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Evidence for hydrodynamic electron flow in PdCoO2

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Science  04 Mar 2016:
Vol. 351, Issue 6277, pp. 1061-1064
DOI: 10.1126/science.aac8385

Electrons that flow like a fluid

Electrons inside a conductor are often described as flowing in response to an electric field. This flow rarely resembles anything like the familiar flow of water through a pipe, but three groups describe counterexamples (see the Perspective by Zaanen). Moll et al. found that the viscosity of the electron fluid in thin wires of PdCoO2 had a major effect on the flow, much like what happens in regular fluids. Bandurin et al. found evidence in graphene of electron whirlpools similar to those formed by viscous fluid flowing through a small opening. Finally, Crossno et al. observed a huge increase of thermal transport in graphene, a signature of so-called Dirac fluids.

Science, this issue p. 1061, 1055, 1058; see also p. 1026

Abstract

Electron transport is conventionally determined by the momentum-relaxing scattering of electrons by the host solid and its excitations. Hydrodynamic fluid flow through channels, in contrast, is determined partly by the viscosity of the fluid, which is governed by momentum-conserving internal collisions. A long-standing question in the physics of solids has been whether the viscosity of the electron fluid plays an observable role in determining the resistance. We report experimental evidence that the resistance of restricted channels of the ultrapure two-dimensional metal palladium cobaltate (PdCoO2) has a large viscous contribution. Comparison with theory allows an estimate of the electronic viscosity in the range between 6 × 10–3 kg m–1 s–1 and 3 × 10–4 kg m–1 s–1, versus 1 × 10–3 kg m–1 s–1 for water at room temperature.

In a quantum fluid without an associated lattice, such as 3He, the momentum of the fluid is conserved except where it interacts with the walls of a channel through which it is flowing. As the temperature decreases and the quasiparticle-quasiparticle mean free path within the fluid increases (because of the decrease of its quasiparticle scattering rate), interactions with the walls become more probable and the viscosity and flow resistance increase. This is intuitively at odds with the behavior seen for electrons moving in a crystalline lattice, whose flow resistance decreases as increases. The resolution of this apparent paradox is that coupling to the lattice and its excitations means that the large majority of collisions in the electron fluid (electron-impurity, normal electron-phonon, Umklapp electron-electron, and Umklapp electron-phonon) relax momentum, taking the fluid far from the hydrodynamic limit. At least some of these momentum-relaxing collisions are inevitable in any real material. Strictly speaking, momentum of the electron fluid can never be conserved, even in a bulk sample for which boundary scattering is negligible. However, this does not mean that the electronic viscosity plays no role in determining electrical resistance. A pragmatic benchmark is whether momentum-conserving processes are faster or slower than momentum-relaxing ones. If the electron fluid’s momentum is relaxed slowly, it can be thought of as being quasi-conserved, and hydrodynamic signatures might be observable (19).

The search for hydrodynamic effects in electrons in solids has been given extra impetus by the introduction of the “holographic correspondence” to condensed matter physics (10). This technique introduced the concept of a minimum viscosity, argued to be applicable to strongly interacting fluids as diverse as the quark-gluon plasma and cold atomic gases (11). Hydrodynamic effects have also been postulated to be at the root of the temperature-linear resistivity of the high-temperature superconductors (6, 7), but because momentum-relaxing scattering is strong in those materials, it is difficult to perform an analysis of the experimental data that unambiguously separates the two effects. In a pioneering experiment, unusual current-voltage relationships in a semiconductor wire were convincingly ascribed to hydrodynamic effects (3), but that avenue of research has not been widely pursued, even though the large difference between transport and electron-electron scattering rates in semiconductors was later demonstrated by direct nonequilibrium measurements (12).

To investigate whether a hydrodynamic contribution to electrical transport can be clearly separated from the more standard contributions from momentum-relaxing processes, we sought a material in which momentum-relaxing scattering is anomalously suppressed. The material that we chose was PdCoO2, a layered metal with a number of unusual properties (1321). Its electronic structure is deceptively simple, with one highly dispersive band, dominantly of Pd 4d/5s character, crossing the Fermi level (2226). Its Fermi volume corresponds to one electron per formula unit to high accuracy (18), and the ratio of in-plane to out-of-plane resistivity is approximately 103, justifying the use of a two-dimensional approximation in treating the in-plane properties.

The electrical conductivity of PdCoO2 is remarkable. At room temperature, its resistivity is just 2.6 × 10–6 ohm·cm, 30% lower per carrier than that of elemental copper. Below 15 K, the resistivity is essentially independent of temperature; in the best single crystals, the resistivity is below 1 × 10–8 ohm·cm (18). This striking behavior might be attributable to phonon drag, in which the phonons follow the electrons into an out-of-equilibrium distribution when an electric field is applied. In PdCoO2, the activation temperature for Umklapp electron-phonon processes is at least 160 K, unusually high for a metal (18).

These properties make PdCoO2 a good candidate for a search for hydrodynamic effects. Below 15 K, momentum-relaxing processes are far slower than those observed in most metals. Better still, if phonon drag is indeed taking place, the normal electron-phonon processes that usually impede electrons in solids from approaching the hydrodynamic limit are now helping that process instead, because they contribute a source of momentum-conserving scattering. An important independent hint that the balance between momentum-conserving and momentum-relaxing scattering is both unusual and favorable for the observation of hydrodynamic effects comes from the ratio of the mean free paths deduced, respectively, from the resistivity, which is sensitive to processes that efficiently relax momentum, and from analysis of the de Haas–van Alphen (dHvA) effect, whose amplitude is sensitive to a wider range of scattering processes. The inverse of that ratio is 5 to 10%, an unusually small value approximately an order of magnitude lower than that seen in ordinary metals (18).

In a purely hydrodynamic fluid, flow resistance in channels is determined entirely by momentum-relaxing boundary scattering, the efficiency of which is determined by the fluid’s viscosity. Boundary scattering also contributes to the resistance in thin wires or channels of metals in which hydrodynamic effects play no observable role, but the standard theory of this phenomenon is well established (27). We therefore set out to construct a series of PdCoO2 “wires,” study how their resistance varied with the channel width, and investigate whether there were observable deviations from the predictions of standard theory. The wires were produced from flux-grown single crystals by focused ion beam (FIB) etching (Fig. 1A). Six such devices were made and shown to have consistent properties (28). We discuss the nature of the boundaries produced by FIB processing in (28) and show that damage is restricted to approximately 20 nm from the edge of the channel. Taking all uncertainties into account, the undamaged channel width can be determined to an accuracy of ±80 nm. For the main experiment, measurements were done on a single successively etched wire from the same crystal, to remove as many experimental uncertainties as possible. A second crystal (Fig. 1B) was etched into a meander channel ideally suited for a measurement of the Shubnikhov–de Haas (SdH) effect. For each experiment, we studied the magnetoresistance of the wire in magnetic fields B ranging from –14 T to +14 T. We also fabricated a multicontact device to verify that our data are length-independent at constant width (28).

Fig. 1 FIB-prepared devices of PdCoO2 crystals.

(A) The crystal used for our channel-thinning experiment, after the first two processing steps. An initial channel 120 μm wide has been reprocessed to produce a narrower conducting channel of width 60 μm. The same channel was subsequently reprocessed seven further times, narrowing it by approximately half each time, until it was 0.7 μm wide. (B) A meander channel processed in a second crystal for use in a search for the Shubnikhov–de Haas effect. Its approximate width is 6 μm in the narrowed parts; thickness is 17 μm. Its resistance in a magnetic field of 14 T at a temperature of 2 K is 5.7 × 10–3 ohms.

Data obtained in the channel-narrowing experiment are shown in Fig. 2A for widths ranging from 60 μm to 0.7 μm at a measurement temperature of 2 K. Data from the meander channel are shown in Fig. 2B. Consistent with previous measurements on other single crystals (18), the resistivity in zero field for the 60-μm sample is 9 × 10–9 ohm·cm. The value of the momentum-relaxing mean free path MR is a crucial parameter in the analysis of the resistance of restricted channels; the single-band electronic structure and well-known Fermi surface shape and volume (16, 18) allow an accurate calculation of MR = 18.5 + 1.5 μm (28). The wire widths W used for the experiment therefore cover the range 0.3 ≤ MR/W≤ 26, enabling study of the crossover between a nearly bulk regime and one in which a sample dimension falls far below the bulk mean free path.

Fig. 2 Effect of channel width W on magnetotransport.

(A) Magnetoresistance data from the sample shown in Fig. 1A taken at 2 K after successive channel-thinning steps. (B) Magnetoresistance data from the sample shown in Fig. 1B taken at 2 K. (C) Bmax [indicated by arrows in (A)] varies as α(ħkF/We) as the device shown in Fig. 1A is successively thinned. Both the functional form and prefactor are in good agreement with theory and with previous measurements on semiconductor wires.

We adopt the usual solid-state physics convention of describing the transport properties of our channels in terms of the resistivity ρ. For a channel of width W, length L, thickness T, and resistance R, ρ ≡ RTW/L. Conceptually, ρ is a bulk property of the material, so in the absence of boundary effects it should be independent of W. In contrast, at low fields, the overall channel resistivity ρ increases by more than an order of magnitude as the wire width is decreased (Fig. 2). Because this involves repeated exposure to ion beam etching, it is natural to wonder whether this trend is caused by beam damage increasing the scattering in the bulk of the wire. However, extending the data to higher fields proves that this is not the case. First, we note that at high fields, the resistivity is similar at all widths above 0.7 μm. Second, the pronounced maximum seen at fields Bmax in the magnetoresistance for W ≤ 30 μm is a well-known phenomenon from the study of narrow conducting channels for which the bulk mean free path is on the order of the channel width or larger (2931). For each channel width, the rise in the magnetoresistivity at low fields is stopped when the cyclotron orbit radius falls to less than the channel width, because the helical pitch of the motion of the drifting electrons becomes so tight that boundary scattering is suppressed, and eventually the bulk, width-independent resistivity is recovered at high fields. Specifically, Bmax = α(ħkF/We) (where ħ is Planck’s constant divided by 2π, kF is the Fermi wave vector, and e is the electronic charge) with constant α = 0.55 has been reported for restricted channels of two-dimensional electron gas (30), and a similar functional form with α = 0.9 was observed in recent work on graphene (31). The data shown in Fig. 2C therefore provide further evidence that the overall scale of the resistivity is increasing because of additional boundary scattering rather than additional bulk scattering. Very low field peaks in the magnetoresistance can still be seen in the 30-μm channel, providing further direct geometrical evidence that MR is long. Further evidence that extremely high bulk purity is retained after ion beam etching comes from the data from the meander sample. Its Bmax is consistent with the measured width of 6 μm (Fig. 2, B and C), and at high fields, clear SdH oscillations are seen, with frequencies in agreement with those seen in dHvA oscillations from bulk crystals (18). High frequencies are particularly hard to see by the SdH effect; the 30-kT frequencies shown in Fig. 3B are among the highest ever reported in SdH measurements.

Fig. 3 Shubnikhov–de Haas oscillations.

(A and B) SdH oscillations from the patterned meander track shown in Fig. 1B. (C) The frequencies of the SdH oscillations extracted by Fourier analysis of the data in (A).

The data presented in Figs. 2 and 3 provide strong evidence that boundary rather than bulk scattering dominates the rise in resistivity seen as we reduce the channel width. As discussed above, boundary scattering is expected even in the absence of hydrodynamic effects as the channel width falls to less than the mean free path and the system enters the ballistic transport regime. The relevant theory (4, 27) can be expressed in a useful dimensionless form, shown as a blue line in Fig. 4. If ρ is normalized to the bulk resistivity ρ0 of an infinitely wide sample and plotted against MR/W, the prediction has no free parameters. At MR/W = 25, ρ is calculated to be 10.3 ρ0. Our measured value for ρ/ρ0 is more than 50% larger than this prediction, and our data also show a functional form that is at odds with the simple theory.

Fig. 4 Hydrodynamic effect on transport.

(A and B) The measured resistivity of PdCoO2 channels normalized to that of the widest channel (ρ0), plotted against the inverse channel width 1/W multiplied by the bulk momentum-relaxing mean free path MR (solid black circles). Blue solid line: Prediction of a standard Boltzmann theory including boundary scattering but neglecting momentum-conserving collisions. Red line: Prediction of a model that includes the effects of momentum-conserving scattering (see text). Error bars stem from uncertainties of sample dimension (28). (C) Predictions of the hydrodynamic theory over a wide range of parameter space.

To examine whether the large deviations of the data from the predictions of standard transport theory are linked to electronic hydrodynamics, we carefully studied the predictions of a more sophisticated theory that takes momentum-conserving scattering into account (4). Originally formulated to analyze the current-dependent hydrodynamic signatures reported in (3), the theory encodes momentum-relaxing scattering via the role of impurities and momentum-conserving scattering via normal electron-electron scattering processes. In PdCoO2, the scale of momentum-conserving electron-electron scattering is uncertain because of Fermi surface faceting (26), and phonons dragged out of equilibrium are also likely to be making a contribution to the momentum-conserving processes. In this sense, even this more sophisticated theory is oversimplified, and it would be dangerous to use it to predict temperature-dependent transport in PdCoO2. However, its results at any fixed temperature depend only on the ratio of a momentum-conserving mean free path MC to the momentum-relaxing mean free path MR, and the microscopic origin of the scattering that produced that ratio is not relevant. The theory is therefore a useful guide to the consequences of including hydrodynamic effects in experiments performed at constant temperature, such as the one shown in Fig. 4, A and B. For any value of MCMR, it predicts a unique functional form and overall magnitude for ρ/ρ0 versus MR/W, with no free fitting parameters.

As can be seen from the red lines in Fig. 4, A and B, the hydrodynamic theory for MCMR = 0.1 produces an excellent match to our data. In Fig. 4C we show how the predictions of the theory change as a function of MCMR, marking the blue and red lines of Fig. 4, A and B, on the contour plot for reference. For large MCMR (i.e., weak momentum-conserving scattering), it limits rapidly to the prediction of the standard theory. At its other limit of strong momentum-conserving scattering (black line), the prediction for ρ/ρ0 is approximately quadratic as a function of MR/W. In this regime, viscous effects dominate, and the W–2 dependence of ρ is equivalent to the W–3 prediction for flow resistance that is obtained from the purely hydrodynamic Navier-Stokes equation (28).

Our value for MCMR (red line in Fig. 4C) sits between these limiting cases. Because viscosity is inversely proportional to MC, the initial ~W–2 rise is steeper than for MCMR = 0.01. As MR/W rises toward 10, MC is no longer much less than W, and the Navier-Stokes prediction evolves smoothly to a solution in which viscous effects are important but the channel constriction is such that we leave the purely hydrodynamic regime. Physically the system is in a hybrid situation in which traditional impurity and boundary scattering mix with viscosity-stimulated boundary scattering to produce the overall evolution of resistivity with channel width. Fig. 4C also shows that the hydrodynamic prediction is insensitive to the precise choice of MCMR. Our choice of 0.1 was not the result of fitting, but simply an estimate motivated by the ratio of scattering rates deduced from measurements of resistivity and the dHvA effect. Choices that were larger or smaller by a factor of 2 would give a similar level of agreement with the data.

The data and predictions shown in Fig. 4 and discussed in (28) provide strong evidence that we have observed a substantial hydrodynamic contribution to electrical transport in a bulk material. Further analysis of the theory in its Navier-Stokes limit allows a quantitative estimate of the electronic viscosity itself as a function of the hydrodynamic contribution to ρ/ρ0. As a result, we are able to estimate the dynamic viscosity η of the electronic fluid in PdCoO2 as lying in the range between 6 × 10–3 kg m–1 s–1 and 3 × 10–4 kg m–1 s–1. For comparison, the η values of two well-known fluids, water at room temperature and liquid nitrogen at 75 K, are 1 × 10–3 kg m–1 s–1 and 1 × 10–4 kg m–1 s–1, respectively.

Although the bounds we can place on the electronic viscosity of PdCoO2 are not very precise, we stress that this is an issue of theory, not experiment. We believe that there is such a large hydrodynamic contribution to our data that they contain all the information required to estimate the viscosity precisely, and we hope that our experiment motivates further work on this issue. As discussed above, PdCoO2 is a weakly scattering system, so it is not surprising that a comparison of our estimate of η with measurements of the entropy density s (15, 18) yields η/s ≈ 106 ħ/kB, far from a proposed minimum viscosity limit (11). It will also be interesting to reexamine a possible role of hydrodynamic effects in explaining the resistivity in systems in which the momentum-conserving scattering is extremely strong (6, 7, 32). In principle, a range of viscosities is to be expected in different electronic fluids, and even the attainment of turbulent electronic flow may be possible.

Supplementary Materials

www.sciencemag.org/content/351/6277/1061/suppl/DC1

Materials and Methods

Figs. S1 to S6

References (3339)

References and Notes

  1. See supplementary materials on Science Online.
  2. Acknowledgments: We thank J. C. Davis, C. Geibel, A. G. Green, S. A. Hartnoll, C. W. Hicks, S. A. Kivelson, Y. Maeno, T. Oka, J. W. Orenstein, and S. H. Simon for stimulating discussions and the UK Engineering and Physical Sciences Research Council for financial support. The FIB work was supported by the SCOPE-M center for electron microscopy at ETH Zürich. We thank P. Gasser, J. Reuteler, and B. Batlogg for FIB support. The data and/or materials supporting this publication can be accessed at http://dx.doi.org/10.17630/cdc6c488-2ea4-440e-a68d-e2a426709171.
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