Abstract
Cooling oxygendeficient strontium titanate to liquidhelium temperature leads to a decrease in its electrical resistivity by several orders of magnitude. The temperature dependence of resistivity follows a rough T^{3} behavior before becoming T^{2} in the lowtemperature limit, as expected in a Fermi liquid. Here, we show that the roughly cubic resistivity above 100 K corresponds to a regime where the quasiparticle meanfreepath is shorter than the electron wavelength and the interatomic distance. These criteria define the MottIoffeRegel limit. Exceeding this limit is the hallmark of strange metallicity, which occurs in strontium titanate well below room temperature, in contrast to other perovskytes. We argue that the T^{3}resistivity cannot be accounted for by electronphonon scattering à la Bloch–Gruneisen and consider an alternative scheme based on Landauer transmission between individual dopants hosting large polarons. We find a scaling relationship between carrier mobility, the electric permittivity and the frequency of transverse optical soft mode in this temperature range. Providing an account of this observation emerges as a challenge to theory.
Introduction
The existence of welldefined quasiparticles 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 MottIoffeRegel (MIR) limit is attained when the meanfreepath of a carrier falls below its Fermi wavelength or the interatomic distance.^{1} In most metals, resistivity saturates when this limit is approached. But in “bad”^{2} or “strange”^{3} metals, it continues to increase.^{4} It is indeed strange when the meanfreepath 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.^{5} recently noticed that the Tlinear scattering rate in numerous metals, either ordinary or strange, has a similar magnitude. This observation suggests the relevance of a universal Planck timescale (\({\tau _P} \sim \frac{\hbar }{{{k_B}T}}\), where k _{ B } is the Boltzmann and ħ the reduced Planck constants) to electronic dissipation and motivated a theoretical attempt to quantify an upper limit to the diffusion constant in incoherent metals.^{6} During the past years, several theories of charge transport with no need for welldefined quasiparticles were proposed.^{7,8,9,10,11,12,13,14} More recently, direct measurements of thermal diffusivity^{15} have provided additional evidence for quasiparticlefree thermal transport at room temperature in cuprates.
We show here that ndoped 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.^{16,17,18,19,20} In the lowtemperature limit, resistivity follows a Tsquare behavior,^{21, 22} as expected in a Fermi liquid. Above 100 K, however, the temperature dependence is close to cubic.^{17, 22} In this regime, the meanfreepath 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 (See Table 1). Since quasiparticles in this system are not welldefined in a time scale long enough to allow the existence of a scattering time, charge transport cannot be reasonably described by a phononscattering picture. This was the approach in previous attempts^{16,17,18,19, 23,24,25} 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 meanfreepath. 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.
Results
Metallicity beyond the MIR limit in strontium titanate
Figure 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 (\(T \ll 100\) K), the system displays a Tsquare resistivity^{21, 22} as expected for electronelectron scattering^{26} in a Fermi liquid. Plotting resistivity as a function of T^{2} (see Fig. 2 in ref. 22), one can see that inelastic resistivity becomes Tsquare in the lowtemperature limit. A wide range of dense Fermi liquids display Kadowaki–Woods scaling^{27} of their Tsquare resistivity prefactor and their electronic specific heat. Since the latter is proportional to carrier concentration, the case of lowdensity systems such as doped SrTiO_{3} is different.^{28} 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.^{22}
Figure 1b shows the temperature dependence of the extracted Hall mobility, \(\mu = \frac{1}{{ne\rho }}\). 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.^{17} Figure 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 lowtemperature (T < 50 K) but smoothly rises with warming. One does not expect the Tsquare resistivity to survive at the Fermi degeneracy temperature. Therefore, the electronelectron scattering origin of the Tsquare term in the dilute limit has been put into question.^{29, 30} We note, however, that the smooth variation of α, makes the detection of the upper boundary of Tsquare resistivity impossible. As one can see in the figure, α peaks around 150 K at a value slightly exceeding three 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.^{31} As a consequence, quantum oscillations are easily detectable in moderate magnetic fields.^{32,33,34} The large magnitude of defectlimited mobility at low temperature and its evolution with carrier concentration can be explained in a simple model^{35} invoking the long effective Bohr radius of the quantum paraelectric^{36} 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^{16,17,18,19} attributed the T^{−3} variation of mobility to the scattering of carriers by phonons. There was a controversy between Fredrikse et al.^{18} (who invoked longitudinal optical phonons) and Wemple et al.^{19} (who argued in favor of transverse optical phonons). Neither during this early debate,^{16,17,18,19} nor in more recent theoretical accounts of the temperature dependence of mobility in ndoped SrTiO_{3},^{23,24,25, 37, 38} the magnitude of the meanfreepath was discussed.
In the Drude–Boltzmann picture, the conductivity, σ, of a metal depends on fundamental constants and a number of materialrelated properties. These are either, the scattering time τ, and the effective mass, m ^{*}, or the Fermi wavevector, k _{ F } and the electron meanfreepath, \({\ell _e}\). Namely:
The Fermi surface of ndoped 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.^{21, 33, 39} The threefold degeneracy of this band is lifted by tetragonal distortion and spinorbit 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^{21} and confirmed by experimental studies of quantum oscillations.^{32,33,34} 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.^{33, 34, 39} Therefore, k _{ F } can be quantified in a straightforward manner. Figure 2a shows the temperature dependence of the meanfreepath 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. 17, 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.^{34} Therefore, there is little ambiguity regarding carrier concentration.
As seen in Fig. 2a, the meanfreepath becomes shorter than the inverse of the Fermi wavevector \({\it{k}}_F^{  1}\) 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.^{32} As one sees in the figure, this is the temperature at which de Broglie thermal wavelength, \({\it{\Lambda }} = \frac{h}{{\sqrt {2\pi {m^*}{k_B}T} }}\) becomes comparable to the interelectron distance n ^{−1/3}. Above this temperature, the electrons become nondegenerate and follow a Maxwell–Boltzmann distribution. As a consequence and thanks to thermal excitation, the typical carrier momentum is larger than ħk_{ F }. In other words, the typical velocity is not the Fermi velocity, but the thermal velocity, \({v_T} = \sqrt {\frac{{2{k_B}T}}{{{m^*}}}} \). Let us define the meanfreepath of electrons with thermal velocity as \(\ell _e^\prime = \tau {v_T} = {\ell _e}{n^{  1/3}}{\rm{/}}{\it{\Lambda }}\). As seen in the figure, because of \(\ell _e^\prime >{\ell _e}\), the crossover temperatures shift upward. However, the picture remains qualitatively the same. In the nondegenerate regime, the relevant wavelength is Λ and not the Fermi wavelength. As seen in the figure, the meanfreepath becomes shorter than Λ too.
A roomtemperature resistivity of 2.4 mΩ cm implies a scattering time of \(\tau \simeq 6\,fs\). This is shorter than the Planck time^{5} \(\left( {{\tau _P} = \frac{\hbar }{{{k_B}T}}} \right)\) at room temperature, τ _{ P }(300 K) = 25.3 fs. 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 \(\left( {\frac{{\it{\Lambda }}}{{{v_T}}} = 1.77{\tau _P}} \right)\). By all accounts, roomtemperature quasiparticles in dilute metallic strontium titanate do not live long enough to qualify for a scatteringbased picture.
A color plot of the meanfreepath 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 \(_F\ell \) 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 welldefined quasiparticles. This is the first principal message of this paper.
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^{40} has shown the relevance of the BlochGrüneisen picture of electronphonon 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 wavevector 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^{40} 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 temperaturedependence of resistivity in ndoped 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^{41} reveals a similar discrepancy. As one can see in Fig. 3b, cooling phosphorousdoped silicon leads to a modest change in resistivity, much less than what is seen in doped SrTiO_{3}. As a consequence, while at lowtemperature, 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 meanfreepath, 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},^{19, 42} and on lead chalcogenide salts (PbTe, PbSe and PbS)^{43, 44} indicate that hightemperature mobility does not vary with carrier concentration, but shows a strong temperature dependence (μ ∝ T ^{−λ}, with \(\lambda \simeq 2.5  3\)). Both these features are similar to what we see in strontium titanate. Only in strontium titanate, however, the magnitude of the room 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 meanfreepath and follows a Bloch–Gruneisen behavior (a slow Tlinear 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 meanfreepath 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.
Discussion
Conduction as transmission in a doped polar semiconductor
Figure 4 sketches two alternative descriptions of charge conduction. In the semiclassic picture of metallic charge transport (Fig. 4a), conductivity increases with decreasing temperature, because the scattering events relaxing the momentum of a chargecarrying quasiparticle become less frequent with decreasing temperature. Now, consider an alternative picture (Fig. 4b), where each dopant is hosting a polaron, ‘an electron trapped by digging its own hole’^{45} in a polar semiconductor. The polaron picture has been frequently invoked in the case of strontium titanate.^{46, 47}
In this second picture, conductivity is set by transmission among dopant sites and there is no need for welldefined quasiparticles. 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.^{36} 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. Figure 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 lowtemperature effective mass of m^{*} = 1.8m_{ e } seen by quantum oscillations,^{32} one finds that the de Broglie thermal wavelength is \({\it{\Lambda }} \simeq 3  6\,nm\) in this temperature range. Thus, the system is mostly nondegenerate. Electric permittivity, \(\epsilon \), is large (\(\epsilon {\rm{(}}300\,K{\rm{)}} \simeq 300\))^{36, 48} and increases steadily with cooling. As a consequence, the effective Bohr radius \(\left( {{\rm{a}}_B^* = \frac{{4\pi \epsilon {\hbar ^2}}}{{{m^*}{e^2}}}} \right)\) varies between 8 and 40 nm. This puts the system above the Mott criterion for metallicity (n^{1/3} \({\rm{a}}_B^* >0.25\) ^{49}), which, one shall not forget, is relevant to degenerate electrons.^{50} These length scales are in the same range of magnitude. They are all longer than the lattice parameter (0.39 nm). The question is how to picture conduction as transmission^{51} along a random network of donors.
In Drude picture, σ = neμ. In the present case, the mobility, μ does not depend on carrier concentration above 100 K. Therefore, it is tempting to express conductivity as:
Here, \({G_0} = \frac{{2{e^2}}}{h}\), 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 \(\left( {\mu = 2\frac{e}{h} < T >} \right)\).
The temperature dependence of < T >, extracted from the experimental data using Eq. (2) is shown in Fig. 5b. As seen in the figure, a hundredfold 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.^{51} It is therefore surprising that the averaged transmission coefficient, < T >, is insensitive to such a large change in the average interdopant distance.
Figure 5b also compares the magnitude of < T > with Λ ^{2}. the latter represents the areal uncertainty in the spatial localisation of a nondegenerate 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 temperaturedependent 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 temperaturedependent polaronic mass. However, there is no experimental evidence for this. At low temperatures, quantum oscillations^{33, 34} resolve a modest mass enhancement (≤4). This is also the case of infrared conductivity.^{47} Thus, we cannot simply picture conductivity as ballistic charge transport along Landauer wires. Let us now consider a diffusive picture.
Mobility, diffusivity and their link to electric permittivity
In a nondegenerate semiconductor, mobility, and diffusion constant, D, of carriers are linked through Einstein’s equation^{52}:
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 1 cm^{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.^{53}
Hartnoll recently proposed a lower bound to diffusion constant in incoherent metals^{6}:
As one can see in Fig. 6a, the experimental D decreases faster than T^{−1}. Hartnoll boundary was obtained for degenerate systems. Ours is nondegenerate 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.^{36, 48} We have found that in our temperature range of interest D is proportional to \({\epsilon ^{{\rm{1}}{\rm{.5}}}}\). This empirical observation is the second principal message of this paper.
The scaling between static electric permittivity, \({\epsilon _0}\), and the frequency of the Transverse Optic (TO) soft mode,^{54, 55} ω _{ TO } in strontium titanate is wellknown. Yamada and Shirane^{55} showed that \({\omega _{TO}} = 194.4\sqrt {{\epsilon ^{  1}}} \) within experimental resolution. In a polar crystal, there is a theoretical link between the ratio of static, \({\epsilon _0}\), to highfrequency, \({\epsilon _\infty }\), 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^{56, 57}:
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,^{58} the starting point of the polarization theory of ferroelectricity,^{59} explains the temperature dependence of ω _{ TO } 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^{60, 61} 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,^{55} one finds that:
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.
Concluding remarks
The persistence of Tsquare resistivity in dilute metallic strontium titanate is puzzling.^{22} However, its context is familiar. Electrons are degenerate and \({{\it{k}}_F}\ell \gg 1\). We now see that this lowtemperature puzzle is only the tip of an iceberg. Above 100 K, the application of the Drude picture of conductivity yields a paradox: a carrier meanfreepath shorter than all conceivable length scales. Both the magnitude and the functional behavior of the temperatureinduced change in resistivity are very different from what is expected according to the BlochGrüneisen 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.
Methods
Oxygen vacancies were introduced to commercial SrTiO_{3} single crystals by heating them in vacuum (pressure 10^{−6}–10^{−7} mbar) to temperatures of 775–1100 °C. Ohmic contacts have been realized prior to oxygen removal by evaporation of gold contact pads. The electrical conductivity was measured in a Quantum Design Physical Property Measurement System between 1.8 and 300 K. Carrier density of each sample was determined by measuring its Hall coefficient. In addition to the samples listed in ref. 22, three others were measured up to 400 K. The resistivity and the Hall coefficient of two PbTe single crystals were also measured for comparison.
Data availability
All data supporting the findings of this study are available within the paper.
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Acknowledgements
It is a pleasure to thank Annette Bussmann–Holder, Sean Hartnoll, and Dmitrii Maslov for stimulating discussions. This work has been supported by an Ile de France regional grant, by Fonds ESPCIParis and by JEIP Collège de France. It has been funded in part by a QuantEmX grant from ICAM and the Gordon and Betty Moore Foundation through Grant GBMF5305 to KB.
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X.L. and C.W.R. prepared oxygendeficient samples of strontium titannate and measured their resistivity. L.B. and A.J. performed resistivity measurements on PbTe. They were all assisted in the measurements by B.F. K.B. wrote the paper. All authors discussed the results and contributed to the manuscript.
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Lin, X., Rischau, C.W., Buchauer, L. et al. Metallicity without quasiparticles in roomtemperature strontium titanate. npj Quant Mater 2, 41 (2017). https://doi.org/10.1038/s4153501700445
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DOI: https://doi.org/10.1038/s4153501700445
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