Abstract
We examine the carrier density dependence of the scattering rate in two and threedimensional electron liquids in SrTiO_{3} in the regime where it scales with T^{n} (T is the temperature and n ≤ 2) in the cases when it is varied by electrostatic control and chemical doping, respectively. It is shown that the scattering rate is independent of the carrier density. This is contrary to the expectations from Landau Fermi liquid theory, where the scattering rate scales inversely with the Fermi energy (E_{F}). We discuss that the behavior is very similar to systems traditionally identified as nonFermi liquids (n < 2). This includes the cuprates and other transition metal oxide perovskites, where strikingly similar densityindependent scattering rates have been observed. The results indicate that the applicability of Fermi liquid theory should be questioned for a much broader range of correlated materials and point to the need for a unified theory.
Introduction
A complete understanding of materials whose transport and thermodynamic properties deviate from Landau Fermi liquid theory remains one of the central problems in physics^{1,2}. NonFermiliquid behavior is usually identified via powerlaws in the temperature (T) dependence of properties such as electrical resistance. For example, the resistance of nonFermi liquids scales with T^{n}, where n < 2. In many cases, including some very extensively characterized materials, such as the cuprate superconductors, it is still under debate if the observed behavior can be described by a modified Fermi liquid theory or if a completely new description of such metallic states is required^{1,2,3,4}.
Conversely, a correlated electron system is usually identified as a classic Fermi liquid when n = 2, which is the exponent predicted from the electronelectron scattering contribution to the resistance. There may, however, exist another case, namely systems that appear to behave like a Fermi liquid, exhibiting n = 2, but on closer inspection fail to be described by Fermi liquid theory. This case is important, because many of the systems that exhibit nonFermi liquid behavior and exotic states, such as unconventional superconductivity, emerge from “parent phases” that are thought to be Fermi liquids, because n = 2.
The fact that the resistance follows T^{2} behavior up to unusually high temperatures (T » T_{F}, where T_{F} is the Fermi energy) in many correlated electron systems may already hint at difficulties for Fermi liquid theory. Examples include highly doped SrTiO_{3}^{5,6,7,8}, cuprate superconductors that show T^{2} behavior to room temperature in some parts of their phase diagrams^{9,10} and rare earth nickelates, which follow n = 2 to room temperature^{11,12}. Welldefined T^{2} resistivity is found in many other 3d, 4d and 5d transition metal perovskites^{13,14,15}. Another phenomenon that is difficult to reconcile with Fermi liquid quasiparticles^{16,17,18} is a pronounced separation of the 0K Hall and longitudinal transport scattering rates in electron liquids in SrTiO_{3} quantum wells even though n ≈ 2 indicates a Fermi liquid^{19}.
In this work, we examine the applicability of Fermi liquid theory by tuning the carrier density in two different SrTiO_{3}based electron liquids: (i) chemically doped SrTiO_{3} and (ii) electrostatically gated twodimensional electron liquids (2DELs) in thin SrTiO_{3} quantum wells sandwiched between two insulating SmTiO_{3} layers. Uniformly doped SrTiO_{3} exhibits a T^{2} dependence of the resistance over an exceptionally wide doping range^{5,6,7,8,20}. As the doping is increased, optical phonon mode scattering, which operates at high temperatures, is screened out and the T^{2} regime expands to ever higher temperatures^{7}. This behavior is consistent with electronelectron scattering dominating the transport. The quantum wells contain about 7 × 10^{14} cm^{−2} mobile carriers that are introduced by the charge (polar) discontinuity between SrTiO_{3} and SmTiO_{3}^{21}. Electrical transport is dominated by the T^{n} contribution (n ≤ 2) to room temperature^{7,22}. A drop from n ~ 2 to n ~ 5/3 occurs at a critical quantum well thickness^{19}. The endpoint (SmTiO_{3} without any embedded SrO layer) is an antiferromagnetic Mott insulator.
We examine the temperature coefficient A, the residual resistance R_{0} and the exponent n in the temperature dependence of the electrical sheet resistance, R_{xx}, as the 2D or 3D carrier density is varied:
We use electric field gating to modulate the charge carrier densities in the 2DELs. This has the advantage of avoiding alloy disorder and allows for continuous modulation of the carrier density by the applied electric voltage. We show that Fermi liquid theory faces difficulties in providing a quantitative explanation of a nearly carrier density independent scattering rate in R_{xx} for both the 2DELs and the uniformly doped SrTiO_{3}.
Results
Figure 1(a) shows a schematic of the gated Hall bar device structure. An optical micrograph is shown in Fig. 1(b). A 6nm layer of the wideband gap insulator SrZrO_{3} served as a gate dielectric. The thin SrTiO_{3} quantum well, containing one unit cell of SrTiO_{3} (2 SrO layers), is confined by insulating SmTiO_{3} layers. At this thickness of the SrTiO_{3} quantum wells a nonFermi liquid exponent n is expected^{19,23}.
Figure 1(c) shows the temperature dependence of the quantum well sheet resistance, R_{xx}, at different gate voltages, V_{G}. The normalized [(R/R(V_{G} = 0)] sheet resistance is shown in Fig. 1(d) across the entire measured T range (2–300 K). The resistance modulation between V_{G} = −1 and +1 V was 3–4% and was reversible. The gate leakage current, I_{G}(V_{G}), showed nonlinear behavior at all temperatures [Fig. 1(e)]. The highest leakage was 15 nA, below 1 nA for most V_{G}’s and in all cases orders of magnitude below the sourcedrain current of 40 μA.
The electrostatic gate effect was independently confirmed by Hall effect measurements as a function of V_{G}. The Hall resistance was linear in magnetic field, allowing for extraction of the carrier density N, which is shown in Fig. 1(f). The changes in N were consistent with accumulation (depletion) of charge carriers for positive (negative) V_{G}, respectively and the observed resistance change. The total modulation between V_{G} = −1 and +1 V was ΔN = 1.62 × 10^{13} cm^{−2}. Despite this being a fairly large amount of charge modulated, due to the very high carrier density (~3 × 10^{14} cm^{−2} per interface, corresponding to the theoretically expected ½ of an electron per interface unit cell^{21}), the fractional modulation is only 2.1%. This was, however, sufficient to investigate the changes in the nonFermi liquid parameters.
Discussion
Figure 2 shows an example of a fit of R_{xx}(T) at V_{G} = 0 V with n = 1.645. The results for all V_{G} are shown in Fig. 3. The fit range was restricted to 150–240 K, but the description was valid in the 125–300 K range. Below 125 K, R_{xx}(T) exhibited a logarithmic upturn, which is not yet understood. When R_{0}, A and n were used as free parameters, n was insensitive to V_{G} and thus the carrier density modulation, within experimental error [see error bars in Fig. 3(a)]. We note that these results are fully consistent with a much larger, systematic study of resistivity and Hall effect in ungated RTiO_{3}/SrTiO_{3}/RTiO_{3} (R = Sm, Gd) quantum wells as a function of SrTiO_{3} thickness presented elsewhere^{19}.
The nonFermi liquid exponent n ~ 5/3 has been observed in several other systems, such as electrondoped cuprates^{24} and rare earth nickelates^{11,12}. All systems are in proximity to an antiferromagnetically ordered state. Nevertheless, n is at variance with theoretical predictions of two and threedimensional antiferromagnetic quantum critical fluctuations (n = 1 and 3/2, respectively^{25}). In principle, n could be modified from a theoretical value by disorder^{26}. However, while disorder (interface roughness) is clearly reflected in R_{0}(V_{G}) (see below), it does not measurably change the nonFermi liquid exponent n. This points to an intrinsic origin of this particular value of n.
For reliable analysis of the trends in the A parameter, n was fixed to 1.645. As shown in Fig. 3(b,c), both R_{0} and A depend on V_{G} and therefore N, as expected. We discuss R_{0}(V_{G}) first. Specifically, R_{0} = m^{*}Γ_{0}/Ne^{2}, where m^{*} is the effective electron mass, Γ_{0} is the scattering rate at T = 0 (a measure of disorder level) and e the electron charge^{27}. Figure 3(d) shows R_{0}(V_{G}), A(V_{G}) and N^{−1}(V_{G}) normalized to their values at V_{G} = 0 V. R_{0}(V_{G}) varies by a factor of two more than N(V_{G}), implying that Γ_{0} also changes with V_{G}. Γ_{0} decreases upon charge carrier accumulation. This can be explained with asymmetries in interfacial roughness. Figure 4 shows a highangle annular darkfield scanning transmission electron microscopy (HAADFSTEM) image, with interfaces indicated by the dashed lines. Overlaid on the image are averaged intensities of each column of atoms, integrated parallel to the interface. HAADF intensities are sensitive to atomic numbers present in the columns and thus provide a measure of intermixing/roughness. The intensities are relatively constant within the SrTiO_{3} and SmTiO_{3} layers. Interfacial regions on either side of the quantum well, marked by arrows, show a less abrupt intensity change for the bottom interface of the quantum well (right side in Fig. 5), indicating a rougher (more intermixed) interface. Asymmetries in top/bottom interface roughness are ubiquitous in superlattices^{28,29}. In thin quantum wells, the dominant scattering contributing to R_{0} is interface roughness^{30}. For V_{G} > 0, charge carriers accumulate at the smoother top interface, thereby decreasing Γ_{0} relative to V_{G} < 0.
We next discuss the dependence of A in the carrier density N. Within the Drude model, the AT^{n} term in Eq. (1) can be written as^{31}:
where Γ(T^{n}) is the scattering rate. From Figs 3(d) and 5(a), one observes A ~ N^{−1}, in accordance with the prefactor in Eq. (2). This implies that the scattering rate Γ(T^{n}) is independent of N. This observation is at odds with the standard Fermi liquid theory of the electronelectron scattering contributions to the resistance, where the relevant energy scale is the Fermi energy, E_{F}. From dimensional analysis and n = 2^{31}:
where B is a dimensionless factor that incorporates scattering event probabilities and k_{B} is the Boltzmann constant. For a Fermi liquid, E = E_{F}^{31}. For an arbitrary exponent (1 ≤ n ≤ 2), Eq. (3) becomes:
In the case of the gated quantum well, n = 1.645 and Γ(T^{1.645}) ~ 1/E^{0.645}. Details of B depend on the mechanism(s) giving rise to a momentum change and other assumptions^{6,32,33,34}. In particular, Umklapp processes are necessary to relax the momentum^{35} and their contribution is strongly dependent on N. For instance, according to refs. 33,36, A ~ 1/N^{5/3}. The importance of Umklapp processes has been discussed recently^{8} for electronelectron scattering in SrTiO_{3} doped to very low concentrations, where Umklapp processes are thought to be negligible (i.e., with much lower concentrations than the quantum wells studied here). There is debate in the literature as to whether a normal process can also relax momentum under certain conditions^{37,38,39}. Independent of this question, however, from Luttinger’s theorem, Eqs. (3, 4) should result in a pronounced carrier density dependence of Γ(T^{n}) if E = E_{F}, contrary to what is observed. The insensitivity of Γ(T^{n}) to a change in N upon gating therefore suggests that E is not E_{F}, but some other, N independent, energy scale.
To check if the scattering rate and the relevant energy scale E remain insensitive to N when it is varied over many orders of magnitude, we consider uniformly doped SrTiO_{3}. Figure 5(b) shows A as a function of N (which is the 3D carrier density in this case) for different samples from the literature in the regime where R_{xx}(T) ~ AT^{2}. Again, we observe A ~ N^{−1}, implying that Γ(T^{2}) is independent of N, at odds with Fermi liquid theory. The fact that this relationship is maintained over such a wide range of N appears to also rule out any accidental cancellations by any carrier dependencies in B. We note weak steps in A, possibly associated with additional bands being filled^{8}, which was also observed in the temperature coefficient of the mobility^{7} and could be due to an associated change in the density of states and/or multiband effects, both of which increase A. Nevertheless, the A ~ N^{−1} relationship is maintained in each regime, even though the temperature exponent would suggest a classic Fermi liquid, i.e. n = 2.
In systems with Tlinear behavior, such as the cuprates, E in Eq. (4) becomes irrelevant as can be seen by setting n = 1. Furthermore, B ~ 1^{40,41}. This has sometimes been attributed to an underlying quantum critical point that causes the relaxation time to become independent of the microscopic processes so that the temperature is the only energy scale in the system^{42}. The results here suggest something far more general, namely, a breakdown of the Fermi liquid state in certain materials, such as doped SrTiO_{3}, far from any (at least magnetic) quantum critical point. In these materials the energy scale in the scattering rate, which is not E_{F}, becomes increasingly irrelevant as a system transitions from an n = 2 regime to n < 2, as can be seen from Eq. (4).
It is remarkable that the “anomalous Fermi liquid” behavior is observed despite the fact that the materials studied here have welldefined Fermi surfaces^{43,44,45} at low temperature and show features consistent with filling of successive bands with increasing doping^{43,45}. It is instructive to plot Γ, defined through Eq. (2), taking m* = m_{e}, the free electron mass, as shown in Fig. 6. We see that it is always much smaller than the bandwidth (several eV). Furthermore, ħΓ » k_{B}T at high temperature, which is clearly outside Fermi liquid theory. Moreover, if we estimate an order of magnitude for the energy scale by combining Eqs. 2 and 3, E = (k_{B}T)^{2}/ħΓ, we find E ~ 10 meV, which is smaller than E_{F}, at least at high carrier densities.
These observations compound the difficulty of conventional Fermi liquid theory in addressing the T^{2} resistivity and suggest that some distinctly stronger scattering process must be involved. Future studies should address this mechanism and the nature of the energy scale E in Eqs. (3, 4) in strongly correlated materials, such as the cuprates, nickelates and titanates, all of which show a wide T^{2} regime (as well as deviations from it). We note that in several materials with n < 2, which include the cuprates and the SrTiO_{3} quantum wells, the Hall scattering rate maintains a T^{2} dependence^{18,19,46}. It has been pointed out that the nature of the energy scale (E) for the Hall scattering rate in such systems is unclear^{17}, but, interestingly, it has been found to have a weak doping dependence, i.e., similar to what we observe for R_{xx}^{9,17}. A doping independent scattering rate has also been recently reported in the cuprates^{10} and several other perovskites^{15}. Finally, we note that the continuity of the A ~ 1/N behavior across four orders of magnitude of N is inconsistent with a sudden, sharp increase in Umklapp scattering around N ~ 2 × 10^{20} cm^{−3}, as predicted in refs 20 and 8.
In summary, the broad implication of this study is that some materials with a robust T^{2} resistivity are not conventional Fermi liquids and their transport is likely just as anomalous as that of those that are traditionally identified as nonFermi liquids (when 1 ≤ n ≤ 2). This leaves a formidable challenge to theory, namely a unified understanding of the entire T^{n} resistivity regime, with a carrier density independent energy scale in the scattering rate and allowing for intermediate exponents between the T^{2} and the Tlinear limits.
Methods
The SmTiO_{3}(10 nm)/SrTiO_{3}(2 SrO layers thick)/SmTiO_{3}(1.2 nm)/SrZrO_{3}(6 nm) stack was grown on a (001)LSAT substrate by hybrid molecular beam epitaxy, as described in detail elsewhere^{47,48,49}. The devices were processed using standard contact photolithography techniques. The Pt (100 nm) gate contact was deposited by electron beam evaporation. The mesa was defined by dry etching. Prior to the deposition of the Ti(400 nm)/Au(3000 nm) Ohmic contacts, SrZrO_{3} was selectively removed with a wet etch in buffered HF diluted 1:10 in water. Electrical measurements from 2 to 300 K were performed in a Quantum Design Physical Property Measurement System (PPMS). Cross section samples for HAADFSTEM imaging were prepared by focused ion beam thinning and imaged using a FEI Titan S/TEM operated at 300 kV, with a convergence angle of 9.6 mrad. Atomic column positions were obtained by fitting each column to a twodimensional Gaussian. Integrated intensities were obtained by averaging a circular region with radius ¼ of the unit cell around each atomic column^{50}.
Additional Information
How to cite this article: Mikheev, E. et al. Carrier density independent scattering rate in SrTiO_{3}based electron liquids. Sci. Rep. 6, 20865; doi: 10.1038/srep20865 (2016).
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Acknowledgements
This work was supported in part by FAME, one of six centers of STARnet, a Semiconductor Research Corporation program sponsored by MARCO and DARPA and by the U.S. Army Research Office (grant no. W911NF1410379). L.B. and S.S. thank the MRSEC Program of the National Science Foundation under Award No. DMR1121053, which supported the film growth experiments and the central facilities of the UCSB MRL used in this work. The microscopy studies (J. Y. Z.) were supported by the DOE (DEFG0202ER45994). J.Y.Z. also received support from the Department of Defense through an NDSEG fellowship.
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S.R., P.B.M. and A.P.K. performed the film growth. E. M. processed the devices and carried out the electrical measurements and data analysis. J.Y.Z. performed the STEM analysis. E.M., L.B. and S.S. wrote the manuscript.
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Mikheev, E., Raghavan, S., Zhang, J. et al. Carrier density independent scattering rate in SrTiO_{3}based electron liquids. Sci Rep 6, 20865 (2016). https://doi.org/10.1038/srep20865
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