Xiao Lin

Carl Willem Rischau

Lisa Buchauer

Alexandre Jaoui

Benoît Fauqué

Kamran Behnia

Metallicity without quasi-particles in room-temperature strontium titanate

I. INTRODUCTION

The existence of well-defined quasi-particles is taken for granted in the Boltzmann-Drude picture of electronic transport. In this picture, carriers of charge or energy are scattered after traveling a finite distance. The Mott-Ioffe-Regel(MIR) limit is attained when the mean-freepath of a carrier falls below its Fermi wavelength or the interatomic distance [START_REF] Hussey | Universality of the Mott-Ioffe-Regel limit in metals[END_REF]. In most metals, resistivity saturates when this limit is approached. But in "bad" [START_REF] Emery | Superconductivity in bad metals[END_REF] or "strange" [START_REF] Sachdev | Bekenstein-Hawking Entropy and Strange Metals[END_REF] metals, it continues to increase [START_REF] Gunnarsson | Colloquium: Saturation of electrical resistivity[END_REF]. It is indeed strange when the mean-free-path persists to fall after attaining its shortest conceivable magnitude and often unexpected behavior is considered bad.

In most cases, bad metals present a linear temperature dependence beyond the MIR limit. Bruin et al. [START_REF] Bruin | Similarity of scattering rates in metals showing T-linear resistivity[END_REF] recently noticed that the T-linear scattering rate in numerous metals, either ordinary or strange, has a similar magnitude. This observation suggests the relevance of a universal Planck timescale (τ P ∼ k B T , where k B is the Boltzmann and the reduced Planck constants) to electronic dissipation and motivated a theoretical attempt to quantify an upper limit to the diffusion constant in incoherent metals [START_REF] Hartnoll | Theory of universal incoherent metallic transport[END_REF]. During the past years, several theories of charge transport with no need for well-defined quasi-particles were proposed [START_REF] Mukerjee | Statistical theory of transport by strongly interacting lattice fermions[END_REF][START_REF] Lindner | Conductivity of hard core bosons: A paradigm of a bad metal[END_REF][START_REF] Andreev | Hydrodynamic Description of Transport in Strongly Correlated Electron Systems[END_REF][START_REF] Davison | Holographic duality and the resistivity of strange metals[END_REF][START_REF] Principi | Violation of the Wiedemann-Franz law in hydrodynamic electron liquids[END_REF][START_REF] Pakhira | Shear viscosity of strongly interacting fermionic quantum fluids[END_REF][START_REF] Limtragool | Power-law optical conductivity from unparticles: Application to the cuprates[END_REF][START_REF] Perepelitsky | Transport and optical conductivity in the Hubbard model: A hightemperature expansion perspective[END_REF]. More recently, direct measurements of thermal diffusivity [START_REF] Zhang | Anomalous Thermal Diffusivity in Underdoped YBa2Cu3O6+x[END_REF] have provided additional evidence for quasi-particle-free thermal transport at room temperature in cuprates.

We show here that n-doped SrTiO 3 is an unnoticed case of strange or bad metallicity. The resistivity of this dilute metal decreases by orders of magnitude upon cooling to cryogenic temperatures [START_REF] Frederikse | Electronic Transpoort in Strontium Titanate[END_REF][START_REF] Tufte | Electron Mobility in Semiconducting Strontium Titanate[END_REF][START_REF] Frederikse | Hall Mobility in SrTiO3[END_REF][START_REF] Wemple | Electron Scattering in Perovskite-Oxide Ferroelectric Semiconductors[END_REF][START_REF] Spinelli | Electronic transport in doped SrTiO3: Conduction mechanisms and potential applications[END_REF]. In the low-temperature limit, resistivity follows a T-square behavior [START_REF] Van Der Marel | Common Fermi-liquid origin of T 2 resistivity and superconductivity in n-type SrTiO3[END_REF][START_REF] Lin | Scalable T 2 resistivity in a small single-component Fermi surface[END_REF], as expected in a Fermi liquid. Above 100 K, however, the temperature dependence is close to cubic [START_REF] Tufte | Electron Mobility in Semiconducting Strontium Titanate[END_REF][START_REF] Lin | Scalable T 2 resistivity in a small single-component Fermi surface[END_REF]. In this regime, the mean-free-path of carriers becomes shorter than their wavelength. At room temperature, it falls well below the interatomic distance, the lowest conceivable length scale. Strontium titanate is the strangest known metal according to the criteria for strangeness widely used. Since quasi-particles in this system are not well-defined in a time scale long enough to allow the existence of a scattering time, charge transport cannot be reasonably described by a phonon-scattering picture. This was the approach in previous attempts [16-19, 24, 25, 27] to understand charge transport. We argue that metallicity is caused by the temperature dependence of the transmission coefficient along Landauer canals between dopants. Such a picture does not require a scattering time or a mean-free-path. We find an empirical link between mobility and permittivity, which gives a reasonable account of the experimental data. Providing a quantitative account of this observation emerges as a specific challenge to transport theory.

II. RESULTS

A. Metallicity beyond the MIR limit in strontium titanate

Fig. 1a shows the temperature dependence of resistivity in SrTiO 3-δ as the carrier concentration is changed from 2.4 10 17 cm -3 to 3.5 10 19 cm -3 . At low temperature (T 100K), the system displays a Tsquare resistivity [START_REF] Van Der Marel | Common Fermi-liquid origin of T 2 resistivity and superconductivity in n-type SrTiO3[END_REF][START_REF] Lin | Scalable T 2 resistivity in a small single-component Fermi surface[END_REF] as expected for electron-electron scattering [28] in a Fermi liquid. Plotting resistivity as a function of T 2 (see Fig. 2 in ref. [START_REF] Lin | Scalable T 2 resistivity in a small single-component Fermi surface[END_REF]), one can see that inelastic resistivity becomes T-square in the lowtemperature limit. A wide range of dense Fermi liquids display Kadowaki-Woods scaling [START_REF] Kadowaki | Universal relationship of the resistivity and specific heat in heavy-fermion compounds[END_REF] of their T-square I: How the MIR limit of metallicity is exceeded in different perovskytes-The table compares the Fermi wave-vector, the resistivity and the mean-free-path at 400 K in four different systems using equation 1. The table lists an underdoped, an optially-doped cuprate and strontium ruthenate and compares them with doped strontium titanate. In the case of YBa2Cu4O8, we assume that there are four Fermi surface pockets with a kF corresponding to the square root of the frequency of quantum oscillations. In contrast to other systems, the MIR limit in doped strontium titanate occurs well below room temperature.

Compound kF (nm

1 1 0 1 0 0 3 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 -4 1 0 -3 1 0 -2 1 0 -1 1 0 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 2 . 0 2 . 5 3 . 0 n ( 1 0 1 7 c m -3 ) = m o b i l i t y ( c m 2 / ( V . s ) T ( K ) T -3 b ) 2 . 4 4 . 1 8 . 0 1 2 1 4 2 6 3 2 5 4 8 3 1 7 0 2 9 0 ρ ( Ωc m ) S r T i O 3-δ a ) α T ( K ) ρ= ρ 0 + T α T A F D c )
FIG. 1: Resistivity and mobility in doped SrTiO3: a) Temperature dependence of resistivity in SrTiO 3-δ as the carrier concentrations is tuned from 10 17 to 10 19 cm -3 . Roomtemperature resistivity is several hundred times higher than low-temperature resistivity. b) Hall mobility as a function of temperature. Above 100 K, the mobility does not depend on carrier concentration and is roughly cubic in temperature. c) Assuming that inelastic resistivity follows a power law, i.e. ρ = ρ0 + AT α , one can extract the exponent, α by taking a logarithmic derivative: α = dln(ρ -ρ0)/dlnT , as in the case of cuprates [START_REF] Cooper | Anomalous Criticality in the Electrical Resistivity of La2xSrxCuO4[END_REF]. At low temperature, α 2. Above 50K, it starts a significant shift upward and exceeds 3 around 150K, before steadily decreasing afterwards. The antiferrodistortive transition [START_REF] Tao | Nonmonotonic anisotropy in charge conduction induced by antiferrodistortive transition in metallic SrTiO3[END_REF] is the source of the small anomaly at 105K. resistivity prefactor and their electronic specific heat. Since the latter is proportional to carrier concentration, the case of low-density systems such as doped SrTiO 3 is different [START_REF] Hussey | Non-generality of the Kadowaki-Woods Ratio in Correlated Oxides[END_REF]. However, the scaling between the resistivity prefactor and Fermi energy persists. Indeed, the magnitude of this prefactor decreases by several orders of magnitude with changing carrier concentration [START_REF] Lin | Scalable T 2 resistivity in a small single-component Fermi surface[END_REF].

Fig. 1b shows the temperature dependence of the extracted Hall mobility, µ = 1 neρ . Above 100 K, it displays a temperature dependence close to T -3 and barely varies with carrier concentration. Our data is in agreement with the less extensive data reported previously by Tufte and Chapman [START_REF] Tufte | Electron Mobility in Semiconducting Strontium Titanate[END_REF]. Fig. 1c shows how the exponent of inelastic resistivity evolves as a function of temperature. The total resistivity can be expressed as ρ = ρ 0 + AT α . As seen in Fig. 1c, α is 2 at very low-temperature (T < 50K) but smoothly rises with warming. One does not expect the T-square resistivity to survive at the Fermi degeneracy temperature. Therefore, the electron-electron scattering origin of the T-square term in the dilute limit has been put into question [START_REF] Swift | Conditions for T 2 resistivity from electron-electron scattering[END_REF][START_REF] Maslov | Optical response of correlated electron systems[END_REF]. We note, however, that the smooth variation of α, makes the detection of the upper boundary of T-square resistivity impossible. As one can see in the figure, α peaks around 150 K at a value slightly exceeding 3 and then decreases with increasing temperature.

In the zero temperature limit, the magnitude of electron mobility, set by defect scattering can become remarkably large in doped strontium titanate [START_REF] Son | Epitaxial SrTiO3 films with electron mobilities exceeding 30,000 cm 2 V -1 s -1[END_REF]. As a consequence, quantum oscillations are easily detectable in moderate magnetic fields [START_REF] Lin | The Fermi surface of the most dilute superconductor[END_REF][START_REF] Allen | Conduction-band edge and Shubnikov-de-Haas effect in low-electron-density SrTiO3[END_REF][START_REF] Lin | Critical Doping for the Onset of a Two-Band Superconducting Ground State in SrTiO 3-δ[END_REF]. The large magnitude of defect-limited mobility at low temperature and its evolution with carrier concentration can be explained in a simple model [START_REF] Behnia | On mobility of electrons in a shallow Fermi sea over a rough seafloor[END_REF] invoking the long effective Bohr radius of the quantum paraelectric [START_REF] Müller | SrTiO3: An intrinsic quantum paraelectric below 4 K[END_REF] parent insulator.

The focus of the present paper is charge transport above 100 K where mobility presents a strong temperature dependence and almost no variation with carrier concentration over two orders of magnitude (Fig. 1b).

Early studies [START_REF] Frederikse | Electronic Transpoort in Strontium Titanate[END_REF][START_REF] Tufte | Electron Mobility in Semiconducting Strontium Titanate[END_REF][START_REF] Frederikse | Hall Mobility in SrTiO3[END_REF][START_REF] Wemple | Electron Scattering in Perovskite-Oxide Ferroelectric Semiconductors[END_REF] attributed the T -3 variation of mobility to the scattering of carriers by phonons. There was a controversy between Fredrikse et al. [START_REF] Frederikse | Hall Mobility in SrTiO3[END_REF] (who invoked longitudinal optical phonons) and Wemple et al. [START_REF] Wemple | Electron Scattering in Perovskite-Oxide Ferroelectric Semiconductors[END_REF] (who argued in favor of transverse optical phonons). Neither during this early debate [START_REF] Frederikse | Electronic Transpoort in Strontium Titanate[END_REF][START_REF] Tufte | Electron Mobility in Semiconducting Strontium Titanate[END_REF][START_REF] Frederikse | Hall Mobility in SrTiO3[END_REF][START_REF] Wemple | Electron Scattering in Perovskite-Oxide Ferroelectric Semiconductors[END_REF], nor in more recent the- . Also shown are the de Broglie thermal wavelength, Λ, the interelectron distance n -1/3 , the inverse of the Fermi wavelength (k -1 F ) and the lattice parameter, a=0.39 nm. The system remains metallic even at 400 K, in spite of exceeding the MIR limit by all conceivable criteria. b) A color plot of e , extracted from the resistivity data of Fig. 1 in the (T, n) plane. Different crossovers are shown. The MIR limit is exceeded in a high-temperature window narrowing down with increasing concentration. oretical accounts of the temperature dependence of mobility in n-doped SrTiO 3 [START_REF] Baratoff | Mechanism of superconductivity in SrTiO3[END_REF][START_REF] Verma | Intrinsic Mobility Limiting Mechanisms in Lanthanum-Doped Strontium Titanate[END_REF][START_REF] Himmetoglu | First-principles study of the mobility of SrTiO3[END_REF][START_REF] Mikheev | Limitations to the room temperature mobility of two-and three-dimensional electron liquids in SrTiO3[END_REF][START_REF] Zhou | Electron-soft phonon scattering in n -type SrTiO3[END_REF], the magnitude of the mean-free-path was discussed.

In the Drude-Boltzmann picture, the conductivity, σ, of a metal depends on fundamental constants and a number of material-related properties. These are either, the scattering time τ , and the effective mass, m * , or the Fermi wave-vector, k F and the electron mean-free-path, e . Namely:

σ = ne 2 τ m * = 1 3π 2 e 2 k 2 F e (1)
The Fermi surface of n-doped SrTiO 3 in the dilute limit is known. It is located at the center of the Brillouin zone, because a band originating from Ti atoms' t 2g orbitals has a minimum at the Γ-point [START_REF] Van Der Marel | Common Fermi-liquid origin of T 2 resistivity and superconductivity in n-type SrTiO3[END_REF][START_REF] Allen | Conduction-band edge and Shubnikov-de-Haas effect in low-electron-density SrTiO3[END_REF][START_REF] Tao | Nonmonotonic anisotropy in charge conduction induced by antiferrodistortive transition in metallic SrTiO3[END_REF]. The threefold degeneracy of this band is lifted by tetragonal distortion and spin-orbit coupling. As the system is doped, the three bands are filled successively and three concentric Fermi surfaces will emerge and grow in size one after the other. This is a picture based on band calculations [START_REF] Van Der Marel | Common Fermi-liquid origin of T 2 resistivity and superconductivity in n-type SrTiO3[END_REF] and confirmed by experimental studies of quantum oscillations [START_REF] Lin | The Fermi surface of the most dilute superconductor[END_REF][START_REF] Allen | Conduction-band edge and Shubnikov-de-Haas effect in low-electron-density SrTiO3[END_REF][START_REF] Lin | Critical Doping for the Onset of a Two-Band Superconducting Ground State in SrTiO 3-δ[END_REF]. Below 10 18 cm -3 , the first critical doping for the emergence of secondary pockets, there is a single Fermi surface with moderate (1.6) anisotropy [START_REF] Allen | Conduction-band edge and Shubnikov-de-Haas effect in low-electron-density SrTiO3[END_REF][START_REF] Lin | Critical Doping for the Onset of a Two-Band Superconducting Ground State in SrTiO 3-δ[END_REF][START_REF] Tao | Nonmonotonic anisotropy in charge conduction induced by antiferrodistortive transition in metallic SrTiO3[END_REF]. Therefore, k F can be quantified in a straightforward manner. Fig. 2a shows the temperature dependence of the mean-free-path extracted from Eq. 1 in a dilute sample whose resistivity was measured up to 400 K. The carrier density is the one given by the Hall coefficient. As previously reported in ref. [START_REF] Tufte | Electron Mobility in Semiconducting Strontium Titanate[END_REF] , the Hall coefficient does not display any notable temperature dependence. Moreover, the carrier density according to the magnitude of the Hall coefficient matches the Fermi surface volume according to the frequency of quantum oscillations [START_REF] Lin | Critical Doping for the Onset of a Two-Band Superconducting Ground State in SrTiO 3-δ[END_REF]. Therefore, there is little ambiguity regarding carrier concentration.

As seen in Fig. 2a, the mean-free-path becomes shorter than the inverse of the Fermi wave-vector k -1 F and then the lattice parameter a. Now, at this carrier concentration, with an effective mass of m * =1.8m e , the Fermion degeneracy temperature is as low as 15 K [START_REF] Lin | The Fermi surface of the most dilute superconductor[END_REF]. As one sees in the figure, this is the temperature at which de Broglie thermal wavelength, Λ = h √ 2πm * k B T becomes comparable to the interelectron distance n -1/3 . Above this temperature, the electrons become non-degenerate and follow a Maxwell-Boltzmann distribution. As a consequence and thanks to thermal excitation, the typical carrier momentum is larger than k F . In other words, the typical velocity is not the Fermi velocity, but the thermal velocity, v T = 2k B T m * . Let us define the mean-free-path of electrons with thermal velocity as e = τ v T = e n -1/3 /Λ. As seen in the figure, because of e > e , the crossover temperatures shift upward. However, the picture remains qualitatively the same. In the non-degenerate regime, the relevant wavelength is Λ and not the Fermi wavelength. As seen in the figure, the mean-free-path becomes shorter than Λ too.

A room-temperature resistivity of 2.4 mΩ.cm implies a scattering time of τ 6f s. This is shorter than the Planck time [START_REF] Bruin | Similarity of scattering rates in metals showing T-linear resistivity[END_REF] (τ P = k B T ) at room temperature, τ P (300K) = 25.3f s. This is the typical time it takes for a carrier with a thermal velocity, v T , to move as far as its de Broglie thermal wavelength ( Λ v T = 1.77τ P ). By all accounts, room-temperature quasi-particles in dilute FIG. 3: Comparison with other dilute metallic systems: a) R of a one Fermi-wavelength-thick sheet of SrTiO 3-δ compared with graphene [START_REF] Efetov | Controlling Electron-Phonon Interactions in Graphene at Ultrahigh Carrier Densities[END_REF] with comparable areal carrier density. b) Temperature-dependence of resistivity in SrTiO 3-δ and in metallic silicon [START_REF] Yamanouchi | Electric Conduction in Phosphorus Doped Silicon at Low Temperatures[END_REF] with comparable volume carrier concentration. c) Temperature-dependence of resistivity in SrTiO 3-δ , metallic PbTe and metallic KTaO 3-δ [START_REF] Wemple | Some Transport Properties of Oxygen-Deficient Single-Crystal Potassium Tantalate (KTaO3)[END_REF] in similar range of carrier concentration. In silicon and graphene, the change in resistance caused by phonon scattering of electrons is a modest part of the total resistance. In all other systems, which are close to a ferroelectric instability, resistivity changes by several orders of magnitude.

metallic strontium titanate do not live long enough to qualify for a scattering-based picture.

A color plot of the mean-free-path in the (T, n) plane produced from the data of Fig. 1 and Eq. 1 is presented in Fig. 2b. The MIR limit is exceeded in a hightemperature window, widening with the decrease in carrier concentration. The hitherto unnoticed specificity of SrTiO 3 is that it becomes a strange metal well below room temperature. The data in Table 1 gives an idea of the "strangeness" or the "badness" of SrTiO 3 compared to two other perovskytes. As seen in the table, at 400 K, k F in this system is much lower than in cuprates or in Sr 2 RuO 4 . Contrary to the two other systems, the passage beyond the MIR limit occurs well below room temperature implying the necessity of a transport picture with no reference to well-defined quasi-particles. This is the first principal message of this paper.

B. Comparison with graphene and metallic silicon

The temperature dependence of resistivity above 100 K and the magnitude of inelastic resistivity are very different from dilute metallic systems subject to phonon scattering. Let us compare strontium titanate with two exemplary systems. A recent study of charge transport in graphene [START_REF] Efetov | Controlling Electron-Phonon Interactions in Graphene at Ultrahigh Carrier Densities[END_REF] has shown the relevance of the Bloch-Grüneisen picture of electron-phonon scattering in our temperature window of interest. In this picture, above a characteristic temperature (usually, but not always, the Debye temperature) resistivity displays a linear temperature dependence. This happens because all phonon modes capable of scattering electrons are thermally populated and the phase space for scattering is proportional to the thermal width near the Fermi level. Far below the characteristic temperature, the resistivity is expected to display a much stronger T 5 (in 3D) or T 4 (in 2D) temperature dependence because the typical wave-vector of the thermally excited phonons becomes smaller with decreasing temperature. As a consequence, the capacity of scattering phonons to decay the momentum flow decays rapidly with temperature. Efetov and Kim [START_REF] Efetov | Controlling Electron-Phonon Interactions in Graphene at Ultrahigh Carrier Densities[END_REF] showed that this picture gives a very satisfactory account of inelastic resistivity in graphene as the carrier density is tuned from 1.36 to 10.8 10 13 cm -2 . Let us note that in gated graphene, inelastic (that is, temperaturedependent) resistivity is far from dominating the total resistivity. This is seen in Fig. 3a, which compares the temperature-dependence of resistivity in n-doped SrTiO 3 and graphene at an identical interelectron distance. As one can see in the figure, resistivity changes by 200 in SrTiO 3 and by 1.6 in graphene. Charge conductivity is higher in graphene by a factor of two at low temperature and by two orders of magnitude at room temperature. A comparison with the case of metallic silicon [START_REF] Yamanouchi | Electric Conduction in Phosphorus Doped Silicon at Low Temperatures[END_REF] reveals a similar discrepancy. As one can see in Fig. 3b, cooling phosphorous-doped silicon leads to a modest change in resistivity, much less than what is seen in doped SrTiO 3 . As a consequence, while at low-temperature, strontium titanate has a significantly higher mobility, the opposite is true at room temperature. In contrast to strontium titanate, in both graphene and silicon, one can use Eq. 1 one to extract a meaningful mean-free-path, which increases by a factor of two or less as the system is cooled down from room temperature to helium liquefaction temperature.

It is also instructive to compare SrTiO 3 with two other dilute metals close to a ferroelectric instability. As one can see in Fig. 3c, both PbTe and KTaO 3 show comparably large changes in their resistivity upon cooling. Reports on KTaO 3 [START_REF] Wemple | Electron Scattering in Perovskite-Oxide Ferroelectric Semiconductors[END_REF][START_REF] Wemple | Some Transport Properties of Oxygen-Deficient Single-Crystal Potassium Tantalate (KTaO3)[END_REF] and on lead chalcogenide salts (PbTe, PbSe and PbS) [START_REF] Petritz | Mobility of Electrons and Holes in the Polar Crystal, PbS[END_REF][START_REF] Allgaier | Mobility of Electrons and Holes in PbS, PbSe, and PbTe between Room Temperature and 4.2[END_REF] indicate that hightemperature mobility does not vary with carrier concentration, but shows a strong temperature dependence (µ ∝ T -λ , with λ 2.5 -3). Both these features are similar to what we see in strontium titanate. Only in strontium titanate, however, the magnitude of the room Conductivity increases because the time interval (and the spatial distance) between two successive scattering events increases with cooling. Such a picture requires a well-defined quasi-particle across the distance in time and in space separating two scattering events. b) In a Landauer picture of transport, each dopant (each oxygen vacancy here) is a reservoir connected to other adjacent reservoirs by conducting channels. Conductivity increases upon cooling because the transmission coefficient enhances. Such a picture does not require well-defined quasi-particles.

temperature mobility is low enough to make it a clearly strange metal below room temperature.

To sum up, in silicon and graphene, the temperatureinduced change in resistivity implies a modest change in mean-free-path and follows a Bloch-Gruneisen behavior (a slow T-linear behavior at high temperature and a fast T 4 or T 5 behavior at low temperature). On the other hand, doped SrTiO 3 and other metals close to a ferroelectric instability do not fit within this picture. The apparent mean-free-path of carriers changes by orders of magnitude. It becomes very short at high temperature, and in the case of strontium titanate, too short to be physically meaningful.

III. DISCUSSION

A. Conduction as transmission in a doped polar semiconductor

Fig. 4 sketches two alternative descriptions of charge conduction. In the semi-classic picture of metallic charge transport (Fig. 4a), conductivity increases with decreasing temperature, because the scattering events relaxing the momentum of a charge-carrying quasi-particle become less frequent with decreasing temperature. Now, consider an alternative picture (Fig. 4b), where each dopant is hosting a polaron, 'an electron trapped by dig- ging its own hole' [START_REF] Mott | Electronic Processes in Ionic Crystals[END_REF] in a polar semiconductor. The polaron picture has been frequently invoked in the case of strontium titanate [START_REF] Eagles | Polaron coupling constants in SrTiO3[END_REF][START_REF] Van Mechelen | Electron-phonon interaction and charge carrier mass enhancement in SrTiO3[END_REF]. In this second picture, conductivity is set by transmission among dopant sites and there is no need for well-defined quasi-particles. Can conductivity increase with decreasing temperature in such a picture and generate a metallic behavior? This can happen if cooling increases the transmission probability between adjacent dopant sites. In the case of strontium titanate, static permittivity increases with cooling [START_REF] Müller | SrTiO3: An intrinsic quantum paraelectric below 4 K[END_REF]. Therefore, the lower the temperature, the shallower the donor potential. As a consequence, polarons become less bound with decreasing temperature and conductivity is metallic.

Let us now consider different length scales of the system. Fig. 5a shows them in our temperature range of interest where mobility becomes independent of carrier concentration. When carrier density changes from 2.4 × 10 17 cm -3 to 1.7 × 10 19 cm -3 , the distance between electrons varies between 16 to 4 nm. Taking the low-temperature effective mass of m * =1.8m e seen by quantum oscillations [START_REF] Lin | The Fermi surface of the most dilute superconductor[END_REF], one finds that the de Broglie thermal wave-length is Λ 3 -6nm in this temperature range. Thus, the system is mostly non-degenerate. Electric permittivity, , is large ( (300K) 300) [START_REF] Müller | SrTiO3: An intrinsic quantum paraelectric below 4 K[END_REF][START_REF] Lowndes | Stabilization of the paraelectric phase of KTaO3 and SrTiO3 by strong quartic anharmonicity[END_REF] and increases steadily with cooling. As a consequence, the effective Bohr radius (a * B = 4π 2 m * e 2 ) varies between 8 and 40 nm. This puts the system above the Mott criterion for metallicity (n 1/3 a * B > 0.25 [START_REF] Edwards | Universality aspects of the metal-nonmetal transition in condensed media[END_REF]), which, one shall not forget, is relevant to degenerate electrons [START_REF] Mott | Metal-insulator transitions[END_REF]. These length scales are in the same range of magnitude. They are all longer than the lattice parameter (0.39nm). The question is how to picture conduction as transmission [START_REF] Imry | Conductance viewed as transmission[END_REF] along a random network of donors.

Since mobility is found to be independent of carrier concentration above 100K, we can express conductivity as:

σ = ρ -1 = G 0 n < T > (2)
Here, G 0 = 2e 2 h , is the quantum of conductance and < T > represents the average transmission between a dopant site and its immediate neighbors. Note that in three dimensions, < T > is proportional to mobility (µ = 2 e h < T >).

The temperature dependence of < T >, extracted from the experimental data using Eq. 2 is shown in Fig. 5b. As seen in the figure, a hundred-fold enhancement in carrier concentration does not affect < T >. One would have expected that the Landauer transmission between one site and its more or less distant neighbors would depend on the length of the trajectory and the details of the structural landscape. It is therefore surprising that the averaged transmission coefficient, < T >, is insensitive to such a large change in the average interdopant distance.

Fig. 5b also compares the magnitude of < T > with Λ 2 . the latter represents the areal uncertainty in the spatial localisation of a non-degenerate electron in three dimensions. As seen in the figure, < T > decreases faster than Λ 2 . There is little experimental information regarding the effective mass in this temperature range. A larger temperature-dependent effective mass would make Λ 2 smaller and its decrease faster. Thus, one can bridge the gap between < T > and Λ 2 with a hypothetical temperature-dependent polaronic mass. However, there is no experimental evidence for this. At low temperatures, quantum oscillations [START_REF] Allen | Conduction-band edge and Shubnikov-de-Haas effect in low-electron-density SrTiO3[END_REF][START_REF] Lin | Critical Doping for the Onset of a Two-Band Superconducting Ground State in SrTiO 3-δ[END_REF] resolve a modest mass enhancement(≤ 4). This is also the case of infrared conductivity [START_REF] Van Mechelen | Electron-phonon interaction and charge carrier mass enhancement in SrTiO3[END_REF]. Thus, we cannot simply picture conductivity as ballistic charge transport along Landauer wires. Let us now consider a diffusive picture.

B. Mobility, diffusivity and their link to electric permittivity

In a non-degenerate semiconductor, mobility and diffusion constant, D, of carriers are linked through Einstein's equation [START_REF] Ashcroft | Solid State Physics[END_REF]:

D = µk B T e (3)
The temperature dependence of D extracted from the mobility data of the sample with a carrier concentration of 2.4 10 17 cm -3 is shown in Fig. 6a. Note that since mobility does not show any detectable variation with carrier concentration, this curve is representative of D for carrier concentrations in the range of 10 17 -10 19 cm -3 . The magnitude of D obtained in this way is remarkably low, well below 1cm 2 /s. For comparison, the diffusion constant of electrons in silicon rises from 35 cm 2 /s at room temperature to ∼100 cm 2 /s at 100 K [START_REF] Brunetti | Diffusion coefficient of electrons in silicon[END_REF].

Hartnoll recently proposed a lower bound to diffusion constant in incoherent metals [START_REF] Hartnoll | Theory of universal incoherent metallic transport[END_REF]:

D H = v 2 F k B T (4)
As one can see in Fig. 6a, the experimental D decreases faster than T -1 . Hartnoll boundary was obtained for degenerate systems. Ours is non-degenerate and the typical velocity is thermal velocity and not v F . The Fermi velocity changes drastically with carrier density, but the experimental D does not. As defined in Eq. 4 , D H , does not appear to be directly relevant to our observation.

An obvious piece of the puzzle of charge transport in strontium titanate is the fact that the insulating parent is a quantum paraelectric with a strongly temperaturedependent electric permittivity [START_REF] Müller | SrTiO3: An intrinsic quantum paraelectric below 4 K[END_REF][START_REF] Lowndes | Stabilization of the paraelectric phase of KTaO3 and SrTiO3 by strong quartic anharmonicity[END_REF]. We have found that in our temperature range of interest D is proportional to 1.5 . This empirical observation is the second principal message of this paper.

The scaling between static electric permittivity, 0 , and the frequency of the Transverse Optic (TO) soft mode [START_REF] Fleury | Electric-Field-Induced Raman Scattering in SrTiO3 and KTaO3[END_REF][START_REF] Yamada | Neutron Scattering and Nature of the Soft Optical Phonon in SrTiO3[END_REF], ω T O in strontium titanate is well-known. Yamada and Shirane [START_REF] Yamada | Neutron Scattering and Nature of the Soft Optical Phonon in SrTiO3[END_REF] showed that ω T O = 194.4 √ -1 within experimental resolution. In a polar crystal, there is a theoretical link between the ratio of static, 0 , to highfrequency, ∞ , permittivity and the ratio of longitudinal, ω L , to transverse, ω T , phonon frequencies. This is the expression first derived by Lyddane, Sachs, and Teller [START_REF] Lyddane | On the Polar Vibrations of Alkali Halides[END_REF][START_REF] Cochran | Dielectric constants and lattice vibrations[END_REF]:

0 ∞ = ( ω L ω T ) 2 (5)
When the transverse mode is soft, the system begins to show a large static polarizability. As one can see in Fig. 6b, the available experimental data confirm the scaling between these two distinct measurable quantities. A theory assuming an anisotropy in the polarizability of the oxygen ion [START_REF] Migoni | Origin of Raman Scattering and Ferroelectricity in Oxidic Perovskites[END_REF], the starting point of the polarization theory of ferroelectricity [START_REF] Bilz | Theory of ferroelectricity: The polarizability model[END_REF], explains the temperature dependence of ω T O in strontium titanate quantitatively. The soft mode and the large permittivity are properties of the insulating parent. However, they are known to survive in presence of dilute metallicity. Indeed, neutron scattering studies have explored the way the soft mode responds to the introduction of mobile electrons [START_REF] Bäuerle | Soft modes in semiconducting SrTiO3: II. The ferroelectric mode[END_REF][START_REF] Bussmann-Holder | A polarizability model for the ferroelectric mode in semiconducting SrTiO3[END_REF] and found negligible shift in frequency for carrier densities below 7.8 10 19 cm -3 . As one can see in Fig. 6a, comparing our data with those reported by Yamada and Shirane [START_REF] Yamada | Neutron Scattering and Nature of the Soft Optical Phonon in SrTiO3[END_REF], one finds that:

µk B T ∝ ω -3 T 0 ∝ /ω T 0 (6)
An explanation for this empirical link between carrier mobility and lattice properties is beyond the scope of this paper and emerges as a challenge to theory.

C. Concluding remarks

The persistence of T-square resistivity in dilute metallic strontium titanate is puzzling [START_REF] Lin | Scalable T 2 resistivity in a small single-component Fermi surface[END_REF]. However, its context is familiar. Electrons are degenerate and k F >> 1. We now see that this low-temperature puzzle is only the tip of an iceberg. Above 100 K, the application of the Drude picture of conductivity yields a paradox: a carrier mean-free-path shorter than all conceivable length scales. Both the magnitude and the functional behavior of the temperature-induced change in resistivity are very different from what is expected according to the Bloch-Grüneisen law and what is seen in silicon and graphene. The aim of this paper is to attract theoretical attention to a hitherto unnoticed case of strange metallicity. It remains to be seen if electron viscosity plays any role in diffusive transport. On the experimental side, extending the available data to higher temperatures and lower doping concentration would map the boundaries of metallicity and the MIR limit in this system as well as other polar semiconductors.

It is a pleasure to thank Sean Hartnoll and Dmitrii Maslov for stimulating discussions. This work has been supported by an Ile de France regional grant, by Fonds ESPCI-Paris and by JEIP-Collège de France and funded in part by a QuantEmX grant from ICAM and the Gordon and Betty Moore Foundation through Grant GBMF5305 to KB.

FIG. 2 :

 2 FIG. 2: Breakdown of the Mott-Ioffe-Regel limit: a) Temperature dependence of the mean-free-path in dilute metallic SrTiO 3-δ sample up to 400 K. Blue circles represent e, extracted from resistivity using Eq.1. Red diamonds represent the mean-free-path in the non-degenerate regime: e = e n -1/3 Λ

FIG. 4 :

 4 FIG.4: Two pictures of temperature-dependent conductivity: a) In a Drude-Boltzmann picture of transport, cooling the system leads to a rarefication of scattering events. Conductivity increases because the time interval (and the spatial distance) between two successive scattering events increases with cooling. Such a picture requires a well-defined quasi-particle across the distance in time and in space separating two scattering events. b) In a Landauer picture of transport, each dopant (each oxygen vacancy here) is a reservoir connected to other adjacent reservoirs by conducting channels. Conductivity increases upon cooling because the transmission coefficient enhances. Such a picture does not require well-defined quasi-particles.

FIG. 5 :

 5 FIG.5: Length scales and transmission a) Temperature dependence of different length scales at two different carrier densities. The de Broglie thermal wavelength, Λ, was calculated assuming an effective mass of m * =1.8me, found by quantum oscillations at low temperature. The effective Bohr radius was calculated using this mass and the permittivity data reported in[START_REF] Müller | SrTiO3: An intrinsic quantum paraelectric below 4 K[END_REF]. b) Transmission coefficient extracted from resistivity data for different carrier concentrations compared to Λ 2 .

3 )FIG. 6 :

 36 FIG. 6: Diffusivity, ferroelectric soft mode and permittivity a) Blue solid circles represent the diffusion constant extracted from mobility. The dashed line represents a T -1 temperature dependence. Open circles represent the soft mode frequency, ω -3T O according to ref.[START_REF] Fleury | Electric-Field-Induced Raman Scattering in SrTiO3 and KTaO3[END_REF]. The two quantities scale with each other in this temperature range (D ∝ ω -3T O ). b) The inverse of electric permittivity, -1[START_REF] Müller | SrTiO3: An intrinsic quantum paraelectric below 4 K[END_REF] and ω 2T O in the same temperature range. The scaling is -1 ∝ ω 2 T O .

TABLE


Clément Collignon

Benoît Fauqué

Antonella Cavanna

Ulf Gennser

Dominique Mailly

Kamran Behnia

Superfluid density and carrier concentration across a superconducting dome: The case of strontium titanate

teaching and research institutions in France or abroad, or from public or private research centers.

I. INTRODUCTION

In many superconductors with an insulating parent, the critical temperature is a non-monotonic function of carrier concentration. The very existence of such a superconducting dome raises a fundamental question. How does the superfluid density, n S , evolve in such a context? Does it remain equal to the concentration of electrons in the normal state? Or does it follow the non-monotonic variation of the critical temperature? In the case of high-T c cuprates, the doping dependence of the superfluid stiffness [1][2][3] has remained the subject of an intense debate, focused on the link between critical temperature and superfluid stiffness [4][5][6][7][8]. A recent subject of debate has been the correlation between superfluid density and critical temperature in overdoped cuprates. It has been interpreted as incompatible with the standard Bardeen-CooperSchrieffer (BCS) description [8] or in good agreement with the dirty BCS theory [9]. However, a comparison between the magnitude of n S with the normal-state carrier concentration, n H was absent in this debate. To the best of our knowledge, and in spite of abundant experimental data, such a textbook [START_REF] Tinkham | Introduction to superconductivity[END_REF] link has never been verified in any superconductor.

Among doped semiconductors with a superconducting ground state [START_REF] Bustarret | [END_REF], SrTiO 3 [12][13][14][15][16] distinguishes itself. With only 10 -5 electron per formula unit (f.u.), it becomes a superconductor and when carrier density exceeds 0.02/f.u., it ceases to be so. The existence of this superconducting dome raises many questions: how does superconductivity persist in the dilute limit in spite of a hierarchy inversion between Fermi and Debye temperatures? Why does it disappear on the overdoped side despite the steady increase in the electronic density of states? Do plasmons play a role in binding Cooper pairs in the extreme dilute limit [17,18]? The vicinity to a ferroelectric instability has motivated theoretical scenarios invoking ferroelectric quantum criticality [19,20], which have found support in a number of recent experiments [21,22].

On the other hand, the case of SrTiO 3 offers a unique opportunity to explore the behavior of superfluid density when the critical temperature is not a monotonic function of carrier density. The superconducting instability occurs in a well-documented Fermi surface in which carrier concentration is known with a reliable accuracy [16] and can be tuned across orders of magnitude. Here, we present an extensive study of the lower critical field in SrTi 1-x Nb x O 3 across the superconducting dome with a focus on the relative magnitude of superfluid and normalcarrier density. In the underdoped regime, we find that the superfluid density extracted from the magnitude of the lower critical field is in agreement with the carrier concentration in the normal state. Deep in the overdoped regime, a mismatch between the extracted superfluid density and the concentration of normal electrons is detectable and steadily increases with doping. We show that this mismatch is concomitant with the passage from clean to dirty limit. However, we fail to achieve a quantitative account in the dirty limit in a single-band picture. This is most probably because the multiplicity of the electronic bands significantly affects the n S /n H ratio. The results have implications beyond the case of strontium titanate. Comparing SrTiO 3 with a dense s-wave superconductor, namely niobium, we find that when a superconductor is clean, the magnitude of the penetration depth correlates with its carrier density. It is also in-structive to compare optimally-doped YBCO with these two systems. Its reported penetration depth in the ab plane happens to be where it is expected according to its carrier density, the effective mass of its carriers and the BCS theory.

II. EXPERIMENTAL

We measured the lower critical field, H c1 , with Hall probes realized using a high-mobility AlGaAs/GaAs heterostructure with a two-dimensional electron gas (2DEG) 160 nm below the surface. The 2DEG has a mobility of 320.000 cm 2 /Vs and a carrier density of 2 10 11 cm 2 at liquid helium temperature. The devices were fabricated using electron beam lithography and 250V argon ions to define the mesa. As shown in the inset of Fig. 1.a, each probe is a 5×5 µm 2 square, 100 µm spaced from its immediate neighbor.

Such a device can monitor local magnetization at micron scale and was used before to study vortex avalanches in superconducting niobium [23]. Similar Hall microprobes have been used before to measure H c1 in heavyfermion [24] and in iron-based [25] superconductors.

The Hall resistance of a 2DEG probe yields the local magnetic field, B :

R xy,probe = B 2 /ne (1)
Here n is the carrier density of the electron gas. In our case with 1/ne ∼ 0.3Ω/G, we could easily resolve very small variations in magnetic field. We checked that the 1/ne coefficient does not change with temperature below 1 K. To measure H c1 , the sample is laid on the array as depicted in the inset of Fig. 1a. At low field, the sample is in the Meissner state and the probes below do not feel any magnetic field, which is screened by the sample. When the first vortex penetrates the sample, the microprobe detects a rise in the measured magnetic field. The field value at which it occurs gives the penetration field H p . The data for one sample at one temperature, presented in Fig. 1a, illustrates how clearly one can detect H p . Each isothermal curve was obtained by two sets of sweeps from zero to positive and from zero to negative fields. Between the two sweeps the sample was heated up to a temperature significantly larger than the critical temperature and back to the measuring temperature. The intrinsic lower critical field, H c1 is proportional H p . In the case of a platelet, the geometrical factor is:

H c1 = 1 tanh( 0.36t/w) H p (2)
Where t and w are respectively the thickness and the width of the slab [26]. This geometrical factor is given for each sample in Table I. We monitored the Hall resistance of several probes in the proximity of the sample edges. Our data is based on a probe located below the sample and about 150 µm distant from the edge. We chose this option to avoid two phenomena, which can disrupt an accurate determination of H c1 . Vortices can penetrate by the corner of the sample below H p [27]. Therefore, the sample corners are to be avoided. On the other hand, vortex pinning may generate a non-uniform distribution of magnetic field [28]. Like a previous study [29], the probe was located close to the edge in order to avoid this. The sharpness of the increase in R xy,probe at H p in our data indicates that these phenomena do not contaminate our measurements.

A perfect surface may forbid the flux lines to penetrate the sample and thus artificially increase the penetration field [30]. Experimentally, this surface barrier effect will be manifested as a slow increase above H p and then a sharp rise when the barrier is overcome [27]. Such effect is absent from our data (see Fig. 1a). Most probably, this is because the surface roughness of our samples is of the same order as the penetration depth [31].

Another experimental challenge is the presence of residual and Earth magnetic field. Their magnitude are not negligible compared to the lower critical field measured here. We compensated them by applying a small magnetic field. As seen in Fig. 1, the magnitude of H p was identical for the two field polarities. The small asymmetry for the most overdoped sample quantifies the limits of our compensation method.

Still using the Hall probes, we were also able to quantify the upper critical field of our samples. At a given fixed field, H app , when the temperature is decreased below T c (H app ) the sample start to expel the magnetic field. Thus, for a probe under the sample, the signal will be constant at T > T c (H app ) and then start to drop in the superconducting phase. By repeating this measurement at several fixed fields, we can finally extract the field dependency of T c , or the other way around, the temperature dependency of H c2 . In practice, for applied field larger than 10 Oe, the expelled field is two or three orders smaller than the applied one. Hence, the drop we want to observe in R xy,probe is below the resolution of the range we have to use to not get a saturation of the signal. We consequently measured the difference between the signals of two probes, one under the sample and another outside of it, in order to cancel the large and uninteresting signal from the applied field. This method give us directly a signal proportional to the magnetic susceptibility.

We measured six SrTi 1-x Nb x O 3 samples (Source: CrysTec GMbH) with labeled niobium concentrations of 0.2, 0.8, 1, 1.4, 2 and 2 %. Two slabs were cut from each sample: one to measure resistivity and Hall effect and one for the magnetometry measurement of approximate size 1 × 1 × 0.5 mm 3 (except for one sample with a thickness of 1 mm). We performed resistivity temperature sweeps from T = 300 to 2 K at zero field and Hall effect field sweeps from H = 0 to ±12 T at T = 2 K via a Quantum Design Physical Property Measurement System (PPMS). The extracted ρ 0 and n H values are listed in Table I. Similar samples were studied pre- viously using multiple experimental techniques, such as electrical resistivity [32], thermal conductivity [33], thermoelectric response [34] and specific heat [33]. Thanks to their high mobility, quantum oscillations can be detected in a moderate magnetic field [16,34]. Their Nb content was checked by secondary ion beam mass spectroscopy (SIMS) [33].

x

III. RESULTS

We measured the lower critical field H c1 of all six samples and the upper critical field H c2 of three of them. These two sets of data allowed us to extract the Ginzburg parameter, κ. The penetration depth, λ can be computed using κ and H c1 . The penetration depth combined with effective mass yields superfluid density, which is the quantity we wish to put under scrutiny.

A. Lower critical field Hc1

For each sample, we performed measurements at different temperatures extending from T base ∼ 30 -50 mK up to T c and above, and extracted H p (T ). The raw data for n H = 2.1 10 20 cm -3 is shown in Fig. 1b. The measured local field, B, can be used to extract magnetization through M = B/µ 0 -H. In all six samples, magnetization is proportional to -H at low field, sharply drops at H p and decreases smoothly afterwards (See Fig. 2). Superconducting slabs with a small Ginzburg parameter typically behave in this way [35]. For each temperature, H p is taken to be the magnetic field at which M deviates significantly from -H (See Fig. 2). The temperature dependence of ±H p for opposite orientations of magnetic field in the six samples is shown in the same figure. The small difference between +H p and -H p indicates that residual (including Earth) magnetic field has been compensated in a satisfactory manner.

The temperature dependence of H c1 for all six samples is exposed in Fig. 3. We note that the temperature dependence of H c1 is somewhat different among different samples. The observed variety in the temperature dependence of lower critical field in different samples may be a consequence of multigap superconductivity, which is known to produce additional structure in the temperature dependence of H c1 [36]. One may expect that the curvature of H c1 is set by the way disorder tunes the contribution of different bands. This variety in the temperature dependence has little effect on the extraction of a reliable H c1 (0), which is the purpose of the present study. We note an agreement between our data and what was reported in an early study on the lower critical field of two SrTi 1-x Nb x O 3 samples near optimal doping [37]. As one can see in Fig. 3b, the doping dependence of H c1 and T c are similar to each other, both presenting a dome-like structure.

B. Upper critical field Hc2

We managed to measure the upper critical field H c2 for three of our samples with carrier densities n H = 2.1, 3.2, and 3.5 10 20 cm -3 , with the methods explained Part I. The signal difference between one probe under the sample and one far from, proportional to the magnetic susceptibility ξ, is flat at high temperature and start to drop at

T c +2H 02H Q + FP +2H Q + FP +2H Q + FP +2H 02H Q + FP +2H Q + FP +2H Q + FP 7P. + S 2H Q + FP 7P. Q + FP 7P. Q + FP 7P. + S 2H Q + FP 7P. Q + FP
7P. as seen in the upper panels of figure 4. The field dependence of T c allows us to plot the temperature dependence of H c2 depicted in the lower panel of figure 4. We then proceeded to fit the data points with the Werthamer, Helfand and Hohenberg (WHH) function [38] in order to extract the value in the T = 0 limit. Those fits give the values reported in Table III.

The upper critical field is set by the superconducting coherence length, Ξ, while the lower critical field is set by the magnetic penetration depth, λ. Roughly speaking, at H c1 the whole magnetic flux is contained by a single vortex of radius λ. A more elaborate treatment take into consideration energy corrections due to the internal structure of the vortex and the Ginzburg parameter κ, the ratio of the penetration depth over the superconducting coherence length. When κ is large, the ratio H c1 /H c2 is given with negligible error by the following approximation [START_REF] Tinkham | Introduction to superconductivity[END_REF]39,40]:

H c1 H c2 = ln(κ) + 0.5 2κ 2 (3)
Noting that H c2 = φ 0 /2πξ 2 , where φ 0 is the quan-tum of magnetic flux, this leads to the following relation between H c1 and λ :

H c1 = φ 0 4πλ 2 (ln(κ) + 0.5) (4)
This widely-used expression is to be used with caution given its implicit assumption of a large κ. The question is particularly relevant in the case of SrTi 1-x Nb x O 3 where the κ is relatively low. Harden and Arp [41] made numerical calculations for κ values ranging from 0.3 to 100. They found that when κ = 5, H c2 /H c1 = 22.44, which is to be compared with tH c2 /H c1 = 23.74 deduced from Eq. 3. This represents an error of 6%. At κ = 10 the same comparison yields an error of 1.4%. Therefore, if the ratio H c1 /H c2 ratio plugged in to Eq. 3 yields a κ larger than 5, then we can legitimately use Eq. 4 to extract the penetration depth, λ.

From the measured bulk H c2 in three of our samples (combined with unpublished AC susceptibility data in the case of n H = 2.6 10 20 cm -3 sample [42]), we extracted κ. When carrier concentration is low and the system is in the clean limit κ 8.5. It increases to 17 at high concentration when the mean-free-path shortens 7P.

+ F 2H 7 F . WKLVVWXG\ [[ >@ Q + FP + F 2H 6U7L [ 1E [ 2 D E F Q + FP FIG.
3. a: Lower critical field vs temperature for all six samples deduced from Hp and geometrical correction of the demagnetization field. b: Doping dependence of the critical temperature determined by Hall probe magnetometry (blue disks) and thermal conductivity (black squares) [33], the blue line is a guide to the eye. c: Doping dependence of Hc1, the blue line is a guide to the eye.

and pulls down the coherence length. This allowed us to extract the penetration depth from H c1 using Eq. 3.

Since κ > 5, our use of Eq. 4 is legitimate.

IV. DISCUSSION

A. Penetration depth

The superfluid density and the penetration depth are intimately linked through the London equation [START_REF] Tinkham | Introduction to superconductivity[END_REF]:

λ -2 = µ 0 e 2 n s m (5)
The doping dependency of λ -2 is shown in Fig. 5. It also presents a dome-like structure, reminiscent of the case of cuprates [6]. Uemura and co-workers [1] were the first to notice that the correlation between the superfluid DX 4. Upper panels show the temperature dependence of the difference in signal between one probe under the sample and one probe far from it at various fixed fields and for three dopings. The difference is proportional to the magnetic susceptibility ξ. The lower panel shows the temperature dependence of the upper critical fields Hc2 extracted from the upper panels. The lines are WHH fits [38] to the data points.

Q + FP * * Q + FP * * Q + FP * * 7P. + F 2H Q + FP FIG.
stiffness (∝ ns m ) and the critical temperature in cuprates.

B. Comparison with other superconductors

The penetration depth of optimally-doped strontium titanate is 870nm. Let us now compare this with two other superconductors (Fig. 4b). In niobium, it is 31.5nm [43]. Given the large difference in their carrier concentration, this is not surprising. The figure compares λ(n H ) for these two systems with what is expected according to Eq. 5, and assuming n H = n s . One can see that the data points fall close to their expected position. It is instructive to compare the measured λ ab in optimally-doped YBCO (103 nm [44]) with these two systems. As seen in Fig. 4b, assuming the carrier density given by the Hall coefficient in the vicinity of optimal doping [45] (See table II for details), this value of the penetration depth is what is expected in this system. This confirms a key expectation of the BCS theory : Absent disorder, what sets the magnitude of the penetration depth in a given superconductor is its carrier density (and not its critical temperature). As far as we know, such an experimental verification of this expectation by a comparison of densities across two orders of magnitude was not done before. II. The dashed lines represent the expectation of Eq. 5, assuming nH = ns. The penetration depth in cuprates is the

ab plane one λ ab System T c λ m n S n H (K) (nm)
(m e ) (10 20 cm -3 ) (10 20 cm -3 ) STO 0.3 952 4.2 1.3 2.1 Nb 9.3 31.5 [43] 1.5-4 [46] 520 620 [47] YBCO 89 103 [44] 3.6 [48] 96 50 [45] TABLE II. Penetration depth, superfluid density, normalstate carrier-density in three superconductors. Note that in the case of YBCO, the largest effective mass [48] was measured at p=0.15, below optimal doping (p=0.18) and it is the in-plane penetration depth, λ ab , which is considered.

C. Disorder

Disorder can reduce superfluid density through two distinct mechanisms. The first mechanism is relevant to all superconductors. The energy scale of Drude metallic conductivity, /τ , increases with increasing disorder and decreasing scattering time, τ . When the superconducting gap, ∆, is much larger than this energy scale, the condensate density and the density of normal quasi-particles are comparable. In the opposite limit (when /τ ∆), a small fraction of normal-state charge carriers have subgap energies and one expects n s n H [49,50].

Quantitatively the superfluid density can be described by the following equation [49,50]:

n s 2m πe 2 2∆/ 0 dωσ 1 (ω) (6)
Where σ 1 (ω) is the real part of the optical conductivity.

If we assume that σ 1 (ω) behave as a Lorentzian of width 1/τ and zero frequency value σ 0 = ne 2 τ /m as described by the Drude model we can simply integrate Eq.6:

n s 2n π arctan 2π∆τ (7)
Note that in the dirty limit (i.e. ∆τ → 0), leading to the Homes law which states that n s /m ∝ σ 0 T c [2,3]. Another distinct disorder-driven mechanism is specific to nodal superconductors like cuprates and is associated with non-magnetic pair breaking. Note that in both cases, the crucial parameter is ratio of /τ to ∆. Since experiments indicate that n-doped SrTiO 3 is s-wave [33,51], only the first mechanism is relevant here.

D. Superfluid density

According to band calculations [52], metallic strontium titanate has three distinct bands. This was confirmed by an extensive study of quantum oscillations [16], finding new frequencies emerging above two critical doping levels. Early tunneling studies [15] and more recent thermal conductivity measurements [33] detected two distinct gaps. The cyclotron masses of the three bands are different. The lower band (or the outer sheet of the Fermi surface) is heavier (m 1 = 3.85±0.35) compared to the higher bands (m 2,3 = 1.52 ± 0.25). These numbers are consistent with what was found by ARPES (1.5m e and 6m e ) at a higher concentration [53]. In our window of interest, three-fourth of all carriers reside in the lower band [16]. The electronic specific heat, γ = 1.55 mJ mol -1 K -2 (at n H = 2.6 10 20 cm -3 ) is known. Assuming that all carriers reside in one band, one obtains m = 4.2m e [33].

Let us assume a single band with an effective mass of 4.2 m e and n H extracted from ( Hall coefficient, which is close to the total number of carriers residing in different Fermi surface pockets and detected by quantum oscillations [16]. The results can be seen in Fig. 6a, which shows the ratio of the extracted n s over the normal state carrier density n H for our six samples. As seen in this figure, the superfluid density and the normal carrier density match each other at low doping. A deviation starts at optimal doping before drastically increasing at higher doping levels. Thus, the superfluid stiffness (that is λ -2 ) follows the doping dependence of the critical temperature, because n S becomes lower than n H in the overdoped regime.

In order to document the passage to the dirty limit, we extracted the scattering time [43,56] and dirty [54,55] niobium.

resistivity (1/ρ 0 = ne 2 τ /m ) and estimated the magnitude of the superconducting gap using the BCS relation ∆ ∼ 1.76k B T c [15,57]. The product of the two tells us on which side of the clean/dirty limit the system is. Note that /ξ = π∆τ / . When π∆τ / < 1, one enters the dirty limit. Fig. 6b. compares the evolution of n s /n H and π∆τ / . The two quantities deviate from unity concomitantly, but are not proportional to each other in the dirty limit as one may expect in the crudest conceivable approximation. It is also instructive to compare our data with available data on Nb from three different studies [43,54,55]. Figure 6.c shows the variation of n s /n H with π∆τ / in SrTi 1-x Nb x O 3 and in Nb. One can see that in both systems, as theoretically expected [3], the clean-to-dirty crossover and the loss of superfluid density are concomitant. However, in the case of strontium titanate, n s /n H and π∆τ / are not simply proportional to each other. This is a presumably due to the inadequacy of a single-band approach.

E. Comparison with the interface superconductor

Let us compare our quantification of superfluid density in the bulk superconducting strontium titanate with the case of the superconducting LaAlO 3 /SrTiO 3 interface. Bert et al. [58] studied how a gate voltage tunes the two-dimensional superfluid density in this system. The maximum superfluid density found was n 2D s = 3 × 10 12 cm -2 . This is much more dilute than the maximum three-dimensional superfluid density found here, which is n 3D s = 1×10 20 cm -3 . The two densities can be contrasted by comparing the average distance between carriers. In their case, the peak n 2D s corresponds to d ee = 5.7nm.

In our case, n 3D s corresponds to d ee = 2.15nm. In other words, the peak density for the interface system is significantly more dilute than in the bulk system. The most plausible explanation for this difference is the presence of additional disorder in the interface superconductor. As discussed above, the larger the disorder, the lower the superfluid density.

V. SUMMARY

In summary, we found that superconducting SrTi 1-x Nb x O 3 has a dome-like λ -2 reminiscent of cuprates. Comparing three different systems, one sees that when a superconductor is clean, its λ is primarily set by its carrier density and not by its critical temperature. The density mismatch between superconducting and normal states is concomitant with the entry to the dirty limit. Since in all superconducting domes, the T c and the gap eventually vanish, the mismatch is expected in any superconducting dome far from optimal doping. Finally, we notice that a quantitative account of disorder-driven loss of superfluid density needs to take into account the multiplicity of the electronic bands and the superconducting gaps. Q + FP

Q V FP Q + FP Q V Q + Q V Q + D 6U7L [ 1E [ 2 E 6U7L [ 1E [ 2 F 6U7L [ 1E [ 2 1E
FIG. 6. a: Evolution of superfluid density as a function of carrier density. The straight line represents nS = nH . b: The ratio of superfluid density over normal-state carrier density (red discs) compared to the product of the scattering time and the superconducting gap (blue squares) as a function of doping. Lines are guides to the eye. c: The evolution of superfluid density in Nb (blue squares [43,54,55]) and in Nbdoped SrTiO3 (red discs). The continuous black line is the analytic solution of the Ferrell Glover Tinkham sum rule Eq.7.

FIG. 1 .

 1 FIG. 1. a: Typical field dependence of the Hall resistivity of a microprobe with a superconducting sample above. The measured signal, directly proportional to the local magnetic field, suddenly rises when the first vortex penetrates the sample. The sharpness indicates the absence of surface-barrier effects or gradual vortex leak from sample corners[27]. The inset shows, in false colors, a picture of the array of Hall microprobes used for this experiment and a scheme of how a sample is mounted on the array of probes. b: Raw data for the nH = 2.1 10 20 cm -3 sample at different temperatures.

FIG. 2 .

 2 FIG.2. Upper panels show magnetization M as a function of magnetic field at different temperatures for the six samples. Lower panels show the extracted penetration fields Hp for opposite orientations of magnetic field by taking the crossing point of M(H) at a given temperature with the dashed lines shown in the panels. Black dash lines are empirical fits to Hp(T ) = Hp(0) (1 -(T /Tc) α ).

FIG. 5 .

 5 FIG.5. a: Doping dependence of the penetration depth plotted as λ -2 vs nH . It exhibits a dome-like structure comparable to Tc, as in the case of cuprates[6]. b: Penetration depth plotted as a function of nH /m in three different superconductors with m beeing unitless. The values are detailed in TableII. The dashed lines represent the expectation of Eq. 5, assuming nH = ns. The penetration depth in cuprates is the ab plane one λ ab

  τ from low-temperature

	Sample	n H (10 20 cm -3 ) (mK) (Oe) (Oe) T c H c1 H c2 κ	λ (nm) (10 20 cm -3 ) (µΩ.cm) (ps) n s ρ 2K τ	ns n H	π∆τ
		0.41	183 1.6	-	-1647	0.44	49	7.43 1.07 0.980
		1.9	300 4.8	-	-943	1.34	53	1.50 0.70 0.326
	STO	2.1 2.6	303 4.7 240 8.2 952 382 6.05 480 10.5 881	1.31 1.53	71 109	1.34 0.62 0.294 0.52 0.58 0.144
		3.2	192 2.8 210 10.5 1292	0.72	56	0.84 0.22 0.116
		3.5	118 0.5 85 17 3305	0.11	45	0.95 0.03 0.081
		620	9.3 K 1800 4000 0.85 31.5	520	2 10 -4	5.8 0.84	38
	Nb	620	9.3 K -	-	-	44	264	1.2	8 10 -2 0.42 0.53
		620	8.3 K -	-	-	90	63	3.9	3 10 -2 0.10 0.20

TABLE III .

 III Parameter for SrTi1-xNbxO3 , and for clean

VI. ACKNOWLEDGMENTS

We thank A. J. Millis, J. Orenstein and L. Taillefer for helpful discussion and comments. This work is supported by Agence Nationale de la Recherche through the QUANTUM LIMIT project, by Fonds-ESPCI-Paris and by JEIP-Collège de France.