier concentration obtained by measuring the Hall coefficient (#1 n H = 3.1 × 10 20 cm -3 and #2 n H = 1.8 × 10 20 cm -3 ). Fig S1 compares the temperature-dependence of their resistivity with previous data on SrTi 1-x Nb x O 3

FIG. S2 .

 S2 FIG. S2. Sketch of the setup measurement : T1 T2 and T3 are three Cernox CX-1030 thermometers. The heater is a 1 KΩ chip resistor. All connections were made using silver paste. 6 tin pads (white rectangles) are soldered on the sample for ohmic contacts. This setup allows us to obtain all transport coefficients.

Fig

  Fig S2 shows a sketch of the setup used to measure the electrical and thermal resistance reported in the main manuscript. Longitudinal (ρ xx = Ex J e ) and transverse resistivity (ρ xy = Ey J e ) are measured in absence of heat current (J Q =0 and J e ̸ =0). Longitudinal (W xx = -∇Tx J Q ) and transverse (W xy = -∇Ty J Q ) thermal resistivity are measured

Fig

  FigS3 shows the field-dependent electric and thermal conductivity in sample #1. The variation of thermal and electric conductivity caused by the magnetic field are similar. This is true for both the longitudinal and transverse response. The Lorentz force is thus the main source for the variation of thermal conductivity. The field-dependent Lorenz ratio in longitudinal and transverse are shown in FigS3 (c) and (f). The finite temperature departure of the Wiedemann Franz law does not change with the magnetic field. This confirms our assumption of a field-independent Lorenz number.

κ

  FIG. S3. Field dependence of the electric and thermal conductivity: (a) Longitudinal thermal conductivity (∆κxx = κxx(0)-κxx(B)) divided by temperature, (b) Longitudinal electric conductivity (∆σxx = σxx(0)-σxx(B)) and (c) Longitudinal Lorenz ratio (Lxx/L0 = ∆κxx/T L0∆σxx) as a function of the magnetic field for three temperatures. (d), (e) and (f) are the same as (a), (b), and (c) in transverse configuration. The Lorenz ratios Lxx and Lxy are constant in field but smaller than L0.

  FIG. S4. Seebeck effect and Hall angle: (a) Seebeck coefficient (S), (b) Hall angle as a function of temperature for sample #2 (n=1.8e10 20 cm -3) and in a lightly doped SrTiO3-σ (n=1.6e10 18 cm -3 ). No phonon drag peak is observed in sample #2. The Hall angle in sample #1 is much lower than in lightly doped SrTiO3-σ.

TABLE I :

 I T-square resistivity in metals-Residual resistivity ρ 0 , and electrical (A) and thermal (B) T-square prefactors and their ratio in several Fermi liquids. In the case of CeCoIn 5[61], the Fermi liquid behavior appears only in presence of a magnetic field larger than the upper critical field of the superconductor.B A is always found to be larger than unity, varying between 1.5 and 6.

	System	κ T (W.m) vF (m/s) kF (nm -1 ) EF (K) τκT 2 (ns.K 2 )	ζ
	3 He (saturating vapor pressure) 2.9 ×10 -4	60	7.9	1.8	3.9 ×10 -4	35
	3 He (melting pressure)	7.3 ×10 -5	32.4	8.9	1.1	1.4 ×10 -4	58
	SrTi3-xNbxO3 (n=3.1×10 20 cm -3 )	27 ±6	6 ×10 4	2.1	480	1.4	2.6 ±0.6
	SrTi3-xNbxO3 (n=1.8×10 20 cm -3 ) 13 ±1.5	5 ×10 4	1.7	330	0.8	3.2 ±0.3

TABLE II :

 II Thermal and electronic properties of 3 He and SrTi 3-x Nb x O 3 : Thermal conductivity, fermion-fermion scattering and Fermi liquid properties in a strongly correlated and a weakly correlated fermionic system.

  3 . The (electric or thermal) conductivity tensor is the inverse of the (electric or thermal) resistivity tensor. Hence the longitudinal and transverse electric and thermal conductivity are respectively equal to σ xx = ρxx

				ρ 2 xx +ρ 2 xy	, σ xy =	-ρxy ρ 2 xx +ρ 2 xy
	and κ xx = Wxx W 2 xx +W 2 xy	, κ xy =	-Wxy W 2 xx +W 2 xy	.
			S2. Field independence of the Lorenz ratio

We thank L. Hechler, M. Feigel'man, X. Li, A. Marguerite, D. Maslov, D. Vollhardt P. Wölfle, and Z. Zhu for discussions. This work was supported by the Agence Nationale de la Recherche (ANR-19-CE30-0014-04), by Jeunes Equipes de l ′ Institut de Physique du Collège de France and by a grant attributed by the Ile de France regional council. S.J. acknowledges a grant from China Scholarship Council.