Scalable T2 resistivity in a small single-component Fermi surface

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Science  28 Aug 2015:
Vol. 349, Issue 6251, pp. 945-948
DOI: 10.1126/science.aaa8655

Trying to break a stubborn law

The electrical resistivity of most metals at low temperatures has a characteristic quadratic dependence on temperature. This law is typically ascribed to the scattering of electrons off each other in the presence of a crystal lattice. By measuring the resistivity of SrTiO3 with varying dopant concentrations, Lin et al. test the applicability of the law for metals with low carrier densities. The law persists down to the lowest carrier concentrations, into the regime where the conditions for this behavior were previously thought to break down.

Science, this issue p. 945


Scattering among electrons generates a distinct contribution to electrical resistivity that follows a quadratic temperature (T) dependence. In strongly correlated electron systems, the prefactor A of this T2 resistivity scales with the magnitude of the electronic specific heat, γ. Here we show that one can change the magnitude of A by four orders of magnitude in metallic strontium titanate (SrTiO3) by tuning the concentration of the carriers and, consequently, the Fermi energy. The T2 behavior persists in the single-band dilute limit despite the absence of two known mechanisms for T2 behavior: distinct electron reservoirs and Umklapp processes. The results highlight the absence of a microscopic theory for momentum decay through electron-electron scattering in various Fermi liquids.

Warming a metal enhances its resistivity because scattering events along the trajectory of a charge-carrying electron become more frequent with increasing temperature (T). In most simple metals, the dominant mechanism is scattering by phonons, leading to a T5 dependence of resistivity. In 1937, Baber identified electron-electron scattering as the origin of T2 resistivity observed in many transition metals (1). During the past few decades, it has been firmly established that, at low temperatures, resistivity (ρ) in a Fermi liquid follows a quadratic temperature dependence expressed as ρ = ρ0 + AT2 (where A is the prefactor of the T2 resistivity) and that correlations among electrons enhance both A and the electronic specific heat, γ. This is often expressed through the Kadowaki-Woods ratio (RKW = A2) (26), which links two distinct properties of a Fermi liquid, each set by the same material-dependent Fermi energy, EF.

The Pauli exclusion principle is the ultimate reason behind both the T-linear specific heat and T-square resistivity in Fermi liquids. Electrons that give rise to both properties are those confined to a width of kBT/EF, where kB is the Boltzmann constant. In the case of resistivity, this is true of both electrons participating in the scattering event, hence the exponent of two. However, electron-electron scattering alone does not generate a finite contribution to resistivity, because such a scattering event would conserve momentum with no decay in the charge current. The presence of an underlying lattice is required in any scenario for generating T2 resistivity from electron-electron scattering. Dimensional considerations implyEmbedded Image (1)Here, ħ is Planck’s constant h divided by 2π; e is the electron charge; and Embedded Image is a material-dependent length scale, which can be set by the Fermi wavelength of electrons, the interatomic distance, or a combination of both. Mott argued that the average distance between two scattering events is proportional to the concentration and the collision cross section of electrons (σcs) (7). ThereforeEmbedded Image (2)Here, kF is the Fermi wave vector, and σcs is set by the specific process governing the decay in charge current due to the presence of a lattice.

There are several types of theoretical proposals for generating T2 resistivity from electron-electron scattering in the presence of a lattice. The first (1) invokes a multiband system with two different electron masses. Momentum transfer between these two distinct electron reservoirs sets the temperature dependence of resistivity, and the mass mismatch leads to a leak of momentum toward the lattice thermal bath. The second invokes Umklapp scattering and the fact that momentum conservation does not prohibit the transfer of a unit vector of the reciprocal lattice (8, 9). In addition to these theories, Pal et al. have recently argued that Fermi liquids lacking Galilean invariance can display T-square resistivity, even in the absence of any Umklapp process (10), due to electron-impurity scattering. In addition to these semiclassic scenarios, quantum interference can also generate a resistivity proportional to T2lnT (10, 11). The relevance of these ideas to the ubiquitous T2 resistivity observed in a wide variety of Fermi liquids has not been settled experimentally.

It has been known for two decades that n-doped SrTiO3 with a carrier density exceeding 0.01 electrons per formula unit follows a T2 resistivity (12). This T2 resistivity provided input for the analysis of the Kadowaki-Woods ratio in low-density Fermi liquids (5) and the Landau quasi-particles of the polaron Fermi liquid (13). More recently, it has been reported that because of its exceptionally long Bohr radius, SrTiO3 keeps a robust metallic resistivity down to very low doping levels (14). Moreover, both oxygen-deficient (15, 16) and La-doped SrTiO3 (17) host a well-defined Fermi surface down to carrier densities as low as 3 × 1017 cm–3 (which corresponds to 2 × 10–5 electrons per formula unit. Such a context provides an opportunity to test the relevance of different theoretical pictures for the origin of T2 resistivity.

Here we present resistivity measurements showing that the T2 resistivity persists when the carrier density becomes two orders of magnitude lower than previously reported values (12, 13). The magnitude of A varies smoothly as a function of EF and becomes comparable to what has been seen in a heavy-fermion metal. The most important finding is the persistence of T2 behavior in the single-band regime, where there is only a single electron reservoir with a Fermi wave vector much too small for any Umklapp process. This severely restrains possible origins of the observed T2 resistivity. The experimental determination of the collision cross section of electrons in a Fermi liquid with a simple and well-documented Fermi surface topology provides a quantitative challenge for theory. Comparing the data obtained on n-doped SrTiO3 with other Fermi liquids, we argue that Embedded Image, the characteristic length scale of electron-electron scattering in each Fermi liquid, is a source of information regarding the microscopic origin of momentum decay.

The evolution of resistivity as carrier density changes from between 1017 and 1020 cm−3 is presented in Fig. 1 [see (18) for details on all 35 samples studied]. In agreement with previous reports (1417, 19), SrTiO3 in this doping range is found to be a dilute metal whose resistivity drops by several orders of magnitude as it is cooled from room temperature to liquid helium temperatures.

Fig. 1 Doping and temperature dependence of resistivity in n-doped SrTiO3.

(A) Evolution of resistivity in SrTiO3–δ with doping across two orders of carrier density. (B) The product of resistivity and carrier density yields the scattering rate, which does not depend on carrier concentration above 100 K. (C) Resistivity plotted as a function of T2 in oxygen-deficient and Nb-doped SrTiO3 samples of comparable carrier concentration displays the same slope but different intercepts. (D to F) Resistivity versus T2 in SrTiO3–δ as the carrier density changes by two orders of magnitude. Straight solid lines represent the best fits to low-temperature data. As doping increases, the slope gradually decreases, and the upward deviation toward the phonon-dominated regime shifts to higher temperatures. Note the change in the vertical and horizontal scales with increasing carrier density.

Above 100 K, the scattering rate extracted from resistivity and carrier concentration [Embedded Image (n, carrier density; me, free electron mass) in Fig. 1B] does not vary with doping and roughly follows a T3 dependence (we are neglecting the mass renormalization, which would lead to a correction between 1.8 and 5 in this doping window). Below 100 K, inelastic resistivity evolves with carrier concentration. Both electron-phonon and electron-electron scattering mechanisms can depend on the size of the Fermi surface. In the case of acoustic phonons, as documented in graphene (20), the Bloch-Grüneisen temperature (Embedded Image, where vs is the sound velocity) separates two regimes. In a degenerate three-dimensional Fermi liquid, the inelastic resistivity caused by phonon scattering is expected to follow T5 below ΘBG and become T-linear above ΘBG. In our case, ΘBG and the Fermi degeneracy temperature are of the same order of magnitude. Therefore, at high temperatures, electrons scattered by phonons are obeying Boltzmann statistics. Here we focus on the T2 inelastic resistivity emerging at low temperatures, which has been attributed to the scattering of electrons off of each other (5, 12, 13).

As seen in Fig. 1, D to F, the slope of the ρ-versus-T2 plots for SrTiO3–δ (δ quantifies the departure from stoichiometry due to the introduction of oxygen vacancies) smoothly decreases with increasing carrier concentration. In all cases, there is a deviation upward from the T2 behavior toward a regime with a higher exponent. This is in contrast to the case of Fermi liquids with strong correlation, in which quasi-particles are destroyed by warming well below the degeneracy temperature. In SrTiO3, the temperature at which the deviation occurs increases with doping. We found similar behavior in Nb-doped and La-doped SrTiO3 (18). Fig. 2A shows the magnitude of A as a function of carrier concentration. Our data are compatible with those previously reported for higher carrier concentrations (12, 13). Thus, decreasing carrier concentration is concomitant with a monotonous and uninterrupted increase in the magnitude of A across several orders of magnitude, as expected from Eq. 1. The residual resistivity ρ0 (inset) varies much less with carrier concentration. Figure 1C shows that the magnitude of A is quasi-identical in two samples with identical carrier densities but different residual resistivities. Therefore, the magnitude of A is set by n and not by ρ0.

Fig. 2 Variation of A with carrier concentration and Fermi energy.

(A) The prefactor A of T2 resistivity as a function of carrier concentration on a log-log scale. The data represented by empty circles and diamonds are from (12) and (13), respectively. A dash-dot vertical line marks the first critical doping, above which a second band begins to be filled (13, 16). The evolution of the Fermi surface with increasing concentration is also depicted here. Below nc1, the Fermi surface is a simple squeezed ellipsoid, whereas above nc1 it has two concentric components with growing outer lobes. (Inset) Residual resistivity ρ0 extracted from ρ = ρ0 + AT2 fits. (B) Dispersion of the two bands extracted from quantum-oscillation measurements (16). (C) Dependence of the prefactor A on the Fermi energy measured from the bottom of the lower band. Its dependence is close to EF–2 across nc1, with a deviation emerging at higher energies.

In a Fermi liquid, the Fermi energy is reduced when the Fermi surface shrinks or when the effective mass is enhanced. In both cases, the magnitude of A is expected to increase, according to Eq. 1. Mass enhancement is the origin of the large A in heavy-fermion metals. Our results show that a large A can also be achieved by reducing the sheer size of the Fermi surface. In the extreme dilute limit, A becomes an order of magnitude larger than what is found in heavy-fermion UPt3 (21).

Figure 2A reveals a hump in A(n) near n = 1.2 × 1018 cm−3. According to an extensive study of quantum oscillations (16), at this carrier density (dubbed nc1), a second band begins to be filled and the cyclotron mass of the lowest band suddenly enhances. Figure 2B shows the energy dispersion in the two bands constructed from the frequency and effective mass obtained by quantum oscillations (18). The deviation from parabolicity in the lowest band occurs at k = 0.4 nm−1, close to the expectations from the theoretical band structure, according to which anticrossing between bands generates a downward deviation of the lowest band near this wave vector (13).

The dispersion map of Fig. 2B allows us to determine the Fermi energy of each sample from its carrier density, leading to Fig. 2C, which shows A as a function of the Fermi energy of the lowest band with no visible anomaly near nc1. The dependence remains close to EF−2 over a wide range. This is a strong indication that the nc1 anomaly seen in Fig. 1A is almost entirely caused by deviation from parabolic dispersion in the lowest band, which hosts most of carriers.

As seen in fig. S1 (18), one can clearly detect a correlation between large A and small EF across different materials by comparing the variation of A with Fermi energy in SrTiO3–δ and in other Fermi liquids. The inclusion of dilute Fermi liquids in which the electronic specific heat is set by the ratio of carrier density to Fermi energy is an extension of the Kadowaki-Woods approach.

Using Eq. 1, one can extract Embedded Image, the characteristic length scale associated with electron-electron scattering in SrTiO3–δ. The extracted length (Fig. 3A) shows only a very slight decrease with doping and is not proportional to the Fermi wavelength (λF). A proportional relation between Embedded Image and λF would have led to a n–5/3 dependence of A in conformity with the simplest available treatments of electron-electron scattering (22, 23).

Fig. 3 Characteristic length scale of electron-electron scattering in SrTiO3–δ compared with other Fermi liquids.

(A) The length scale shown here is defined in Eq. 1 and extracted from A and TF (Embedded Image, where G0 = 2e2/h) in SrTiO3–δ, as well as a number of other Fermi liquids (see tables S3 and S4 for details and references). The two horizontal solid lines correspond to the Kadowaki-Woods A2 ratio in heavy fermions (10 microhm·cm mol2 K2 J−2; red) and transition metals (0.4 microhm·cm mol2 K2 J−2; black) (2, 3, 6). LSCO, La2–xSrxCuO4; YBCO, YBa2Cu3Oy. (B) The extracted collision cross section of electrons (see Eq. 2) as a function of Fermi wavelength follows a dependence close to λF1.2.

Figure 3A compares the magnitude of Embedded Image in SrTiO3–δ with that of other Fermi liquids (see tables S3 and S4 for details). In a multicomponent Fermi surface, a complication arises because there is a multiplicity of Fermi energies. When the Fermi surface occupies a large fraction of the Brillouin zone, one can assume that there is roughly one electron per formula unit, and it is possible to extract the Fermi energy from γ. Thus, one can estimate the order of magnitude of Embedded Image in dense heavy-fermion and transition metals. In Fig. 3A, these metal types lie close to the horizontal lines that represent the Kadowaki-Woods ratios in the two families (26). Figure 3A also includes data for the Fermi-liquid unconventional superconductor Sr2RuO4 (24), the heavily doped nonsuperconducting compound La2–xSrxCuO4 (25), and the YBa2Cu3Oy cuprate at a doping level of p = 0.11 [in which resistivity is T2 (26, 27) and the Fermi energy of the small pocket seen by quantum oscillation has been quantified (28)]. We have also included reported data on bismuth (29), graphite (30), and arsenic-doped germanium (31). Figure 3A shows that Embedded Image lies mostly between 1 and 40 nm. Its magnitude can be linked to the microscopic details of momentum decay by scattering in each system.

Using Eq. 2, we have also extracted the collision cross section of electrons in SrTiO3–δ from the magnitude of A and the measured radius of the Fermi surface. Figure 3B shows variation of the collision cross section as a function of their Fermi wavelength. If the electrons were classical objects bouncing off of each other, σcs would have been 2πλF2. Our data are inconsistent with that classical picture; Fig. 3B shows that σcs is much smaller than 2πλF2 and does not follow λF2. Hence, the theoretical challenge of providing a quantitative explanation for this observation remains.

The mechanism by which electron-electron scattering in Fermi liquids causes T2 resistivity is not well understood. Previously, T2 resistivity in Fermi liquids was observed in systems with a large single-component Fermi surface (such as La1.7Sr0.3CuO4) or those with small multicomponent surfaces (such as bismuth or graphite). In each case, one of the scenarios shown in Fig. 4 could be ruled out, but one could still invoke either the multiplicity of reservoirs or the relevance of Umklapp processes. However, in the case of extremely dilute SrTiO3–δ, no room is left for either of the two. An Umklapp event cannot occur unless the largest available Fermi wave vector is one-fourth the size of the smallest vector of the reciprocal lattice, G. By a rough estimation, this corresponds to a carrier density of 2 × 1020 cm−3, and Umklapp scattering may cause the hump in the energy dependence of A(EF) near 10 meV (see Fig. 2C), which corresponds to this carrier concentration. However, we find that A is still growing when kF becomes 30 times smaller than G.

Fig. 4 Theoretical models for T2 resistivity.

(A) The original mechanism (1) requires two distinct reservoirs of electrons with different strengths of coupling to the lattice. (B) Umklapp scattering in which the momentum balance between incoming and outgoing electrons differs by a unit vector (k1i, k2i) of the reciprocal lattice. Such events are possible only when the Fermi wave vector (k1f, k2f) is equal to or larger than one-fourth the width of the Brillouin zone. Embedded Image, reciprocal lattice vector; Embedded Image, wave-vector imbalance between initial and final states. (C) Neither of these scenarios can explain the persistence of T2 resistivity in the case of dilute SrTiO3–δ, in which there is a single tiny Fermi surface at the center of the Brillouin zone. A scenario is required in which (at least some) scattering events between electrons are accompanied by an asymmetric exchange of momentum with the lattice.

In the specific case of doped SrTiO3, an explanation of the T2 resistivity may invoke the polaronic nature of the quasi-particles (13) or the distorted structure of the Fermi surface (17). Beyond this particular case, our results highlight the absence of a microscopic theory for momentum decay through electron-electron scattering in different Fermi liquids. The magnitude of Embedded Image can be experimentally quantified in each Fermi liquid. A potentially important role of phonon-assisted (32) electron-electron scattering should be reconsidered.

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 and S2

Tables S1 to S4

References (3360)

References and Notes

  1. See supplementary materials on Science Online.
  2. Acknowledgments: We thank H. Maebashi, D. Maslov, and K. Miyake for stimulating discussions. This work was supported by Agence Nationale de Recherche through the SUPERFIELD and QUANTUM LIMIT projects and by a grant from the Ile de France region.

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