Observation of the Dirac fluid and the breakdown of the Wiedemann-Franz law in graphene

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Science  04 Mar 2016:
Vol. 351, Issue 6277, pp. 1058-1061
DOI: 10.1126/science.aad0343

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


Interactions between particles in quantum many-body systems can lead to collective behavior described by hydrodynamics. One such system is the electron-hole plasma in graphene near the charge-neutrality point, which can form a strongly coupled Dirac fluid. This charge-neutral plasma of quasi-relativistic fermions is expected to exhibit a substantial enhancement of the thermal conductivity, thanks to decoupling of charge and heat currents within hydrodynamics. Employing high-sensitivity Johnson noise thermometry, we report an order of magnitude increase in the thermal conductivity and the breakdown of the Wiedemann-Franz law in the thermally populated charge-neutral plasma in graphene. This result is a signature of the Dirac fluid and constitutes direct evidence of collective motion in a quantum electronic fluid.

Understanding the dynamics of many interacting particles is a formidable task in physics. For electronic transport in matter, strong interactions can lead to a breakdown of the Fermi liquid (FL) paradigm of coherent quasi-particles scattering off of impurities. In such situations, provided that certain conditions are met, the complex microscopic dynamics can be coarse-grained to a hydrodynamic description of momentum, energy, and charge transport on long length and time scales (1). Hydrodynamics has been successfully applied to a diverse array of interacting quantum systems, from high-mobility electrons in conductors (2) to cold atoms (3) and quark-gluon plasmas (4). Hydrodynamic effects are expected to greatly modify transport coefficients compared with their FL counterparts, as has been argued for strongly interacting massless Dirac fermions in graphene at the charge-neutrality point (CNP) (58).

Many-body physics in graphene is interesting because of electron-hole symmetry and a linear dispersion relation at the CNP (9, 10). Together with the vanishing Fermi surface, the ultrarelativistic spectrum leads to ineffective screening (11) and the formation of a strongly interacting quasi-relativistic electron-hole plasma known as a Dirac fluid (DF) (12). The DF shares many features with quantum critical systems (13): most importantly, the electron-electron scattering time is fast (1417) and well suited to a hydrodynamic description. Because of the quasi-relativistic nature of the DF, this hydrodynamic limit is described by equations (18) quite different from those applicable to its nonrelativistic counterparts. A number of unusual properties have been predicted, including nearly perfect (inviscid) flow (19) and a diverging thermal conductivity, which results in the breakdown of the Wiedemann-Franz (WF) law at finite temperature (5, 6).

Away from the CNP, graphene has a sharp Fermi surface, and the standard FL phenomenology holds. By tuning the chemical potential, we are able to measure thermal and electrical conductivity in both the DF and the FL in the same sample. In a FL, the relaxation of heat and charge currents is closely related, as they are carried by the same quasi-particles. The WF law (20) states that the electronic contribution to a metal’s thermal conductivity κe is proportional to its electrical conductivity σ and temperature T, such that the Lorenz ratio Embedded Image satisfiesEmbedded Image(1)where e is the electron charge, kB is the Boltzmann constant, and Embedded Image is the Sommerfeld value derived from FL theory. Embedded Image depends only on fundamental constants, not specific details of the system such as carrier density or effective mass. As a robust prediction of FL theory, the WF law has been verified in numerous metals (20). At high temperatures, the WF law can be violated due to inelastic electron-phonon scattering or bipolar diffusion in semiconductors, even when electron-electron interactions are negligible (21). In recent years, several nontrivial violations of the WF law—all of which are related to the emergence of non-FL behavior—have been reported in strongly interacting systems such as Luttinger liquids (22), metallic ferromagnets (23), heavy fermion metals (24), and underdoped cuprates (25).

Owing to the strong Coulomb interactions between thermally excited charge carriers, the WF law is expected to be violated at the CNP in a DF. An electric field drives electrons and holes in opposite directions; collisions between them introduce a frictional dissipation, resulting in a finite conductivity even in the absence of disorder (26). In contrast, a temperature gradient causes electrons and holes to move in the same direction, thus inducing an energy current that grows unimpeded by interparticle collisions as the total momentum is conserved. The thermal conductivity is therefore limited by the rate at which momentum is relaxed by residual impurities.

Realization of the DF in graphene requires that the thermal energy be larger than the local chemical potential μ(r), defined at position r: Embedded Image. Impurities cause spatial variations in the local chemical potential, and even when the sample is globally neutral, it is locally doped to form electron-hole puddles with finite μ(r) (2730). At high temperatures, formation of the DF is complicated by phonon scattering, which can relax momentum by creating additional inelastic scattering channels. This high-temperature limit occurs when the electron-phonon scattering rate becomes comparable to the electron-electron scattering rate. These two temperatures set the experimental window in which the DF and the breakdown of the WF law can be observed.

To minimize disorder, we used monolayer graphene samples encapsulated in hexagonal boron nitride (31). All devices used in this study have two terminals, so as to keep a well-defined temperature profile (32), with contacts fabricated using the one-dimensional edge technique (33) to minimize contact resistance. We employed a back-gate voltage Embedded Image applied to the silicon substrate to tune the charge carrier density n = nenh, where ne and nh are the electron and hole densities, respectively (21). All measurements were performed in a cryostat to control the bath temperature Tbath. Figure 1A shows the resistance R versus Vg measured at various fixed temperatures for a representative device [see (21) for all samples]. From this, we used the known sample dimensions to estimate the electrical conductivity σ (Fig. 1B). At the CNP, the residual charge carrier density nmin can be estimated by extrapolating a linear fit of log(σ) as a function of log(n) out to the minimum conductivity (34). At the lowest temperatures, nmin saturates to ~8 × 109 cm–2. Extraction of nmin by this method prompts overestimation of the charge-puddle energy, consistent with previous reports (31). Above the disorder temperature scale Tdis ~ 40 K, nmin increases as Tbath is raised, which suggests that thermal excitations begin to dominate and the sample enters the nondegenerate regime near the CNP.

Fig. 1 Temperature- and density-dependent electrical and thermal conductivity.

(A) Resistance (R) versus gate voltage (Vg) at various temperatures. kΩ, kilohm. (B) Electrical conductivity (blue) as a function of the charge density set by the back gate for different bath temperatures. The residual carrier density at the neutrality point (green) is estimated by the intersection of the minimum conductivity with a linear fit to log(σ), away from neutrality (dashed gray lines). Curves have been offset vertically such that the minimum density (green) aligns with the temperature axis to the right. Solid black lines correspond to 4e2/h. At low temperatures, the minimum density is limited by disorder (charge puddles). Above Tdis ~ 40 K (yellow shaded area), thermal excitations begin to dominate, and the sample enters the nondegenerate regime near the neutrality point. (C to E) Thermal conductivity (red points) as a function of (C) gate voltage and [(D) and (E)] bath temperature, compared to with the WF law, Embedded Image (blue lines). At low temperature and/or high doping Embedded Image, we find the WF law to hold. This is a nontrivial check on the quality of our measurement. In the nondegenerate regime Embedded Image, the thermal conductivity is enhanced and the WF law is violated. Above T ~ 100 K, electron-phonon coupling becomes appreciable and begins to dominate thermal transport at all measured gate voltages. At this temperature, the yellow shaded background ends. All data from this figure are taken from sample S2 [inset in (E)].

Electronic thermal conductivity was measured using high-sensitivity Johnson noise thermometry (JNT) (32, 35). We applied a small bias current through the sample, thus injecting a Joule heating power P directly into the electronic system and inducing a small difference between the temperature of the graphene electrons and that of the bath: Embedded Image. The electron temperature Te was monitored independently of the lattice temperature through the Johnson noise power emitted at 100 MHz, with a 20-MHz bandwidth defined by an inductor-capacitor matching network. We designed our JNT setup to be operated over a wide temperature range, from 3 to 300 K (35). With a precision of ~10 mK, we measured small deviations of Te from Tbath (i.e., Embedded Image). In this limit, the temperature of the graphene lattice is well thermalized to the bath (32), and our JNT setup allows us to sensitively measure the electronic cooling pathways in graphene. When the temperature is low enough, electron and lattice interactions are weak (35, 36), and most of the Joule heat generated in graphene escapes via direct diffusion to the contacts (21). As the temperature increases, electron-phonon scattering becomes appreciable, and thermal transport becomes limited by the electron-phonon coupling strength (3638). The onset temperature of appreciable electron-phonon scattering, Tel-ph, depends on the sample disorder and device geometry: Tel-ph ~ 80 K (35, 36, 39, 40) for our samples. Below this temperature, the electronic contribution of the thermal conductivity can be obtained from P and ΔT using the device dimensions (21).

Figure 1C shows κe(Vg) plotted alongside the simultaneously measured σ(Vg) at various fixed bath temperatures. Here, for a direct quantitative comparison based on the WF law, we plot the scaled electrical conductivity as Embedded Image in the same units as κe. In a FL, these two values will coincide, in accordance with Eq. 1. At low temperatures (T < Tdis ~ 40 K), where the puddle-induced density fluctuations dominate, we find that Embedded Image, monotonically increasing as a function of carrier density with a minimum at the neutrality point, confirming the WF law in the disordered regime. As T increases (T > Tdis), however, we begin to observe violation of the WF law. This violation appears only close to the CNP, with the measured thermal conductivity maximized at n = 0 (Fig. 1C). The deviation is the largest at 75 K, where κe is more than an order of magnitude larger than the value expected for a FL. The nonmonotonicity of κe(T) is consistent with acoustic phonons relaxing momentum more efficiently than impurities as T increases (41). For Embedded Image in our samples, activation of optical phonons introduces an additional electron-phonon cooling pathway (35), and the measured thermal conductivity is larger than κe. This non-FL behavior quickly disappears as Embedded Image increases; κe returns to the FL value and restores the WF law. In fact, away from the CNP, the WF law holds for a wide temperature range, consistent with previous reports (35, 36, 39) (Fig. 1E). For this FL regime, we verify the WF law up to T ~ 80 K.

Our observation of the breakdown of the WF law in graphene is consistent with the emergence of the DF. Figure 2 shows the full density and temperature dependence of the experimentally measured Lorenz ratio, highlighting the presence of the DF. The blue region denotes Embedded Image, suggesting that the carriers in graphene exhibit FL behavior. The WF law is violated in the DF (yellow-red region), with a peak Lorenz ratio 22 times larger than Embedded Image. The green dotted lines show the corresponding nmin(T) for this sample; the DF is found within this regime, indicating the coexistence of thermally populated electrons and holes. Disorder and electron-phonon scattering serve as lower and upper limits, respectively, on the temperature range over which the DF can be observed.

Fig. 2 Breakdown of the WF law in the DF regime.

The Lorenz ratio is shown as a function of the charge carrier density and bath temperature. Near the CNP and for temperatures above the disorder (charge puddle) regime but below the onset of electron-phonon coupling, the Lorenz ratio is measured to be an order of magnitude greater than the FL value of 1 (blue). The WF law is observed to hold outside of the DF regime. All data from this figure are taken from sample S1. Green dotted lines show the corresponding nmin(T) for this sample.

We investigate the effect of impurities on hydrodynamic transport by comparing the results obtained from samples with varying disorder. Figure 3A shows nmin as a function of temperature for three samples used in this study. nmin(T = 0) is estimated as 5 × 109, 8 × 109, and 10 × 109 cm–2, respectively, in samples S1, S2, and S3. All devices show qualitatively similar DF behavior: For samples S1, S2, and S3, the largest values of Embedded Image measured in the DF regime are 22, 12, and 3, respectively (Fig. 3B). Cleaner samples not only have a more pronounced peak but also have a narrower density dependence (Fig. 3C), as predicted (5, 6).

Fig. 3 Disorder in the DF.

(A) Minimum carrier density as a function of temperature for all three samples. At low temperatures, each sample is limited by disorder. At high temperatures, all samples become limited by thermal excitations. Dashed lines are a guide for the eye. (B) Lorentz ratio of all three samples as a function of bath temperature. The largest WF violation is seen in the cleanest sample. (C) The gate dependence of the Lorentz ratio is well fit to the hydrodynamic theory of (5, 6). Fits of all three samples are shown at 60 K. All samples return to the FL value (black dashed line) at high density. The inset graph shows the fitted enthalpy density as a function of temperature and the theoretical value in clean graphene (black dashed line). The schematic inset illustrates the difference between heat and charge current in the neutral Dirac plasma.

More quantitative analysis of Embedded Image in our experiment can be performed by employing a quasi-relativistic hydrodynamic theory of the DF, incorporating the effects of weak impurity scattering (5, 6, 18)Embedded Image(2)whereEmbedded Image(3) and Embedded Image(4)Here, vF is the Fermi velocity in graphene, σmin is the electrical conductivity at the CNP, Embedded Image is the fluid enthalpy density, lm is the momentum relaxation length from impurities, and n0 is the density scale over which ℒ varies appreciably. Two parameters in Eq. 2, lm and Embedded Image, are undetermined for any given sample. For simplicity, we assume that we are well within the DF limit, where lm and Embedded Image are approximately independent of n. We fit Eq. 2 to the experimentally measured Embedded Image for all temperatures and densities in the DF regime to obtain lm and Embedded Image for each sample (Fig. 3C). lm is estimated to be 1.5, 0.6, and 0.034 μm for samples S1, S2, and S3, respectively. For the system to be well described by hydrodynamics, lm should be long compared with the electron-electron scattering length of ~0.1 μm that is expected for the DF at 60 K (19). This is consistent with the pronounced signatures of hydrodynamics in S1 and S2, but not in S3, where only a glimpse of the DF appears in this more disordered sample. We also observe in S1 that Embedded Image dips substantially below Embedded Image: its minimum is Embedded Image. Embedded Image occurs in Eq. 2 for Embedded Image. The inset to Fig. 3C shows the fitted enthalpy density as a function of temperature compared with that expected in clean graphene (dashed line) (19), excluding renormalization of the Fermi velocity. In the cleanest sample, Embedded Image varies from 1.1 to 2.3 eV/μm2 in the hydrodynamic regime. This enthalpy density corresponds to ~20 meV or ~4kBT per charge carrier—about a factor of 2 larger than the model calculation without disorder (19). The sharp temperature and impurity dependence observed in Embedded Image is a prediction of these hydrodynamic models. These effects and the magnitude of Embedded Image are inconsistent with alternative models for WF violations, including bipolar diffusion in graphene (21, 42). Furthermore, recent experiments report monotonic behavior in thermopower as a function of T (43), implying that phonon drag is not responsible for the peak in κe that we observe as a function of T.

To fully incorporate the effects of disorder, a hydrodynamic theory that treats inhomogeneity nonperturbatively is necessary (41, 44). The enthalpy densities reported here are larger than the theoretical estimation obtained for disorder-free graphene, consistent with the picture that chemical potential fluctuations prevent the sample from reaching the Dirac point. Although we find thermal conductivity well described by (5, 6), electrical conductivity increases more slowly than expected away from the CNP, a result consistent with hydrodynamic transport in a viscous fluid with charge puddles (41).

In a hydrodynamic system, the ratio of shear viscosity η to entropy density s is an indicator of the strength of the interactions between constituent particles. It has been suggested that the DF can behave as a nearly perfect fluid (19): η/s approaches Kovtun et al.’s conjecture that Embedded Image (where ħ is Planck’s constant divided by 2π) for a strongly interacting system (45). A nonperturbative hydrodynamic framework can be employed to estimate η (41). A direct measurement of η is of great interest.

Beyond a diverging thermal conductivity and an ultralow viscosity, other peculiar phenomena are expected to arise in this plasma. The massless nature of the Dirac fermions is expected to result in a large kinematic viscosity, despite a small shear viscosity η. Observable hydrodynamic effects have also been predicted to extend into the FL regime (46). The study of magnetotransport in the DF will lead to further tests of hydrodynamics (5, 18).

Supplementary Materials

Materials and Methods

Figs. S1 to S8

Table S1

References (4759)

References and Notes

  1. Materials and methods are available as supplementary materials on Science Online.
Acknowledgments: We thank M. Foster, D. Efetov, and G.-H. Lee for helpful discussions. The major experimental work at Harvard University is supported by the U.S. Department of Energy (grant DE-SC0012260) and at Raytheon BBN Technologies by Internal Research and Development. J.C. acknowledges support from the Function Accelerated nanoMaterial Engineering (FAME) Center, sponsored by Semiconductor Research Corporation MARCO and Defense Advanced Research Projects Agency. K.W. is supported by Army Research Office (ARO) Multidisciplinary University Research Initiative (MURI) (grant W911NF-14-1-0247). J.K.S. is supported by ARO (grant W911NF-14-1-0638) and the Agency for Science, Technology and Research (A*STAR). P.K. acknowledges partial support from the Gordon and Betty Moore Foundation’s EPiQS Initiative (grant GBMF4543) and the Nano Material Technology Development Program through the National Research Foundation of Korea (grant 2012M3A7B4049966). A.L. and S.S. are supported by the NSF under grant DMR-1360789, the Templeton Foundation, and MURI grant W911NF-14-1-0003 from ARO. Research at the Perimeter Institute for Theoretical Physics is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the Ministry of Education, Culture, Sports, Science and Technology, Japan. T.T. acknowledges support from a Grant-in-Aid for Scientific Research (grant 262480621) and a grant on Innovative Areas “Nano Informatics” (grant 25106006) from the Japan Society for the Promotion of Science. T.A.O. and K.C.F. acknowledge support from Raytheon BBN Technologies. This work was performed, in part, at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Infrastructure Network, which is supported by the NSF under award no. ECS-0335765. CNS is part of Harvard University.
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