Quantum-critical conductivity of the Dirac fluid in graphene

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Science  12 Apr 2019:
Vol. 364, Issue 6436, pp. 158-162
DOI: 10.1126/science.aat8687

Electron hydrodynamics in graphene

Electrons can move through graphene in a manner reminiscent of fluids, if the conditions are right. Two groups studied the nature of this hydrodynamic flow in different regimes (see the Perspective by Lucas). Gallagher et al. measured optical conductivity using a waveguide-based setup, revealing signatures of quantum criticality near the charge neutrality point. Berdyugin et al. focused on electron transport in the presence of a magnetic field and measured a counterintuitive contribution to the Hall response that stems from hydrodynamic flow.

Science, this issue p. 158, p. 162; see also p. 125


Graphene near charge neutrality is expected to behave like a quantum-critical, relativistic plasma—the “Dirac fluid”—in which massless electrons and holes collide at a rapid rate. We used on-chip terahertz spectroscopy to measure the frequency-dependent optical conductivity of clean, micrometer-scale graphene at electron temperatures between 77 and 300 kelvin. At charge neutrality, we observed the quantum-critical scattering rate characteristic of the Dirac fluid. At higher doping, we detected two distinct current-carrying modes with zero and nonzero total momenta, a manifestation of relativistic hydrodynamics. Our work reveals the quantum criticality and unusual dynamic excitations near charge neutrality in graphene.

Landau’s theory of the Fermi liquid describes the interacting electrons of a typical metal as an ideal gas of noninteracting quasiparticles (1). This description is expected to fail in monolayer graphene (2): Owing to its linearly dispersing bands and minimally screened Coulomb interactions, graphene near charge neutrality should host a “Dirac fluid” (38)—a quantum-critical (3, 9, 10) plasma of electrons and holes governed by relativistic hydrodynamics (1113).

A surprising consequence of relativistic hydrodynamics is that in lightly doped graphene, current should be carried by two distinct modes with zero and nonzero total momenta (1113), sometimes referred to as energy waves and plasmons (8). The zero-momentum mode (Fig. 1A, left), which characterizes transport in the pure Dirac fluid at charge neutrality, consists of counterpropagating populations of thermally excited electrons and holes with equal but opposite net momenta. This current can be relaxed by charge carrier collisions, predicted to occur at the quantum-critical rate ~kBT/ħ, where kB is the Boltzmann constant, T is absolute temperature, and ħ is the Planck constant divided by 2π (9). The finite-momentum mode (Fig. 1A, right), which emerges upon doping, can be pictured as a fluid of co-propagating electrons and holes. Unbalanced charge in this moving fluid produces an electrical current that cannot be relaxed by the momentum-conserving interactions between charge carriers, as in a Fermi liquid (12). As doping increases, the weight of the zero-momentum mode should decrease while that of the finite-momentum mode increases, smoothly crossing over from Dirac fluid to Fermi liquid behavior (8, 12). The coexistence of these two modes—relaxed by entirely different mechanisms—leads to transport beyond the standard Drude picture.

Fig. 1 Using on-chip terahertz spectroscopy to probe the electrodynamics of graphene.

(A) Current-carrying modes of a graphene sheet. The zero-momentum mode corresponds to a plasma of counterpropagating electrons and holes, and can be relaxed by electron-hole interactions. The finite-momentum mode corresponds to a fluid of co-propagating electrons or holes with nonzero net charge, and cannot be relaxed by charge carrier interactions. The vector J denotes the net current flow. (B) Cartoon of our sample. Photoconductive switches (“emitter” and “detector”) triggered by a pulsed laser emit and detect terahertz pulses (electric field E) within the waveguide. The transmitted pulse is reconstructed by measuring the current collected by the preamplifier (“A”) as a function of delay between laser pulse trains illuminating the emitter and detector. The graphene is optionally excited by a separate pulsed beam (“pump”) to heat the electron system. (C) Photograph of the heterostructure embedded in the waveguide. Few-layer graphene (FLG) electrodes make contact to the monolayer graphene sheet under study and the WS2 gate electrode. Scale bar, 15 μm.

Recent experiments on clean monolayer graphene have observed low-frequency transport phenomena consistent with the hydrodynamic description, including a violation of the Wiedemann-Franz law predicted for the Dirac fluid (4) and viscous flow of electrons (14, 15). Electron-hole collisions have also been shown to limit conductivity in charge-neutral bilayer graphene (16). Yet direct observation of the quantum-critical conductivity of the Dirac fluid has remained elusive. Time-domain terahertz spectroscopy (17) is an ideal probe of the dynamic response over a broad frequency range, but the mismatch between the ~300-μm photon wavelength and ~10-μm lateral size of clean (18) graphene samples has limited its use to lower-quality, large-area films (1925) in which Dirac fluid physics is obscured (7). Our study leverages the subwavelength confinement of a coplanar waveguide (26) to measure the terahertz optical conductivity of 10-μm–scale graphene encapsulated (18) in hexagonal boron nitride (hBN). The measured conductivity at electron temperatures Te between 77 and 300 K confirms the quantum-critical scattering rate near charge neutrality, as well as the coexistence of the zero- and finite-momentum modes at nonzero doping.

Our waveguide (Fig. 1B) consists of two parallel gold traces 16 μm wide and 200 nm thick, separated by 14 μm and extending 8 mm on a fused quartz substrate. Centered beneath the waveguide is the heterostructure under study (Fig. 1C), which from top to bottom consists of a 48-nm hBN flake, monolayer graphene, a 32-nm hBN flake, and a few-layer WS2 gate electrode, chosen for its minimal terahertz absorption (see fig. S2 for sample cross section). The graphene within the w = 14 μm waveguide gap is d = 9 μm wide. Both graphene and WS2 are isolated at dc from the waveguide traces by the hBN flakes, whose impedances at microwave frequencies are small.

Emission and detection of terahertz pulses is accomplished using photoconductive switches (17) made of semiconducting GaAs with a carrier lifetime of ~1 ps. The emitter switch contacts the lower waveguide trace (Fig. 1B) and is biased with a dc voltage. When triggered by a laser pulse (pulse width 150 fs, center wavelength 800 nm, repetition rate 80 MHz), the biased emitter becomes highly conductive for ~1 ps and injects a current pulse into the coplanar waveguide, exciting a quasi-TEM (transverse electromagnetic) mode whose field distribution resembles an electric dipole with opposite charges on each trace (27). The pulse travels with minimal dispersion (26) along the waveguide, interacting with graphene before reaching a detector switch spanning both traces. When the detector is triggered by a laser pulse, a current proportional to the instantaneous local voltage between waveguide traces flows into the upper trace and is collected by a current preamplifier. Recording the current as a function of time delay between laser pulse trains triggering the emitter and detector measures the time-domain profile of the transmitted voltage pulse, V(t). In practice, we achieve lower noise by modulating the optical path length and detecting the resulting modulation in current with a lock-in amplifier, effectively measuring dV/dt (27).

All measurements are performed at a lattice temperature of 77 K unless noted otherwise. We use a separate optical pump pulse (27) (Fig. 1B) to independently heat the electron system as follows. After absorption, the electron system rapidly (~150 fs) thermalizes to a high temperature Te and equilibrates with a bath of strongly coupled optical phonons (28, 29). These phonons decay over 1 to 2 ps (30, 31), returning the lattice to near-equilibrium and lowering Te below the threshold for optical phonon emission. After this rapid cooling, a period of much slower (tens of picoseconds) cooling ensues (3234); we study different values of Te by timing our probing terahertz pulse to arrive at the sample at different times during that slow-cooling period (3234). In conjunction with ac modulation techniques, our optical heating approach enables us to isolate the terahertz response of graphene from that of other materials in the waveguide (27). Optical heating also reduces phonon contributions to the observed scattering rate because the lattice remains near 77 K.

We first investigated the optical conductivity of the Fermi liquid at 77 K. The transmitted waveforms contain sharp, subpicosecond features that evolve with gate voltage (Fig. 2A, inset), with maximum transmission at charge neutrality (27). To extract the optical conductivity from the time-domain data, we model our device as an infinite, lossless transmission line containing a short segment with conductance G (graphene) between the conductors; this approach is justified by finite-element simulations (27). A wave mechanics calculation (27) yields G=2Z01(V˜0/V˜1), where Z0 = 133 ohms is the waveguide impedance determined from simulations, V˜ is the Fourier transform of the measured time-domain pulse transmitted through graphene, and V˜0 is the Fourier transform of the transmitted pulse in the absence of graphene. Because we cannot measure V˜0, we approximate it by the transmission at charge neutrality, incurring small, correctable errors (27). In all measurements, we heavily dope the graphene beneath the waveguide traces to minimize its impedance, so that G = (d/w)σ measures the conductance of the d × w graphene region of conductivity σ between the traces. [At large εF, however, corrections stemming from graphene beneath the traces must still be applied to obtain σ (27).]

Fig. 2 Frequency-dependent optical conductivity of graphene in the Fermi liquid regime.

(A and B) Real (A) and imaginary (B) parts of the extracted optical conductivity for several Fermi energies between 46 and 119 meV (electron doping) at 77 K. Solid curves are Drude fits using only the scattering rate τ–1 as a free fitting parameter for each curve. Inset in (A) shows an example of the time-domain current data used to extract conductivity in the frequency domain; the purple trace shows the transmitted waveform at 119 meV, and the black trace shows the transmitted waveform at the charge neutrality point (CNP), which is used as a reference. Fast Fourier transforms of these traces are shown in fig. S3. Inset in (B) shows the extracted τ–1 at lattice temperatures of 77 K and 300 K.

The extracted optical conductivities σ = σ′ + iσ′′ at various values of εF ≥ 46 meV (Fig. 2, A and B) clearly follow the Drude form, σ = Dgrπ–1–1iω)–1. Using the chemical potential μ determined from the gate capacitance [32 nm hBN, ε = 3 along the c axis (35)] and the known Drude weight of graphene, Dgr = 2(e/ħ)2kBTe log{2 cosh[μ(2kBTe)–1]} (24), Drude fits match the data well with only τ–1 as a free parameter. The extracted scattering rates at 77 K fall between 0.5 and 1 THz (Fig. 2B, inset), indicative of infrequent scattering by disorder and phonons, and consistent with transport studies of similar heterostructures at comparable doping (18). These results confirm the anticipated Fermi liquid behavior.

We probed transport at charge neutrality by observing the change in terahertz transmission ΔV upon optically heating the electron system from T0 = 77 K to different electron temperatures Te, varied by adjusting the delay between optical pump and terahertz probe pulses. The transmission line model yields (27) the corresponding conductivity change Δσ=(w/d)ΔG=2(w/d)Z01ΔV˜/V˜ (Fig. 3, A and B); although this model becomes less accurate as σ decreases, simulations confirm that it introduces only minor errors in our analysis (27). Current in charge-neutral graphene should be carried by the zero-momentum mode (Fig. 1A, left), which is relaxed primarily by electron-electron scattering at a rate τee–1, but also by disorder/phonon scattering at a rate τd–1. Using a relaxation-time approximation, both scattering mechanisms can be captured by the sum τ–1 = τee–1 + τd–1 so that the conductivity follows the simple Drude form Dgrμ=0π–1–1iω)–1, where Dgrμ=0Te is the Drude weight for graphene evaluated at μ = 0 (9, 12, 13). Although not exact, this Drude form is qualitatively reasonable. We thus fit Δσ to a difference between Drude functions at Te and T0: Δσ = Dgrμ=0(Te–1–1iω)–1Dgrμ=0(T0–10–1iω)–1. For each optical pump delay, Te and the associated scattering rate τ–1 are unknown fit parameters, whereas τ0–1, the scattering rate at T0 = 77 K, is a global fit parameter for all curves.

Fig. 3 Quantum-critical scattering rate of the Dirac fluid.

(A and B) Real (A) and imaginary (B) parts of the change in optical conductivity at charge neutrality upon optically heating the electron system to a temperature Te above the equilibrium temperature T0 = 77 K. Each curve corresponds to a different delay between the optical pump pulse (fluence 21 nJ cm–2) and the terahertz probe pulse; see legend in (B). Solid curves are fits to a difference between Drude functions at Te and T0, using Te and the scattering rate τ–1(Te) as free fit parameters for each pair of curves of the complex conductivity. (C) Blue markers indicate the scattering rates and electron temperatures extracted from the fits shown in (A) and (B); error bars indicate SE in the fits. The experimental scattering rate follows τ–1 = τee–1 + τd–1 (dashed curve), where τee–1 = 0.20kBTe (green line) is the scattering rate due to charge carrier interactions, and τd–1nimpTe–1 (dotted curve) is the scattering rate due to unscreened, singly charged impurities with density nimp = 2.1 × 109 cm–2. (D) Real and imaginary parts (open and solid circles, respectively) of σ at different values of Te (i.e., different optical pump delays), replotted as a function of ħω/kBTe. The data for Te = 100 K (21.3 ps delay) do not collapse and are omitted. Solid and dashed curves are fits to the real and imaginary parts of the universal function σU; the fits yield C = 0.23 (see text).

The data are well described by the difference in Drude functions (Fig. 3, A and B, solid curves), providing precise estimates of τ–1 as a function of Te (Fig. 3C). For Te > 130 K, τ–1 grows almost linearly with Te up to the highest accessible temperature, Te = 299 ± 12 K. This linear evolution is a key signature of charge carrier interactions in the quantum-critical Dirac fluid, which are expected to scale as τee–1 = CkBTe, where C is a dimensionless constant (9). Below 130 K, τ–1 deviates from the linear τee–1, likely because of charged impurities, which contribute a scattering rate τd–1nimpTe–1 for a density nimp of unscreened impurities (27). Using the fitted value nimp = 2.1 × 109 cm–2, the experimental τ–1 is quantitatively explained by the sum τ–1 = τee–1 + τd–1 (Fig. 3C, dashed curve). Phonon scattering appears to be unimportant, consistent with expected phonon scattering rates for a phonon bath near T0 = 77 K [~0.6 THz for Te = 300 K, and much smaller scattering rates for lower Te (7, 36)].

The quantum-critical scattering at charge neutrality can be separately visualized in a plot of the conductivity σ as a function of ħω/kBTe: In the limit τee–1 = CkBTe ≫ τd–1, the conductivity should approach the universal curve σU = 4 log(2) e2/h (C + ω/kBTe)–1. We retrieve σ by summing the measured Δσ (Fig. 3, A and B) and the Drude conductivity at T0 (determined by the fit parameter τ0–1). Above 130 K, σ indeed collapses onto a single curve (Fig. 3D). The best fit to the universal curve σU yields C = 0.23, in approximate agreement with the value C = 0.20 obtained by fitting directly to τ–1(Te) (Fig. 3C).

The experimental value C = 0.20 aligns with theoretical expectations. Following the quantum Boltzmann theory of (9), C determines the fine-structure constant of graphene as α=C/3.646=0.23. This value of α compares favorably with theory: Assuming an effective dielectric constant ε = 4 for interactions between carriers in hBN-encapsulated graphene, an unscreened picture finds α0 ≈ 300/(137ε) = 0.55 (we assume a Fermi velocity vF = 106 m/s), whereas accounting for screening via the random phase approximation (RPA) (9, 37) gives αRPA = α0[1 + (πα0/2)]–1 = 0.29. Similar values of α have also been inferred from measurements of plasmons in hBN-encapsulated graphene (38). In total, our results validate the predicted quantum-critical scattering of the zero-momentum mode at charge neutrality.

Away from charge neutrality, current should be carried by both zero- and finite-momentum modes (Fig. 1A). The latter corresponds to a fluid of co-propagating electrons and holes with a net charge; because the total momentum is conserved by charge carrier collisions, this mode is only relaxed by disorder and phonon scattering events occurring with frequency τd–1 ≪ τee–1. The two-component conductivity can be written asσ=DZπ1(τee1+τd1iω)1+DFπ1(τd1iω)1(1)(8), where DZ and DF are the Drude weights of the zero- and finite-momentum modes, respectively, and are known functions of μ and Te (27). For a given εF ≠ 0, the finite-momentum mode carries all of the Drude weight at Te = 0, whereas at high temperatures, μ tends to zero and DZ dominates (Fig. 4A).

Fig. 4 Coexistence of zero- and finite-momentum modes at low doping.

(A) Calculated Drude weights DZ and DF of the zero- and finite-momentum modes (27) in lightly electron-doped (εF = 33 meV) and undoped graphene. (B and C) Real (B) and imaginary (C) parts of the measured change in optical conductivity when charge-neutral graphene in equilibrium (T0 = 77 K) is simultaneously heated to an electron temperature Te (optical pump delay 3 ps, fluence 21 nJ cm–2) and doped to εF = 33 meV. Solid curves show fits using the function described in the text, which sums the changes (dashed curves) in zero- and finite-momentum mode conductivities between (εF = 33 meV, Te) and (εF = 0, T0). (D and E) Real (D) and imaginary (E) parts of the measured change in optical conductivity when charge-neutral graphene at an electron temperature Te (optical pump delay 4 ps, fluence 20 nJ cm–2) is doped to various values of εF. Data at each doping are well fit by a single Drude function (solid curves) describing the conductivity of the finite-momentum mode with free fit parameters Te = 267 ± 3 K and τd–1F) ~ 1 THz. Inset in (D) shows the scattering rate for the finite-momentum mode τd–1 versus Te extracted from fits at varying Te; colors indicate εF as in (D) and (E).

We recorded the complex conductivity change Δσ = σ(εF = 33 meV, Te) – σ(0, T0) upon simultaneously doping from εF = 0 to 33 meV and heating from T0 = 77 K to Te ≈ 300 K (Fig. 4, B and C). The real part, Δσ′, reveals two distinct Drude components: a narrow peak of width Δω < 1 × 1012 rad/s and a slowly varying background on the order of 7e2/h. We attribute these features to the conductivity changes of the finite- and zero-momentum modes, respectively. A numerical fit to the expression Δσ = σ(εF = 33 meV, Te) – σ(0, T0), with σ described by Eq. 1 and σ(0, T0) already known (Fig. 3), confirms our qualitative assignments (Fig. 4, B and C, dashed curves). The fit yields Te = 296 ± 13 K, τee–1 = 6.6 ± 0.3 THz, and τd–1 = 0.8 ± 0.1 THz.

As further evidence of two-mode conductivity, we isolated the contribution from the finite-momentum mode and showed that it behaves as a single Drude peak with the predicted Drude weight DF. More specifically, we measured the conductivity change Δσ = σ(εF, Te) – σ(0, Te) upon light doping under fixed optical pumping conditions (i.e., for Te nearly constant). For sufficiently large Te, Δσ should approximate the conductivity of the finite-momentum mode at εF: Although light doping away from charge neutrality changes DF and DZ by similar amounts (Fig. 4A), the zero-momentum contribution to Δσ is greatly spread out in frequency because τee–1~10τd–1. Using optical pumping conditions that produce Te ≈ 250 K at charge neutrality, the measured values of Δσ at varying εF indeed behave like Drude functions (Fig. 4, D and E). The data are well fit by the expression DFπ–1d–1iω)–1 (Fig. 4, D and E, solid curves), where the only free fitting parameters are τd–1 for each εF and the shared value of Te, which enters the fit implicitly through DF. The fits yield Te = 267 ± 3 K and τd–1 ~ 1 THz for all values of εF.

To highlight the very different scattering mechanisms of the conducting modes, we collected Δσ = σ(εF, Te) – σ(0, Te) for different doping levels and pumping conditions (i.e., varying both εF and Te) and fit all data using Eq. 1 for σ. We assert that τee–1 = 0.20kBTe does not vary with εF; this approximate assignment leaves only Te and τd–1 as free fitting parameters. The extracted values of τd–1 for the finite-momentum mode (Fig. 4D, inset) generally decrease with increasing Te within our temperature window, in sharp contrast to the scattering rate for the zero-momentum mode, which increases approximately linearly with Te.

The quantitative agreement between our experimental results and the relativistic hydrodynamic theory of the Dirac fluid implies that graphene should host relativistic phenomena not seen in typical electron systems, where relativistic hydrodynamics does not apply. For example, electronic sound waves in conventional metals either morph into plasmons or are destroyed by momentum relaxation, but such waves can exist in charge-neutral graphene owing to low disorder and zero coupling to plasmon modes (39). At high temperatures, graphene is also expected to display a collective cyclotron resonance that is primarily damped by collisions between charge carriers, rather than by disorder (13).

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 to S17

References (41, 42)

References and Notes

  1. See supplementary materials.
Acknowledgments: Funding: Terahertz measurements were primarily supported by the Office of Naval Research under award N00014-15-1-2651. Sample fabrication was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division under contract DE-AC02-05-CH11231 (van der Waals heterostructures program, KCWF16). P.G. was supported by the Heising-Simons Junior Fellowship program of the Kavli ENSI at Berkeley. C.-S.Y. was supported by the Ministry of Science and Technology under grant 105-2917-I-564-026. R.K. was supported by the JSPS Overseas Research Fellowship program. Growth of hexagonal boron nitride crystals (K.W., T.T.) was supported by the Elemental Strategy Initiative conducted by the MEXT, Japan, and the CREST (JPMJCR15F3), JST. Author contributions: P.G. and C.-S.Y. built the optical apparatus and collected the data; P.G. performed the data analysis with input from all authors; P.G., C.-S.Y., F.T., and R.K. fabricated the samples; T.L. and H.Z. performed the waveguide simulations; P.G., C.-S.Y., and F.W. designed the experiment; P.G. wrote the manuscript with input from all authors; and K.W. and T.T. grew the hBN crystals. Competing interests: Authors declare no competing interests. Data and materials availability: All data necessary to understand and assess this manuscript are shown in the main text and supplementary materials. Data shown in the main text are also available in tabular form through Zenodo (40).
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