Non-Fermi liquids in oxide heterostructures

Understanding the anomalous transport properties of strongly correlated materials is one of the most formidable challenges in condensed matter physics. For example, one encounters metal-insulator transitions, deviations from Landau Fermi liquid behavior, longitudinal and Hall scattering rate separation, a pseudogap phase, and bad metal behavior. These properties have been studied extensively in bulk materials, such as the unconventional superconductors and heavy fermion systems. Oxide heterostructures have recently emerged as new platforms to probe, control, and understand strong correlation phenomena. This article focuses on unconventional transport phenomena in oxide thin film systems. We use specific systems as examples, namely charge carriers in SrTiO3 layers and interfaces with SrTiO3, and strained rare earth nickelate thin films. While doped SrTiO3 layers appear to be a well behaved, though complex, electron gas or Fermi liquid, the rare earth nickelates are a highly correlated electron system that may be classified as a non-Fermi liquid. We discuss insights into the underlying physics that can be gained from studying the emergence of non-Fermi liquid behavior as a function of the heterostructure parameters. We also discuss the role of lattice symmetry and disorder in phenomena such as metal-insulator transitions in strongly correlated heterostructures.


Introduction
The degree to which the Coulomb interactions among the conduction electrons ('strong correlations') drive novel states of matter in real materials and determine their physical properties lies at the center of many unresolved questions in condensed matter physics. Over the past few years, high-quality thin films and heterostructures have emerged as novel platforms to study strong correlation physics. Heterostructures allow for control of parameters that affect the microscopic physics in ways not possible with bulk materials. For example, carrier densities can be introduced without the use of chemical dopants by modulation doping [1,2] or polar/non-polar interfaces [3,4] and tuned through field effect [5][6][7]. Dimensionality can be controlled through electrostatic confinement coupled with layer-by-layer control. Interfaces can be used to induce magnetism [8] or superconductivity [9], to control lattice symmetry [10], orbital, and charge order [11], phonon coupling, or to create artificial materials in superlattice geometries [12]. Coherency strains in epitaxial structures modify lattice parameters and crystal symmetry, with profound influence on phenomena such as metal-insulator transitions. Advanced thin film growth approaches yield very high quality films [13]. All of these approaches are now being widely applied to complex oxide materials, including those that are strongly correlated, and have led to many interesting new discoveries.
This article focuses on one specific aspect of correlated oxide heterostructures, namely the electric transport properties of itinerant charge carriers. Deviations from conventional metallic behavior are often referred to as 'non-Fermi liquid', 'strange', or 'bad metal' behavior. These terms describe somewhat different phenomena, and we define them in the following sections, but often they appear within the same materials system. They have been studied for many decades in strongly correlated materials, such as the unconventional superconductors and heavy fermion systems [14][15][16][17][18][19], using measurements of properties, such as the electrical resistance, thermal transport, or conductivity at optical frequencies, as a function of temperature (T), pressure, doping, frequency, or magnetic field. Often, studies were motivated by the idea that the normal (non-superconducting) state may provide insights into the pairing mechanism that leads to high-temperature superconductivity. Because they remain so profoundly enigmatic, however, in particular when contrasted with the degree of understanding that has been achieved for 'conventional' metals and semiconductors, the transport properties of correlated materials are of high scientific interest in their own right.
The level of difficulty is substantial. For example, in strongly correlated materials, non-Fermi liquid behavior is typically considered in the limit of extremely high carrier concentrations, near one electron/unit cell, where carriers are subject to on-site Coulomb interactions and the material is near a 'Mott' metal insulator transition. These materials are often in proximity to complex magnetically or electronically ordered states. At the other end of the spectrum lie low-density, 2D electron systems in semiconductors, which show unconventional transport and metal-insulator transitions under the influence of long-range Coulomb interactions [20], with the limiting state being the Wigner crystal. How these two extremes intermesh is a subject of ongoing investigation. Strong coupling to the lattice, polaron formation [21], and magnetic interactions are key ingredients of transport in narrow band materials. Disorder, which is always present, can also contribute to non-Fermi liquid behavior [22,23]. Despite these challenges, several unifying theoretical concepts exist. For example, an underlying quantum critical point can give rise to non-Fermi liquid behavior [14,15]. The degree to which these concepts apply to a given real material remains, however, a subject of intense debate in many cases.
Only recently have there been substantial efforts to explore correlation phenomena in thin films and heterostructures and in a wider class of materials, beyond high-temperature superconductors [24]. This is in contrast to the study of Coulomb interactions in the very dilute systems, where low-dimensional semiconductors have been the primary focus [25,26]. The objectives of this article are to use a few specific examples (see figure 1) to discuss the interesting transport phenomena that can be probed in oxide heterostructures and to discuss the insights and new questions that have emerged from these studies. We discuss results from transport studies of quantum wells of SrTiO 3 , which contain sheet carrier densities on order of 7 × 10 14 cm −2 confined in a few atomic layers and which exhibit correlated phenomena such as itinerant magnetism, pseudogaps, and metal-insulator transitions [27][28][29][30]. We also discuss thin film rare earth nickelates, which exhibit a metalto-insulator transition upon lowering the temperature [31]. We include in the discussion results from transport studies of doped SrTiO 3 for comparison purposes.

Longitudinal resistance
The temperature dependence of the electrical resistance of correlated materials typically can be described by power laws. Often a distinction is made between regimes (as a function of some tuning parameter), where the resistance is proportional to T 2 , which is the behavior expected for a Fermi liquid at low temperatures, and regimes, where it is proportional to T n , where n is a 'non-Fermi liquid' exponent with 1 ⩽ n < 2. While the T 2 Fermi-liquid regime is difficult to detect for conventional metals because it is either overwhelmed by electron-phonon scattering (at high temperatures) or by impurity scattering (at low temperatures) [32], examples of strongly correlated materials that exhibit a pronounced T 2 behavior of the resistance, often to high temperatures, abound. To name a few, they include materials as diverse as V 2 O 3 [33], La 2−x Sr x CuO 4 [34], Ag 5 Pb 2 O 6 [35], NdNiO 3 [36], NiS 2-x Se x [37], and doped SrTiO 3 [38]. In some cases, the quadratic temperature dependence reaches room temperature.
A T 2 scattering rate is often taken to be a signature of a 'Fermi liquid'. In a Fermi liquid, electron-electron scattering can give rise to a T 2 behavior, because the phase space for scattering is limited to an energy range of kT around the Fermi energy, E F , which is a direct consequence of the Pauli exclusion principle. Specifically, for a Fermi liquid [15]: where τ ee is the quasiparticle lifetime, is the reduced Planck's constant, |V| is a model-dependent interaction potential, g 2 F is the density of states at the Fermi surface, and E the energy of the quasiparticle. At finite temperatures, E is set by the temperature, so for quasiparticles near the Fermi surface the scattering rate (1/τ ee ) is proportional to T 2 . A momentum relaxation mechanism, such as Umklapp scattering, or a complex Fermi surface, are required for electron-electron scattering to influence the resistivity [39,40]. Here, we discuss recent studies on thin films that show that taking the T 2 behavior to be a signature of a Fermi liquid may not always be correct.
The specific example we discuss here are electron-doped SrTiO 3 films. At high temperatures, the mobility of lightly doped SrTiO 3 can be described quantitatively by longitudinal optical (LO) phonon scattering of polarons [41][42][43]. At intermediate temperatures, the resistivity and the Hall mobility show a quadratic temperature dependence (figure 2) [38,[43][44][45]. The LO phonon scattering is increasingly screened as the doping concentration is increased and the T 2 behavior can be observed to higher and higher temperatures [43].
The crux of the matter is that SrTiO 3 shows a T 2 scattering rate at temperatures and electron concentrations where the electron gas is likely non-degenerate [46][47][48][49], that is where k B is the Boltzmann constant or the spacing between the carriers, N −1/3 , is much greater than the thermal de Broglie wavelength, h/ √ 2πm * k B T. Here, N is the 3D carrier density, h is Planck's constant, and m * is the effective carrier mass. SrTiO 3 shows metallic behavior (dR/ dT > 0) even in the non-degenerate case because the dielectric constant (κ), and thus the Bohr radius (a B ), is large, so that the Mott criterion, Na 3 B ~ 0.2, is fulfilled, where a B = ħκ/m * e 2 (e is the electron charge). If the Fermi level is low, however, the phase space arguments that lead to a T 2 scattering rate in a Fermi liquid according to equation (1) are not applicable. Indeed, in the non-degenerate case the transport is captured by Maxwell-Boltzmann statistics, there is no well-defined Fermi energy and the chemical potential is below the conduction band edge.
If SrTiO 3 shows a T 2 scattering rate even though it is not a Fermi liquid (it is a classical gas) how about other mat erials? To discuss this question, consider equation (1), which suggests a strong carrier density dependence of 1/τ ee , i.e. via the interaction potential. A simpler, phenomenological equation is often used for the scattering rate of a Fermi liquid [50]: where B is a dimensionless factor that incorporates scattering event probabilities. 1/τ ee is expected to depend on the carrier density due to E F (and B). Assuming the resistivity is described by ρ xx = ρ 0 + AT 2 , where ρ 0 is the residual resistivity, then, according to the Drude model: Measurements of A as a function of carrier density have been reported for several oxides, including cuprates [51][52][53][54], vanadates [55], bulk, doped SrTiO 3 [56], and SrTiO 3 quantum wells [56,57]. In all cases, A ~ 1/N, consistent with expectations from the Drude model, see equation (3). A plot that shows this behavior for doped SrTiO 3 appears in figure 3, where A ~ 1/N can be observed over a very large carrier density range in both the non-degenerate and degenerate doping regimes. The important point is that A ~ 1/N means that the scattering rate, 1/τ ee , is independent of the carrier density, contrary to what is expected from Fermi liquid theory, see either equation (1) or equation (2) [56]. Because A ~ 1/N is also observed in materials such as the cuprates which, unlike lightly doped SrTiO 3 , can be considered strongly correlated materials, this hints strongly at a surprisingly universal T 2 scattering rate that is not due to electron-electron scattering in a Fermi liquid. Oxide thin films and heterostructures discussed in this review and some of the phenomena that have been observed (see references in the text). Those that are discussed in more detail here are in bold. The dashed lines indicate the quasi-two-dimensional electron system at the interface between RTiO 3 (R is a rare earth ion but not Eu) and SrTiO 3 , which forms as a result of a 'polar discontinuity'. Typical carrier densities are indicated. The thickness of the SrTiO 3 in the quantum well structure can be as low as a single SrO plane sandwiched between the interfacial TiO 2 planes.

Hall mobility or Hall angle
Strongly correlated materials often show resistances that depart from the quadratic dependence on temperature (we will discuss these in the next section). In contrast, the Hall angle (cotθ), which is the inverse of the Hall mobility, almost always shows a very robust T 2 temperature dependence [34,53,[58][59][60][61]. This observation has become known as 'two lifetime' behavior, because it suggests that different scattering rates control the Hall mobility and the longitudinal resistance, respectively [62]. In addition to the cuprates, lifetime separation has been observed in a wide range of correlated materials, including V 2 O 3 [63], ruthenates [64,65], heavy Fermion systems [66,67], MgB 2 [68], and in oxide heterostructures [57]. Figure 4 shows an example of the two lifetime phenomenon for a high-carrier-density, quasi-two-dimensional electron system at a SmTiO 3 /SrTiO 3 interface that is close to a metal-insulator transition. Unlike the lightly doped SrTiO 3 films discussed in the previous section, this electron system is degenerate. The resistance is ~ T 5/3 above ~100 K, while the Hall angle is always ~ T 2 [69]. Lifetime separation was also investigated in quantum wells of SrTiO 3 [57,70], such as formed in SmTiO 3 /SrTiO 3 / SmTiO 3 heterostructures, which contain mobile charge densities on order of 7 × 10 14 cm −2 and which exhibit numerous correlated phenomena, such as itinerant magnetism and pseudo gaps, when their thicknesses are reduced to a few atomic planes. These studies showed that the lifetime separation can have two components, namely the different temperature dependencies of Hall angle and longitudinal resistances, as seen in figure 4, but also a divergence in their 0-K residuals [57]. Both lead to a temperature dependence of the Hall coefficient even when there is no major change in the Fermi surface or carrier density. These two contributions to the lifetime separation may have different origins. In the SrTiO 3 quantum wells, the divergence of the 0-K scattering rates appeared only near a quantum critical point near a transition to an itinerant magnetic phase [57]. Further analysis of the 0-K residuals suggested that the proximity to the quantum critical point influenced the residual in the Hall angle, but not that of the Hall mobility as a function of temperature for the film with a carrier density of 2 × 10 18 cm −3 (symbols). As discussed in [43], the mobility can be described completely by just three terms: a temperature-independent impurity scattering term, μ 0 , which dominates at low temperatures, a LO phonon-polaron scattering term, μ LO , which dominates at high temperatures, and a term that is proportional to T −2 . More details about the LO phonon term can be found in [41]. The results for the fit parameters can be found in [43]. The different lines indicate the contribution of each term and their sum describes the experimental data extremely well, see solid red line. For similar fits for films with different carrier densities, see the supplementary information to [43].  [44,45]. The A-coefficient approximately follows a 1/N dependence on the carrier density N over orders of magnitude in N, as can be seen by comparison with the black dotted line. Slight deviations from the 1/N behavior are expected even if the scattering rate is independent of the carrier density, because the carrier mass also changes as higher lying bands are filled. Reprinted from [56]. CC BY 4.0.
longitudinal resistance. In the cuprates, the 0-K divergence in scattering rates has been explained with anisotropies in the elastic scattering rates [71]. In a recent study of SmTiO 3 / SrTiO 3 /SmTiO 3 quantum wells, we showed that disorder affected the residuals in both the Hall angle and longitudinal resistances, but it did not affect their 0-K divergence [70].
There currently exists no generally accepted understanding of lifetime separation [72]. Suggested explanations include a 2D Luttinger liquid (spin-charge separation), complex Fermi surfaces combined with strongly anisotropic scattering rates, as may occur near an antiferromagnetic transition, bipolarons, and quantum critical points [18,54,62,[73][74][75][76][77][78]. Most theoretical models aimed at reproducing R ~ T and cotθ ~ T 2 , which, unfortunately, does not apply to all of the materials for which lifetime separation is observed, such as the SrTiO 3 interfacial electron systems. One point of view is to take the T 2 Hall scattering rate as a signature of an underlying Fermi liquid state even though the resistance departs from simple behavior [53]. In the light of what was discussed above, namely that a T 2 scattering rate may not always indicate a Fermi liquid, this interpretation should be taken with some degree of caution.
To summarize section 2, there are a surprising number of common features in the transport of a wide range of correlated materials and complex oxide heterostructures: a Hall angle (inverse of the Hall mobility) that remains robustly T 2 , independent of the specific temperature dependence of the longitudinal resistance, and a scattering rate that does not appear to be overly sensitive to factors such as the carrier density, electronic structure, or even whether the electron system is degenerate or non-degenerate. Coupling to the lattice should certainly be considered in any future developments of an understanding of the unusual transport properties of oxides, as will also be discussed in the following sections. The quasiparticle mass in SrTiO 3 at low temperatures is enhanced by a factor of two, most likely due to dressing by phonons [79], as is also seen in angle resolved photoemission [80]. Similar to electron-electron scattering in metals [32], the pronounced T 2 scattering rate, which often extends to very high temper atures, appears to be a feature for electrons in conduction bands derived from d-states. For example, a degenerate perovskite oxide system with an s-band conduction band, La-doped BaSnO 3 , does not exhibit a T 2 scattering rate in its resist ance [81]. The findings illustrate the need for a theoretical framework that can explain (i) the universal T 2 scattering rate and, equally important, (ii) its peculiar robustness in the Hall angle. By universal we mean that it encompasses high-carrier-density, correlated materials as well as low-density systems such as SrTiO 3 .

T n power laws in the electrical resistivity
Departures from T 2 behavior in the longitudinal resistance to other power laws of form T n , with 1 ⩽ n ⩽ 2 are ubiquitous. These power laws are often identified as signatures of 'non-Fermi liquids'. The extreme case are the superconducting cuprates, which show T-linear behavior in the normal state, sometime referred to as 'strange metal' behavior.
One possible explanation for non-Fermi liquid behavior are quantum critical points [15]. Near a magnetic or other instability, the associated order parameter fluctuations may lead to strong scattering and non-Fermi-liquid behavior [82]. Oxide heterostructures allow for testing these ideas by using tuning parameters, such as epitaxial strain, to bring a mat erial system close to a quantum critical point. Here, we discuss results from epitaxial NdNiO 3 films.
Metallic rare earth nickelates (except LaNiO 3 ) possess an orthorhombic perovskite-derived crystal structure (space group Pbnm). They exhibit temperature-driven metal-insulator transitions, whose transition temperature depends on the size of the rare earth ion and degree of oxygen octahedral tilts in the orthorhombic structure, which control the bandwidth [31]. and Hall angle (b) of a quasi-two-dimensional electron liquid at the SmTiO 3 /SrTiO 3 interface, which has a sheet carrier density of ~2 × 10 14 cm −2 , due to the polar discontinuity at the interface. The Hall angle, cotθ, is the inverse of the Hall mobility, μ H , Hcotθ = µ −1 H , where H is the magnetic field. The sheet resistance follows a T 5/3 temperature dependence at elevated temperatures, then shows a dip, followed by a small temperature interval of ~T 2 temperature dependence, and an upturn at low temperatures. In contrast to the complicated behavior of the resistance, the Hall angle follows a T 2 temperature dependence almost over the entire temperature range. Note that the LO phonon scattering term at high temperatures, which is clearly seen for the low density, 3D electron gas in figure 1(b), is absent here, due to the screening of the LO phonons at the high electron density. Data is from [69].
The symmetry of the insulating, low-temperature, antiferromagnetic phase is monoclinic (P2 1 n) [83,84]. In this space group, the presence of two inequivalent Ni sites allow for 1:1 charge order [85] (or 'bond-length disproportionation', since the actual charge on the two Ni sites may not be very different [86,87]). The electrical resistivity of the rare earth nickelates above the metal-insulator transition exhibits T n behavior and various exponents of n ~ 4/3, 5/3, or 2 have been reported [36,[88][89][90]. The rare earth nickelates are therefore an interesting system to investigate various power laws in transport in proximity to metal-insulator transitions.
As shown in [36], the temperature dependence of the resistivity of the rare earth nickelates in the metallic phase can be described by the following equation: where ρ sat is a saturation resistivity. As will be discussed in the next section, resistivity saturation is important in modeling transport when the scattering rate is high and ρ ideal becomes large. Although there are materials that do not show saturation, neglecting ρ sat [89,91] in case of the nickelates yields incorrect values for n [36]. In many correlated materials, ρ ideal is described by a power law: Figure 5 shows results for epitaxial NdNiO 3 films, which were coherently strained to substrates having different signs and degrees of lattice mismatch [36]. It was shown that n assumes just two values, either n = 2 or n = 5/3 [36]. Specifically, if the metal-insulator transition was suppressed, which is the case for in-plane compressive strains, then n = 5/3. Electron diffraction studies showed that films under compressive strains adopt space group P2 1 m [92], which is expected from symmetry arguments discussed in [93]. The P2 1 m space group does not allow for 1:1 charge order, which would require further lowering of the symmetry [92], which is not possible in films that are coherently constrained to a substrate. In contrast, films under tensile strain possess orthorhombic symmetry, which allows for the transition to the charge ordered, insulating state [92]. Thus, thin film studies establish the following about this strongly correlated system: (i) the metal-insulator transition in the rare earth nickelates is suppressed if the heterostructure boundary conditions prevent establishment of 1:1 charge order. In other words, 1:1 charge order is a requirement for the insulating state. (ii) The non-Fermi liquid exponent of n = 5/3 is associated with the suppression of the 1:1 charge ordered insulating state. On first sight, (ii) appears consistent with theories that ascribe non-Fermi liquid exponents to order parameter fluctuations; in this case it may be associated with the suppression of charge and antiferromagnetic order, and a nearby quantum critical point. Interestingly, both types of metals (both symmetries) exhibit pseudogaps (a depletion of the single particle density of states), which, in the case when the insulator is not suppressed, develops into a full gap [94]. Pseudogaps are typically associated with incipient antiferromagnetic order. We note, however, that there are just two exponents observed in the entire phase diagram in figure 4. This observation points to a picture of specific, fractional exponents in the temperature dependence of the resistivity that are firmly tied to a specific non-Fermi liquid phase, with a distinct symmetry. A similar case of a 'non-Fermi liquid phase' has been made previously for the itinerant ferromagnet MnSi [95].

Mott-Ioffe-Regel-limit and resistance saturation
At high temperatures, two types of behaviors of the electrical resistance of high-resistance materials can be distinguished: (i) the resistance increases seemingly without limit or (ii) it 'saturates' by slowly approaching a finite value. The non-saturating behavior (i) has become known as 'bad metal behavior' [96]. It has attracted attention because unconventional superconductors in their normal state are non-saturating and it is considered a signature of a novel kind of metallic state [96]. Neither case is, however, fully understood [97][98][99][100][101][102][103][104]. Confusion arises from the fact that the Mott-Ioffe-Regel limit appears in the discussion, so we discuss this briefly first.

Mott-Ioffe-Regel limit
The resistances of thin metal oxide films are often sufficiently high that they reach or exceed the Mott-Ioffel-Regel limit. This limit is the resistance when the scattering of the carriers becomes so strong that lk F ∼ 1, where l is the mean free path of the carriers and k F the Fermi wave vector [105,106]. An approximate value for the Mott-Ioffe-Regel limit in two dimensions is h/e 2 [107]. It is well-established to be relevant in the regime of elastic scattering at low temperatures, i.e. it establishes a criterion where a plane wave description no longer makes any sense [26,104]. A 2D electron system that, for any reason, exceeds a sheet resistance on order of of h/e 2 (25 kΩ/sq.) at low temperatures may thus be expected to be localized. In three dimensions, there is an additional factor k −1 F [107]. In keeping with this expected behavior, there are ubiquitous observations of oxide thin films and heterostructures that exhibit insulating behavior at all temperatures at low film thicknesses, because their resistance exceeds the Mott-Ioffe-Regel limit [108][109][110][111]. Figure 6 shows an example of this behavior for a LaNiO 3 film [112]. The point here is that the observation of insulating behavior in such cases should not be used as evidence of a correlation-induced metal-insulator transition. Often, the resistivity also increases as the film thickness is reduced. This can be due to a number of reasons, such as increased scattering from surfaces and interfaces. Another reason is bandwidth narrowing, which is related to the octahedral tilts in the distorted perovskite structure that these oxides adopt. Connectivity with the substrate and film strain all affect the octahedral tilts [10,113]. For example, thin LaNiO 3 films exhibit octahedral tilt patterns near the interface with a substrate that are different from those in the bulk and which cause a high resistivity, and insulating behavior below a critical thickness as the Mott-Ioffe-Regel limit is exceeded. In contrast, LaNiO 3 films of the same thickness embedded in superlattices remain metallic, which can be explained with their different octahedral tilt patterns [113]. Films further in the metallic regime typically show weak localization behavior [114] at low temperatures, which is manifested as an upturn in the resistance at low temperatures, also seen in figure 6.
One unfortunate consequence of the high resistances in thin films due to the inherently low carrier mobilities of correlated materials, and associated localization when the Mott-Ioffe-Regel limit is exceeded, is that it makes it difficult to study correlation effects in itinerant, low-dimensional systems, which are pre-empted by this 'trivial' localization. An example of interesting phenomena that may emerge when resistances are kept well below the Mott-Ioffe-Regel limit is a novel metal-insulator transition in an interfacial electron system at the SmTiO 3 /SrTiO 3 interface [69].

High temperatures: saturating and non-saturating metals
Returning to the high temperature characteristics, we remark that at high temperatures the scattering is largely inelastic. Multiple scattering and phase coherence, features of transport at the low temperature localization transition are not important at high temperature. Saturating metals are well described by the parallel resistor formula, equation (4). This formula emerges in a transport model that invokes the Mott-Ioffe-Regel limit as a cut-off time, τ 0 , for the mean time between electron scattering, τ > τ 0 = a/v F [115]. The physical basis of this model is that it suggests there should be no transport time scale shorter than the time it takes an electron to move across a unit cell. Following [115], Hussey et al use this approach with some modifications to obtain the parallel resistor model as well as an intuitive frequency dependent conductivity that consists of a Drude response that is broadened and disappears at high temperature leaving a broad incoherent tail that extrapolates to ρ −1 sat at low frequency [104]. Time scales, high frequency dynamics, and incoherent high frequency tails are discussed in [102]. While ρ sat invokes the Mott-Ioffe-Regel criterion, its precise value for any given system is determined by important details. Since ρ sat is an important feature substrates. In bulk, LaNiO 3 is a metal at all temperatures, and does not show a temperature-induced metal insulator transition, unlike the other rare earth nickelates. The thinnest (3 nm) film exceeds the Mott-Ioffe-Regel limit and is insulating at all temperatures. Its sheet resistance is about 15 kΩ at room temperature. The sheet resistance of the nextthinnest film (4 nm thickness) significantly lower, ~2700 Ω. It shows an upturn in the resistivity at low temperatures, which could be due to weak localization (see arrow). The resistances of LaNiO 3 films are very sensitive to modification of octahedral tilt patterns by the substrate and adjacent layers. Ultrathin LaNiO 3 films in superlattice geometries, which have different tilt patterns than those directly adjacent to a rigid substrate, remain metallic (they do not exceed the Mott-Ioffe-Regel limit) [113]. Reprinted with permission from [112]. Copyright (2010), AIP Publishing LLC.
in complex materials, estimates of the expected value of the maximum scattering rate and the other material parameters that determine the saturated conductivity need to be taken with a grain of salt and can differ from reasonable estimates by orders of magnitude.
It is important to appreciate that the effect of scattering rate saturation and ρ sat will emerge in the experimental measurement well below resistivity saturation and must be included in fits to experimental data to extract ρ ideal , as was emphasized in section 3. The degree to which it affects the data depends on the relative values of ρ sat , ρ 0 , and A. Figure 7 shows these values for the NdNiO 3 films discussed in section 3 [36]. ρ sat increases with increasing epitaxial coherency strain. Strains lift the orbital degeneracy of the Ni e g states [116,117], and the magnitude of ρ sat is sensitive to these changes in the electronic structure. Furthermore, while ρ 0 increases sharply with decreasing film thickness, ρ sat is insensitive to it. This results in a critical film thickness for each mismatch strain at which the condition ρ sat ∼ ρ 0 is fulfilled, and these films are insulating. This can be understood from equation (4): if ρ sat ∼ ρ 0 , then the first term becomes small at finite temperatures compared to 1/ρ sat , which represents the incoherent (non-Drude) part, and such a film cannot be a metal anymore.
We finally note that while much theoretical effort has been expended to understand the absence of resistance saturation in some degenerate electron liquids [97][98][99][100]104], so called 'bad metals', the absence of resistance saturation at high temperatures in low-density SrTiO 3 , is probably neither 'strange' or 'bad', contrary to [118]. The system is a non-degenerate electron gas and as such is described by Maxwell-Boltzmann statistics. Assuming a mean free path no shorter than a unit cell dimension, ~a, the shortest effective scattering time, τ , may be related to the thermal velocity, v th , as τ = a/v th . The thermal velocity, v th = (2k B T/m * ) 1/2 increases as T 1/2 and one can estimate the temperature dependent upper bound to the high temperature resistivity by ρ sat ≈ m * (T) /ne 2 We expect no saturation of the high temperature resistivity for this non-degenerate system; 'ρ sat ' would increase with temperature. Furthermore, transport in SrTiO 3 at high temperatures is described by large polarons and one expects a strong increase in effective mass [119][120][121]. The experimental data shows that the resistivity rises substantially faster than that given by the temperature dependence of v th alone, indicating an increase in m * . Taking a mass of ~5m * at room temperature, we estimate a room temperature resistivity bound of ~5 Ω cm at an electron concentration of ~4 × 10 17 cm −3 . The data in [118] does not violate this bound. Alternatively, one can take the room temperature resistivity of ~4 × 10 17 cm −3 , and a thermal velocity based on a mass of ~5m * and extract a mean free path of roughly ~2a. If we accept these assumptions, low-doped SrTiO 3 is a well-behaved semiconductor at room temperature and not a bad or strange metal. We also note that Boltzmann theory applies for phonon scattering in metals at most temperatures [122], even though the room temperature scattering rate is very large, ~k B T/ [123][124][125]. Indeed, the high temperature mobility of doped SrTiO 3 can be quantitatively described by a gas of polarons that scatters on LO phonons [42,43], even though it has a very large scattering rate.

Conclusions
In summary, anchoring our discussion with transport in doped SrTiO 3 , which appears to be a well-behaved electron gas or liquid albeit with a poorly understood T 2 scattering rate, other oxide heterostructures and thin films exhibit many non-Fermi liquid phenomena, such as power laws in the temperature dependence of the resistance, lifetime separation, and resist ance saturation. Because of the additional degrees of freedom, thin film studies The disorder contribution to the resistivity, ρ 0 ~ ρ (0 K) increases with decreasing film thickness, while the saturation resistance is not strongly affected. Both ρ 0 and ρ sat are strain dependent, establishing a connection between the electronic structure and ρ sat . Reproduced from [36]. CC BY 4.0. allow for illuminating key questions, such as the role of coupling to the lattice in the transport behavior. While some phenomena, such as localization as resistances exceed the Mott-Ioffe-Regel limit, can be understood based in classical models, many others cannot be captured even qualitatively within the framework of existing descriptions of transport. These include the robust and surprisingly universal T 2 dependence of the Hall angle and fractionalized non-Fermi liquid exponents that signify new phases. It is obvious that a new description of charge carrier transport and interactions is likely needed that applies and generalizes to a much wider class of materials than those traditionally considered in the existing models of non-Fermi liquid behavior.