www.sciencemag.org/cgi/content/full/science.aat8687/DC1

Supplementary Materials for

Quantum-critical conductivity of the Dirac fluid in graphene Patrick Gallagher, Chan-Shan Yang, Tairu Lyu, Fanglin Tian, Rai Kou, Hai Zhang,

Kenji Watanabe, Takashi Taniguchi, Feng Wang*

*Corresponding author. Email: fengwang76@berkeley.edu (F.W.)

Published 28 February 2019 on Science First Release DOI: 10.1126/science.aat8687

This PDF file includes: Materials and Methods Supplementary Text Figs. S1 to S17 References

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Materials and Methods Sample fabrication Individual flakes of the graphene heterostructure were first prepared on oxidized silicon wafers by mechanical exfoliation from bulk crystals of graphite (NGS, graphit.de), hBN (Watanabe and Taniguchi), and WS2 (HQ Graphene). The heterostructure was assembled using a dry transfer technique (18) and subsequently deposited on a fused quartz substrate.

Photoconductive switches were separately prepared starting from a bulk wafer supplied by BATOP GmbH, which consisted of two epitaxial layers grown on a semi-insulating GaAs substrate. The top epitaxial layer was a 2.6 micron thick GaAs film grown at low temperature (300°C) to achieve a nominal carrier recombination time of 0.5 ps. Beneath this layer was a 500 nm thick etch stop made of Al0.9Ga0.1As. The top layer was etched into squares (50 micron by 50 micron) and rectangles (200 micron by 50 micron) by defining an etch mask (S1818 photoresist, MicroChem) and subsequently etching in a citric acid solution (~10 minutes in 6 parts citric acid monohydrate:6 parts deionized water:1 part hydrogen peroxide). The etch stop layer was then dissolved in 10:1 buffered hydrofluoric acid (~6 hours) followed by a thorough rinse in deionized water and gentle drying with a low nitrogen flow. Etched GaAs squares and rectangles which remained loosely attached to the semi-insulating GaAs substrate were mechanically transferred to our quartz substrate using PDMS (Sylgard 184, Dow Corning) as an adhesive.

The waveguide structure was finally patterned using photolithography (LOR 3A and S1818 photoresists, MicroChem). Electrode metal (5 nm titanium, 200 nm gold) was deposited using an electron-beam evaporator. Figure S1 shows a large-area photograph of the completed device. Figure S2 shows the cross-section of the heterostructure beneath the waveguide. Terahertz measurements Laser pulses illuminating the emitter and detector were split off from the output of a mode-locked Ti:sapphire laser (Atseva) with center wavelength 800 nm, pulse width ~150 fs, repetition rate 80 MHz, and total output power 800 mW. Pulsed optical pump excitation was generated from the same Ti:sapphire output using a supercontinuum fiber (Newport SCG-800); the fiber output was filtered to transmit wavelengths between 1050 nm and 1500 nm to avoid exciting the WS2 gate electrode. All optical pump fluences reported are the fluences absorbed by the graphene sheet, but should be considered order-of-magnitude values, as geometrical factors and local field effects were not precisely measured. The probing electric field applied to graphene was kept below 10 mV/micron so that the energy accumulated between collisions would not exceed 𝑘𝑘B𝑇𝑇e, preserving approximate electron-hole symmetry (9) for our studies of the Dirac liquid. We verified that reducing the probing field by a factor of ~6 does not change our results near charge neutrality.

The lengths of the emitter and optical pump beam paths were adjusted using two mechanical delay stages. All data shown in the main text were collected by fixing the optical pump stage and adjusting the length of the emitter beam path to measure the time-domain waveform (for Fig. 2, the optical pump excitation was blocked). The length of the detector beam path was modulated at 50 Hz with ~50 micron amplitude using a mirror mounted on a piezoelectric shaker, while the optical pump beam was mechanically chopped at 225 Hz. The 50 Hz and 275 Hz components of the current signal were simultaneously collected to measure, respectively, the time derivative of the transmitted pulse 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑 and its change 𝑑𝑑(𝛥𝛥𝑑𝑑)/𝑑𝑑𝑑𝑑 upon optical pump excitation. The 50 Hz and 275 Hz signals are related to 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑 and 𝑑𝑑(𝛥𝛥𝑑𝑑)/𝑑𝑑𝑑𝑑 by

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frequency-dependent calibration constants of order 1; these constants were carefully measured to avoid scaling errors in the extracted 𝛥𝛥𝜎𝜎. Since the photon energy of the optical pump is below the bandgap of WS2 and hBN, the 𝑑𝑑(𝛥𝛥𝑑𝑑)/𝑑𝑑𝑑𝑑 signal exclusively measures the conductivity change of graphene.

The sample was mounted in the vacuum space of an optical cryostat cooled to 77 K by a liquid nitrogen bath. The 300 K measurements shown in the inset of Fig. 2 of the main text were collected by allowing the nitrogen bath to empty and warm to room temperature. Curve-fitting All fits reported in this work were performed using LMFIT (41), an optimization package for Python. Error bars displayed in figures and written in the text indicate standard deviations determined by LMFIT.

Supplementary Text Determination of charge neutrality Since the Dirac fluid behavior under study should only appear at chemical potentials 𝜇𝜇 <𝑘𝑘B𝑇𝑇, we require the sample to be tuned quite precisely to charge neutrality. Fig. S4 shows the evolution with 𝑑𝑑gate of the measured current signals proportional to 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑 and 𝑑𝑑(𝛥𝛥𝑑𝑑)/𝑑𝑑𝑑𝑑 at fixed delay between terahertz emitter and detector. The gate voltage corresponding to the extrema of these curves identifies charge neutrality to within ~10 mV in gate voltage, or ~7 meV in Fermi energy. In the worst case, our experiments use 𝜖𝜖F = 7 meV as charge neutrality, which corresponds to 𝜇𝜇 = 2.6 meV at 77 K and 𝜇𝜇 = 0.7 meV at 300 K. At all experimental temperatures 𝜇𝜇 < 𝑘𝑘B𝑇𝑇, with 𝜇𝜇 ≪ 𝑘𝑘B𝑇𝑇 at high temperatures. Transmission line model of the waveguide

We model the coplanar waveguide as a transmission line with impedance per unit length 𝑍𝑍′ = 𝑅𝑅′ − 𝑖𝑖𝑖𝑖𝑖𝑖′ and admittance per unit length 𝑌𝑌′ = 𝐺𝐺′ − 𝑖𝑖𝑖𝑖𝑖𝑖′, as shown in Fig. S5. Writing expressions for the voltage 𝑑𝑑(𝑧𝑧) and the current 𝐼𝐼(𝑧𝑧) in the circuit and expanding to first order in the infinitesimal element 𝛥𝛥𝑧𝑧 leads us to solutions that are forward and backward moving waves,

𝑑𝑑(𝑧𝑧) = 𝑑𝑑+𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖 + 𝑑𝑑−𝑒𝑒−𝑖𝑖𝑖𝑖𝑖𝑖 𝐼𝐼(𝑧𝑧) = 𝑍𝑍0−1(𝑑𝑑+𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖 − 𝑑𝑑−𝑒𝑒−𝑖𝑖𝑖𝑖𝑖𝑖)

where the propagation constant is

𝑘𝑘 = 𝜔𝜔𝑛𝑛0𝑐𝑐

= 𝑖𝑖√𝑖𝑖′𝑖𝑖′��1 + 𝑖𝑖𝑅𝑅′

𝜔𝜔𝐿𝐿′� �1 + 𝑖𝑖𝐺𝐺′

𝜔𝜔𝐶𝐶′� (S1)

and the characteristic impedance is

𝑍𝑍0 = �𝑅𝑅′−𝑖𝑖𝜔𝜔𝐿𝐿′

𝐺𝐺′−𝑖𝑖𝜔𝜔𝐶𝐶′.

The resistance per unit length of the gold traces is 𝑅𝑅′ ∼ 4 kΩ/m. In contrast, a simple estimate using two parallel wires of circular cross-section finds 𝑖𝑖′ ∼ 5 × 10−7 H/m, which at 𝑖𝑖 = 0.5 × 1012 rad/s yields 𝑖𝑖𝑖𝑖′ ∼ 250 kΩ/m ≫ 𝑅𝑅′. We therefore take 𝑅𝑅′ = 0 for all frequencies. In the region with no graphene, we also have 𝐺𝐺′ = 0, leaving 𝑘𝑘 = 𝑖𝑖𝑛𝑛0/𝑐𝑐 =𝑖𝑖√𝑖𝑖′𝑖𝑖′ and 𝑍𝑍0 = �𝑖𝑖′/𝑖𝑖′. We find 𝑛𝑛0 = 1.56, 𝑍𝑍0 = 133 Ω, and 𝑖𝑖′ = 3.89 × 10−11 F/m to within 4% over our spectral range using 2D mode simulations in COMSOL, assuming the dielectric constant of fused quartz to be 𝜖𝜖 = 3.85 at terahertz frequencies (42).

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Our transmission line thus reduces to a segment of length 𝑑𝑑 (the graphene-containing region) with unknown conductance per unit length 𝐺𝐺′ surrounded by two semi-infinite, lossless regions. This problem maps directly onto the wave optics problem of normal transmission through a slab of thickness 𝑑𝑑 and index 𝑛𝑛1 immersed in a homogeneous medium of index 𝑛𝑛0 (Fig. S6). Following Eq. S1, the unknown index of refraction 𝑛𝑛1 is

𝑛𝑛1 = 𝑛𝑛0�1 + 𝑖𝑖𝐺𝐺′

𝜔𝜔𝐶𝐶′. (S2)

For an incoming right-moving wave with frequency 𝑖𝑖 and amplitude 𝐴𝐴0 immediately to the left of the slab of thickness 𝑑𝑑, the transmitted right-moving wave immediately to the right of the slab has amplitude 𝑑𝑑𝐴𝐴0, where the transmission coefficient is

𝑑𝑑 = 𝜏𝜏01𝜏𝜏10𝑒𝑒𝑖𝑖𝑘𝑘1𝑑𝑑

1+𝜌𝜌01𝜌𝜌10𝑒𝑒2𝑖𝑖𝑘𝑘1𝑑𝑑 (S3)

with 𝜌𝜌01 = 𝜏𝜏01 − 1 = 𝑛𝑛0−𝑛𝑛1

𝑛𝑛0+𝑛𝑛1.

For Fig. 2 of the main text, we experimentally measured the ratio 𝑑𝑑�/𝑑𝑑�0 of transmission at 𝜖𝜖F ≠ 0 to the “reference” transmission at charge neutrality. We extract 𝐺𝐺′ by numerically inverting the expression 𝑑𝑑/𝑑𝑑0 = 𝑑𝑑�/𝑑𝑑�0, where both 𝑑𝑑 and the reference transmission 𝑑𝑑0 are given by Eq. S3, but 𝑑𝑑0 is known from the conductivity at charge neutrality measured in Fig. 3.

The magnitude of 𝑑𝑑0 is very nearly equal to 1 owing to the minimal conductivity of charge-neutral graphene. To proceed analytically, we ignore the conductivity of graphene in the reference configuration, leading to

𝑡𝑡𝑡𝑡0

=2�1+ 𝑖𝑖𝐺𝐺′

𝜔𝜔𝐶𝐶′𝑒𝑒−𝑖𝑖𝜔𝜔𝑛𝑛0𝑑𝑑

𝑐𝑐

2�1+ 𝑖𝑖𝐺𝐺′

𝜔𝜔𝐶𝐶′ cos�𝜔𝜔𝑛𝑛0𝑑𝑑𝑐𝑐

�1+ 𝑖𝑖𝐺𝐺′

𝜔𝜔𝐶𝐶′�−𝑖𝑖�2+𝑖𝑖𝐺𝐺′

𝜔𝜔𝐶𝐶′� sin�𝜔𝜔𝑛𝑛0𝑑𝑑𝑐𝑐

�1+ 𝑖𝑖𝐺𝐺′

𝜔𝜔𝐶𝐶′� . (S4)

In the thin-film limit 𝜔𝜔𝑛𝑛0𝑑𝑑𝑐𝑐

�1 + 𝑖𝑖𝐺𝐺′

𝜔𝜔𝐶𝐶′≪ 1, which is quite valid near charge neutrality and largely

valid in the Fermi liquid regime, we can analytically invert the above equation to find 𝐺𝐺′ = 2

𝑍𝑍0𝑑𝑑�𝑡𝑡0𝑡𝑡− 1�.

We then compute the change in transmission coefficient 𝛥𝛥𝑑𝑑 when 𝐺𝐺′ changes by a small amount 𝛥𝛥𝐺𝐺′ ≪ 𝐺𝐺′ to find

𝛥𝛥𝐺𝐺′ = − 2𝑍𝑍0𝑑𝑑

𝛥𝛥𝑡𝑡𝑡𝑡

, which is used in Fig. 3 and 4 of the main text. Extracting conductivity from measured conductance Extracting the conductivity 𝜎𝜎 of the graphene sheet between the waveguide traces is complicated by the fact that 𝐺𝐺′ obtained from the experiment also includes serial capacitances and resistances beneath the traces. The situation can be modeled as in Fig. S7A. In the sliver of length 𝛥𝛥𝑧𝑧, the graphene beneath the waveguide trace contributes a serial resistance 𝜌𝜌t𝑠𝑠/𝛥𝛥𝑧𝑧 looking along 𝑥𝑥�, where 𝜌𝜌t = 𝜎𝜎t−1 and 𝑠𝑠 are the resistivity and length of the graphene beneath the traces, respectively. The capacitance between sliver and trace is 𝑐𝑐t𝑠𝑠𝛥𝛥𝑧𝑧, where 𝑐𝑐t is the capacitance per unit area of the hBN. By further dividing the sliver into chunks of length 𝛥𝛥𝑥𝑥, forming an RC network of infinitesimal resistors 𝑅𝑅t = 𝜌𝜌t𝛥𝛥𝑥𝑥/𝛥𝛥𝑧𝑧 and capacitors 𝑖𝑖t = 𝑐𝑐t𝛥𝛥𝑥𝑥𝛥𝛥𝑧𝑧 (Fig. S7B), we can model the sliver of length 𝛥𝛥𝑧𝑧 as its own transmission line with characteristic

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impedance 1𝛥𝛥𝑖𝑖 �

𝑖𝑖𝜌𝜌t𝜔𝜔𝑐𝑐t

. Defining 𝑍𝑍ser = �𝑖𝑖𝜌𝜌t𝜔𝜔𝑐𝑐t

and accounting for the regions beneath both traces as

well as the graphene of conductivity 𝜎𝜎 between the traces, we find the total conductance 𝐺𝐺 = 𝐺𝐺′𝑑𝑑 = 𝑑𝑑(2𝑍𝑍ser + 𝑤𝑤𝜎𝜎−1)−1, (S5)

where we have ignored the effects of the finite length 𝑠𝑠. To mitigate the effect of 𝑍𝑍ser on the total conductance, in all measurements we heavily doped the graphene region beneath the waveguide traces. The doping in this region was controlled by adjusting the voltage difference between graphene sheet (at voltage 𝑑𝑑graphene) and the grounded waveguide traces; i.e., the waveguide traces served as a local top gate. All figures in the main text were collected using 𝑑𝑑graphene = −2 V. For our studies of charge neutrality (Fig. 3), this condition doped the region beneath the traces to 𝜖𝜖F = 106 meV, which practically eliminated the effect of 𝑍𝑍ser on 𝐺𝐺. Solid curves in Fig. S8A,B show 𝐺𝐺 = 𝑑𝑑

𝑤𝑤𝜎𝜎

(i.e., 𝐺𝐺 calculated from Eq. S5 when 𝑍𝑍ser = 0) for charge-neutral graphene at electron temperatures 77 K and 300 K, with 𝜏𝜏−1 = 4 THz and 𝜏𝜏−1 = 8.5 THz, respectively. Dotted and dashed curves show 𝐺𝐺 calculated from Eq. S5 under the same conditions, but assuming finite- and infinite-length RC networks beneath the traces, respectively, with nonzero 𝑍𝑍ser. All curves fall nearly on top of each other, demonstrating the negligible effect of 𝑍𝑍ser. On the other hand, at the highest doping levels studied in Fig. 2 of the main text, the additional doping beneath the traces provided by the condition 𝑑𝑑graphene = −2 V does not entirely suppress the effect of 𝑍𝑍ser (Fig. S8C,D). The conductance 𝐺𝐺 computed using a finite-length RC network with 𝑠𝑠 = 16 micron (dotted lines, Fig. S8C,D) furthermore produces oscillations about the infinite-length result (dashed lines) which become increasingly visible for larger 𝜖𝜖F. To extract the conductivity 𝜎𝜎 displayed in Fig. 2 of the main text, we used an infinite RC network for 𝑍𝑍ser, with 𝜎𝜎t under the traces determined by the known doping and an assumed scattering rate of 0.6 THz. We chose an infinite RC network since the irregular shapes of our graphene and WS2 flakes (Fig. 1C of the main text) should largely average out any oscillations resulting from finite length. Validity of the transmission line model

The transmission line model should be quantitatively accurate if the conductivity of graphene is large enough that the electric field lines distribute uniformly across the graphene sheet; in this case, graphene can be modeled as a simple resistive element with conductance per unit length 𝐺𝐺′. This condition will be satisfied if |𝐺𝐺′| ≫ 𝑖𝑖𝑖𝑖′, so that the admittance due to the line capacitance is negligible. In the opposite limit, |𝐺𝐺′| ≪ 𝑖𝑖𝑖𝑖′, the graphene is a small perturbation on the waveguide mode. The mode will accordingly look like that of the empty waveguide, in which 𝐸𝐸𝑥𝑥 between the traces is not constant (Fig. S9). Additional circuit elements would be required to account for the non-uniform field distribution across the sheet. The line capacitance of our waveguide is 𝑖𝑖′ = 1 𝑒𝑒2

ℎ µm−1 THz−1. At low frequencies in

the Fermi liquid regime (Fig. 2 of the main text), |𝐺𝐺′| ≫ 𝑖𝑖𝑖𝑖′, but for 𝑖𝑖 ∼ 6 × 1012 rad/s, 𝑖𝑖𝑖𝑖′ approaches |𝐺𝐺′|. Figure S10 tracks the evolution of the mode profile (simulated with the COMSOL 2D mode solver) with increasing frequency for 𝜖𝜖F = 119 meV between the traces. The field 𝐸𝐸𝑥𝑥 within the graphene between the traces is extremely homogeneous at 𝑖𝑖 =1 × 1012 rad/s, but spatially varies by ~20% at 𝑖𝑖 = 5 × 1012 rad/s. We note that 𝐸𝐸𝑥𝑥 between the traces (Fig. S10D,F) shows the opposite curvature to that of the empty waveguide

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(Fig. S9C), implying that even at high frequencies, the graphene sheet (including the heavily doped region at 𝜖𝜖F = 159 meV beneath the traces) still strongly affects the field distribution.

At charge neutrality, the magnitude of the line admittance either approaches or exceeds |𝐺𝐺′| throughout our spectral range. Figure S11 tracks the evolution of the mode profile with increasing frequency for 𝜖𝜖F = 0 between the traces and the experimentally relevant doping 𝜖𝜖F =106 meV beneath the traces. The field 𝐸𝐸𝑥𝑥 within the graphene between the traces is clearly less homogeneous than it is at higher doping. Here the curvature of 𝐸𝐸𝑥𝑥 (Fig. S11B,D,F) generally trends toward that of the empty waveguide, although the curvature inverts near the waveguide traces, likely owing to the region of heavily doped graphene beneath the traces. These simulated mode profiles qualitatively suggest that the transmission line model should work well to describe the Fermi liquid regime, but that it may be somewhat less accurate at charge neutrality. To quantitatively evaluate our models, we simulated the effective refractive index of the waveguide mode for various Drude conductivities 𝜎𝜎 between the waveguide traces and 𝜎𝜎t beneath the traces. We then used the transmission line model (Eq. S2) and series impedance model (parameterized by 𝜎𝜎t; Eq. S5) to extract the apparent conductivity 𝜎𝜎sim between the traces, and compared 𝜎𝜎sim to the input 𝜎𝜎. Figure S12 shows an example of our results for higher doping. Here 𝜎𝜎 corresponds to 𝜖𝜖F = 119 meV, with 𝜏𝜏−1 = 0.6 THz and 𝑇𝑇 = 77 K. We find that 𝜎𝜎sim matches well with 𝜎𝜎. More generally, we observe this level of agreement between 𝜎𝜎sim and 𝜎𝜎 throughout the Fermi liquid regime discussed in Fig. 2 of the main text, confirming that the transmission line model adequately describes the physics of the waveguide for high doping. Figure S13 compares 𝜎𝜎 and 𝜎𝜎sim at charge neutrality between the waveguide traces for electron temperatures 77 K and 293 K. The difference between 𝜎𝜎 and 𝜎𝜎sim is less than 2 𝑒𝑒2/ℎ, but clearly does depend on frequency. The extent to which this disagreement impacts our fits is examined in the next section. All of the above simulations were performed using a 2D mode solver, even though the real sample is three-dimensional. We verified for a handful of input conductivities that the apparent conductivity 𝜎𝜎sim does not appreciably change if we instead perform a 3D simulation of the transmission and compute 𝜎𝜎sim using Eq. S4. Curve-fitting near charge neutrality The frequency-dependent disagreement between 𝜎𝜎 and 𝜎𝜎sim at charge neutrality (Fig. S13) indicates that the conductivity change 𝛥𝛥𝜎𝜎 shown in Fig. 3 of the main text, extracted using the transmission line model, does not perfectly measure the true change in graphene conductivity upon optical heating. As such, 𝛥𝛥𝜎𝜎 may not exactly obey a difference in Drude functions. We nonetheless find that fitting 𝛥𝛥𝜎𝜎 to a difference in Drude functions remains a reasonable approximation to extract 𝜏𝜏−1 and 𝑇𝑇e. To demonstrate this, we simulated the effective refractive index of the waveguide mode for charge-neutral graphene of conductivity 𝜎𝜎(𝑖𝑖; 𝜏𝜏−1,𝑇𝑇e) =𝐷𝐷gr𝜇𝜇=0(𝑇𝑇e)𝜋𝜋−1(𝜏𝜏−1 − 𝑖𝑖𝑖𝑖)−1 parameterized by a wide range of 𝜏𝜏−1 and 𝑇𝑇e. We then used wave

mechanics (Eq. S3) to compute the simulated transmission 𝑑𝑑sim(𝑖𝑖; 𝜏𝜏−1,𝑇𝑇e) for all 𝜏𝜏−1 and 𝑇𝑇e, interpolating between discrete parameter values. We finally fit the measured relative transmission change 𝛥𝛥𝑑𝑑/𝑑𝑑 to the function 𝛥𝛥𝑑𝑑sim/𝑑𝑑sim = 𝑑𝑑sim(𝑖𝑖; 𝜏𝜏−1,𝑇𝑇e)/𝑑𝑑sim(𝑖𝑖; 𝜏𝜏0−1,𝑇𝑇0) − 1. The free fit parameters were 𝜏𝜏−1, 𝜏𝜏0−1, and 𝑇𝑇e, with 𝑇𝑇0 = 77 K, as in Fig. 3 of the main text. The 𝛥𝛥𝑑𝑑sim/𝑑𝑑sim curves fit the data well (Fig. S14A,B). The extracted temperatures (Fig. S14C) are very similar to those extracted from the fits shown in Fig. 3 (reproduced in Fig. S14D,E,F), while the extracted scattering rates are also similar but slightly lower than those in

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Fig. 3. Most critically, the scattering rates in both cases are well described by the expression 𝜏𝜏−1 = 𝜏𝜏ee−1 + 𝜏𝜏d−1, with 𝜏𝜏ee−1 = 𝑖𝑖𝑘𝑘B𝑇𝑇e/ℏ the quantum-critical scattering rate for charge-carrier interactions and 𝜏𝜏d−1 ∝ 𝑛𝑛imp𝑇𝑇e−1 the scattering rate due to charged impurities. The fits to 𝛥𝛥𝑑𝑑sim/𝑑𝑑sim yield 𝑖𝑖 = 0.16 and 𝑛𝑛imp = 1.3 × 109 cm−2, as opposed to 𝑖𝑖 = 0.20 and 𝑛𝑛imp =2.1 × 109 cm−2 from the fits to a difference in Drude functions. The data shown in Fig. 4 of the main text mostly concern graphene at higher conductivities, for which the transmission line model should work better than it does at charge neutrality. We expect some fitting errors owing to our application of the transmission line model to extract 𝛥𝛥𝜎𝜎, but these errors will not be substantial, and will not change our qualitative demonstration of two-mode conductivity. Validity of the transient heating approach The measured conductivities in Fig. 3 and 4 were collected as the electron system cooled down following optical heating, using an optical pump/terahertz probe technique. Since the terahertz pulse has a nonzero width, the electron system is effectively probed over a (narrow) range of temperatures. In this section, we demonstrate that this technique introduces negligible error in the extracted scattering rates and electron temperatures. We model the sample response using a classical equation of motion, analogous to standard derivations of the Drude response, but with a time-varying scattering rate (denoted 𝜏𝜏s−1(𝑑𝑑)) to model the effect of temperature decay:

𝑚𝑚�̈�𝑥 + 𝑚𝑚�̇�𝑥𝜏𝜏s(𝑡𝑡)

= 𝑒𝑒𝐸𝐸�𝑑𝑑 − 𝑑𝑑probe�, (S6) where 𝑑𝑑probe indicates the arrival time of the incident pulse. To make direct comparisons with our experiment, we choose the functional form 𝜏𝜏s−1(𝑑𝑑) = 𝑖𝑖𝑘𝑘𝑇𝑇s(𝑑𝑑)/ℏ with the experimental value 𝑖𝑖 = 0.20. The temperature decay is given by 𝑇𝑇s(𝑑𝑑) = 77 K + 𝑇𝑇s(0)exp(−𝑑𝑑/𝑑𝑑d) with decay constant 𝑑𝑑d = 8 ps (Fig. S15C), similar to the decay behavior extracted from our data. To calculate the apparent optical conductivity of the model system described by Eq. S6, we numerically solve for �̇�𝑥(𝑑𝑑), using a Gaussian probing pulse 𝐸𝐸�𝑑𝑑 − 𝑑𝑑decay� as shown in Fig. S15A. This pulse is quite realistic for our experimental conditions, as can be judged from Fig. S15B, which compares the time derivative 𝑑𝑑𝐸𝐸/𝑑𝑑𝑑𝑑 of this pulse to the experimentally measured 𝑑𝑑𝐸𝐸/𝑑𝑑𝑑𝑑. The computed velocity �̇�𝑥(𝑑𝑑) is then converted to a current density using the expression

𝑗𝑗(𝑑𝑑) =𝐷𝐷gr𝜇𝜇=0�𝑇𝑇s(𝑡𝑡)�

𝜋𝜋�𝑚𝑚𝑒𝑒� �̇�𝑥(𝑑𝑑), (S7)

where 𝐷𝐷gr𝜇𝜇=0(𝑇𝑇s(𝑑𝑑)) is the temperature-dependent Drude weight of charge-neutral graphene. (We

note that in the standard Drude picture, 𝐷𝐷 = 𝑛𝑛𝑒𝑒2𝜋𝜋𝑚𝑚

; substituting this in for 𝐷𝐷gr𝜇𝜇=0(𝑇𝑇s(𝑑𝑑)), Eq. S7

reduces to the usual formula 𝑗𝑗(𝑑𝑑) = 𝑛𝑛𝑒𝑒�̇�𝑥.) The conductivity for a given time 𝑑𝑑probe is finally computed using the expression

𝜎𝜎(𝑖𝑖) = 𝑗𝑗(𝜔𝜔)𝐸𝐸(𝜔𝜔)

. We computed 𝜎𝜎(𝑖𝑖) for four different delays 𝑑𝑑probe (Fig. S15C). The real and imaginary parts of these computed conductivities (Fig. S15D and S15E) are seen to closely follow the Drude form. Using the Drude weight 𝐷𝐷gr

𝜇𝜇=0(𝑇𝑇) for charge-neutral graphene at electronic temperature 𝑇𝑇, we performed Drude fits to the computed conductivities with the temperature 𝑇𝑇 and scattering rate 𝜏𝜏−1 as fit parameters. The values (𝑇𝑇, 𝜏𝜏−1) retrieved from the fits (Fig. S15F, open circles) are seen to fall very close to the parameters (𝑇𝑇s, 𝜏𝜏s−1) of the equation of motion at

7

time 𝑑𝑑 = 𝑑𝑑probe (Fig. S15F, crosses). These results confirm that the transient cooling effect introduces negligible error in our extraction of the scattering rate and electron temperature. Disorder scattering at charge neutrality In charge-neutral graphene, unscreened and singly charged impurities of density 𝑛𝑛imp produce a scattering rate 𝜏𝜏d−1 given by (12)

𝜏𝜏d−1 = 𝜋𝜋4

27𝜁𝜁(3)𝑛𝑛imp

ℏ� 𝑒𝑒2

4𝜋𝜋𝜖𝜖0𝜖𝜖�2 1𝑖𝑖B𝑇𝑇e

, where 𝜁𝜁(3) ≈ 1.202 and 𝜖𝜖 is the effective dielectric constant of the medium. We assume 𝜖𝜖 = 4 for graphene encapsulated in hBN. Drude weights of the zero- and finite-momentum modes The Drude weight 𝐷𝐷F of the finite-momentum mode is (8, 12)

𝐷𝐷F = 𝜋𝜋(𝑛𝑛𝑒𝑒𝑣𝑣F)2/𝑊𝑊, where 𝑛𝑛𝑒𝑒 is the charge density, 𝑣𝑣F is the Fermi velocity, and 𝑊𝑊 is the enthalpy density:

𝑊𝑊 = 3(𝑖𝑖B𝑇𝑇e)3

𝜋𝜋(ℏ𝑣𝑣F)2 ∫ 𝑘𝑘�2 � 1𝑒𝑒𝑘𝑘�−𝜇𝜇/(𝑘𝑘B𝑇𝑇e)+1

+ 1𝑒𝑒𝑘𝑘�+𝜇𝜇/(𝑘𝑘B𝑇𝑇e)+1

� 𝑑𝑑𝑘𝑘�∞0 .

The Drude weight of the zero-momentum mode is 𝐷𝐷Z = 𝐷𝐷gr − 𝐷𝐷F, where

𝐷𝐷gr = 2 𝑒𝑒2

ℏ2𝑘𝑘B𝑇𝑇e log �2 cosh � 𝜇𝜇

2𝑖𝑖B𝑇𝑇e��

is the total Drude weight of graphene. Data from an additional sample

We studied the scattering rate as a function of electron temperature at charge neutrality on another sample of similar construction, which we refer to as “Sample 2” (see photograph in Fig. S16). Owing to a noisy terahertz emitter, the data from Sample 2 are substantially noisier and more limited than the data from the sample described in the main text (“Sample 1”). Nonetheless, we were able to measure the conductivity change as a function of optical pump fluence to a reasonable degree of precision (Fig. S17A,B). These measurements are similar to those in Fig. 3 of the main text, except that here we have varied the fluence rather than the pump-probe delay time.

Following the procedure used in Fig. 3, we performed fits to a difference in Drude conductivities and extracted the scattering rate 𝜏𝜏−1 as a function of electron temperature 𝑇𝑇e (Fig. S17C). The scattering rates of Sample 2 are again found to scale approximately linearly with electron temperature, with a very similar slope to that observed in Sample 1: fitting directly to the expression 𝜏𝜏ee−1 = 𝑖𝑖𝑘𝑘B𝑇𝑇e/ℏ, we find 𝑖𝑖 = 0.16 (𝛼𝛼 = 0.21) for Sample 2, similar to the value 𝑖𝑖 = 0.20 (𝛼𝛼 = 0.23) found for Sample 1. We note that compared to Sample 1, Sample 2 shows a much less pronounced upturn in the scattering rate at low temperatures. We hesitate to interpret this strongly given the large noise in this dataset.

The conductivity data for Sample 2 (extracted as for Fig. 3) are seen to collapse onto the universal curve 𝜎𝜎U (Fig. S17D) with 𝑖𝑖 = 0.16. We suggest that the quantum-critical scattering observed at charge neutrality in both Samples 1 and 2 is a robust feature of clean graphene.

8

Fig. S1. Large-area photograph of the waveguide device.

Fig. S2. Cross-sectional view of the heterostructure beneath the waveguide electrodes.

Fig. S3. Fast Fourier transforms of the pulses in Fig. 2A, inset.

9

Fig. S4. Determination of charge neutrality. (A) Current 𝐼𝐼 measured at 50 Hz, proportional to the transmitted waveform 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑, as a function of the voltage difference 𝑑𝑑gate − 𝑑𝑑graphene. The emitter-detector delay is held fixed to the peak of the transmitted waveform, and 𝑑𝑑graphene =−2 V. A larger current corresponds to higher transmission. Charge neutrality is reached at 𝑑𝑑gate − 𝑑𝑑graphene = −0.53 V, the gate voltage of maximum transmission. (B) Current 𝛥𝛥𝐼𝐼 simultaneously measured at 275 Hz, proportional to the transmission change 𝑑𝑑(𝛥𝛥𝑑𝑑)/𝑑𝑑𝑑𝑑 upon optically heating the graphene. The minimum of this signal corresponds to charge neutrality. Inset: finer scan of 𝛥𝛥𝐼𝐼 over a smaller gate voltage range, demonstrating that charge neutrality can be determined to within ~10 mV in 𝑑𝑑gate, which translates to ~7 meV in 𝜖𝜖F.

10

Fig. S5. Repeating element of the transmission line model of our waveguide, characterized by impedance 𝑍𝑍′𝛥𝛥𝑧𝑧 along the waveguide traces and admittance 𝑌𝑌′𝛥𝛥𝑧𝑧 between the traces.

Fig. S6. Optical analogue of our experiment. The graphene region of length 𝑑𝑑 and unknown refractive index 𝑛𝑛1 is immersed in a medium of known refractive index 𝑛𝑛0.

11

Fig. S7. Modeling serial impedance beneath waveguide traces. (A) Top-down view of the waveguide traces (gold color) and the graphene flake (blue color). A sliver of length 𝛥𝛥𝑧𝑧, as shown, is analyzed in the text. (B) Cross-sectional view of the sliver of length 𝛥𝛥𝑧𝑧, modeling the region beneath the traces as RC networks with repeating unit of length 𝛥𝛥𝑥𝑥, resistance 𝑅𝑅t = 𝜌𝜌t𝛥𝛥𝑥𝑥/𝛥𝛥𝑧𝑧, and capacitance 𝑖𝑖t = 𝑐𝑐t𝛥𝛥𝑥𝑥𝛥𝛥𝑧𝑧. The resistance between the traces is 𝜌𝜌𝑤𝑤/𝛥𝛥𝑧𝑧, where 𝜌𝜌 = 𝜎𝜎−1 is the resistivity of graphene in this region.

12

Fig. S8. Calculated effect of 𝑍𝑍ser on total conductance 𝐺𝐺. (A) Real and (B) imaginary parts of 𝐺𝐺 calculated from Eq. S5 for our experimental geometry (𝑑𝑑 = 9 micron, 𝑤𝑤 = 14 micron, 𝑠𝑠 = 16 micron). The graphene region between the traces is held at charge neutrality and an electron temperature of 300 and 77 K, with corresponding scattering rates 8.5 and 4 THz, respectively. Different line-styles show different assumptions about 𝑍𝑍ser: solid curves assume 𝑍𝑍ser = 0, dashed curves assume 𝑍𝑍ser modeled by an infinite RC network, and dotted curves assume 𝑍𝑍ser modeled by a 16 micron RC network. The graphene beneath the waveguide traces is characterized by 𝜖𝜖F = 106 meV, 𝑇𝑇 = 77 K, and 𝜏𝜏−1 = 0.6 THz. (C) Real and (D) imaginary parts of 𝐺𝐺 calculated as for (A) and (B), but at 𝜖𝜖F = 119, 46, and 33 meV between the traces. The doping beneath the traces is 159, 116, and 111 meV, respectively, as expected for 𝑑𝑑graphene = −2 V. The scattering rate is 0.6 THz and the temperature is 77 K throughout the sheet.

13

Fig. S9. Mode of the empty waveguide. (A) Vector plot of the propagating odd mode relevant to our experiment, simulated at 𝑖𝑖 = 3 × 1012 rad/s using the COMSOL 2D mode solver. The half-plane 𝑦𝑦 < 0 contains the fused quartz substrate, while the half-plane 𝑦𝑦 > 0 contains vacuum. The waveguide traces, modeled as perfect electrical conductors, are shown as dark rectangles of width 16 micron and thickness 0.2 micron in the region 0 < 𝑦𝑦 < 0.2 micron. The longitudinal electric field 𝐸𝐸𝑖𝑖 is three orders of magnitude smaller than 𝐸𝐸𝑥𝑥 and 𝐸𝐸𝑦𝑦. (B) Colormap of the horizontal electric field 𝐸𝐸𝑥𝑥 in units of 𝑑𝑑/𝑤𝑤, where 𝑑𝑑 is the voltage between the traces and 𝑤𝑤 =14 micron is the gap between the traces. (C) Horizontal cut of 𝐸𝐸𝑥𝑥 between the traces at vertical position 𝑦𝑦 = 0.

14

Fig. S10. Waveguide mode for heavily doped graphene. (A) Colormap of the horizontal electric field 𝐸𝐸𝑥𝑥 at 𝑖𝑖 = 1 × 1012 rad/s in units of 𝑑𝑑/𝑤𝑤, where 𝑑𝑑 is the voltage between the traces and 𝑤𝑤 =14 micron is the gap between the traces. The graphene (dark line) is located at 𝑦𝑦 = 0 and −23 micron < 𝑥𝑥 < 23 micron. The traces of width 16 micron (dark rectangles) are 48 nm above the graphene sheet, following our experimental geometry. The conductivity 𝜎𝜎 of graphene between the traces corresponds to 𝜖𝜖F = 119 meV, while the conductivity 𝜎𝜎t beneath the traces corresponds to 𝜖𝜖F = 159 meV; the scattering rate is 0.6 THz and the temperature is 77 K throughout the sheet. (B) Horizontal cut of 𝐸𝐸𝑥𝑥 within the graphene and between the traces. (C-F) Analogous to (A,B), but at 𝑖𝑖 = 3 × 1012 rad/s (C,D) and 𝑖𝑖 = 5 × 1012 rad/s (E,F). The period of oscillation beneath the traces agrees with the period calculated from our RC network model of the serial impedance.

15

Fig. S11. Waveguide mode for charge-neutral graphene. Analogous to Fig. S10, except that conductivity 𝜎𝜎 of graphene between the traces corresponds to 𝜖𝜖F = 0, 𝜏𝜏−1 = 8.5 THz, and 𝑇𝑇 = 293 K, while the conductivity 𝜎𝜎t beneath the traces corresponds to 𝜖𝜖F = 106 meV, 𝜏𝜏−1 = 0.6 THz, and 𝑇𝑇 = 77 K.

16

Fig. S12. Verifying the transmission line and series impedance models at high doping. (A) Real and (B) imaginary parts of the conductivity 𝜎𝜎 (solid curves) corresponding to 𝜖𝜖F = 119 meV used as input to the simulation, plotted alongside 𝜎𝜎sim extracted from the simulation (filled circles) using the transmission line model and a 16 micron RC network for 𝑍𝑍ser (Eq. S5). The conductivity 𝜎𝜎t beneath the traces, used both as input to the simulation and to calculate 𝑍𝑍ser, corresponds to 𝜖𝜖F =159 meV, with 𝜏𝜏−1 = 0.6 THz and 𝑇𝑇 = 77 K throughout the graphene sheet.

Fig. S13. Error in the transmission line model in studies of charge neutrality. (A) Real and (B) imaginary parts of the conductivity 𝜎𝜎 (solid curves) used as input to the simulation, plotted alongside 𝜎𝜎sim extracted from the simulation (filled circles). The conductivity 𝜎𝜎 between the traces corresponds to 𝜖𝜖F = 0, 𝜏𝜏−1 = 8.5 THz, and 𝑇𝑇 = 293 K (red curves and markers) or 𝜖𝜖F = 0, 𝜏𝜏−1 = 3.9 THz, and 𝑇𝑇 = 77 K (blue curves and markers). The input conductivity 𝜎𝜎t beneath the traces corresponds to 𝜖𝜖F = 106 meV, with 𝜏𝜏−1 = 0.6 THz, and 𝑇𝑇 = 77 K. The series impedance 𝑍𝑍ser is negligible and is ignored in extracting 𝜎𝜎sim.

17

Fig. S14. Fitting temperature-dependent change in transmission at charge neutrality using simulated transmission. (A) Real and (B) imaginary parts of the measured relative change in transmission 𝛥𝛥𝑑𝑑/𝑑𝑑 upon heating the electron system to unknown temperature 𝑇𝑇e with the optical pump; these data are the same as those shown in Fig. 3 of the main text, where the data were interpreted as the conductivity change 𝛥𝛥𝜎𝜎 = − 2

𝑍𝑍0𝑤𝑤𝛥𝛥𝑡𝑡𝑡𝑡

using the transmission line model. Solid curves show fits to a difference in simulated relative transmissions 𝛥𝛥𝑑𝑑sim/𝑑𝑑sim = 𝑑𝑑sim(𝑖𝑖; 𝜏𝜏−1,𝑇𝑇e)/𝑑𝑑sim(𝑖𝑖; 𝜏𝜏0−1,𝑇𝑇0) − 1. The parameters 𝜏𝜏−1, 𝑇𝑇e, and 𝜏𝜏0−1 are free fit parameters, with 𝑇𝑇0 = 77 K. (C) Scattering rate versus temperature extracted from the fits. The scattering rate is well described by a sum of scattering rates 𝜏𝜏−1 = 𝜏𝜏ee−1 + 𝜏𝜏d−1, with 𝜏𝜏ee−1 = 0.16 𝑘𝑘B𝑇𝑇e/ℏ due to charge-carrier interactions and 𝜏𝜏d−1 ∝ 𝑛𝑛imp𝑇𝑇e−1 due to charged impurities (𝑛𝑛imp = 1.3 × 109 cm−2). (D-F) Reproduction of Fig. 3 in the main text, in which the same dataset (interpreted as 𝛥𝛥𝜎𝜎) is fit using a difference in Drude functions 𝜎𝜎(𝑖𝑖; 𝜏𝜏−1,𝑇𝑇e) − 𝜎𝜎(𝑖𝑖; 𝜏𝜏0−1,𝑇𝑇0).

18

Fig. S15. Estimating error due to transient heating approach. (A) Incident Gaussian pulse 𝐸𝐸(𝑑𝑑 − 𝑑𝑑delay) used in the numerical model. (B) Time derivative 𝑑𝑑𝐸𝐸/𝑑𝑑𝑑𝑑 of the pulse used in the model (blue curve), displayed alongside the measured time derivative of the experimental pulse (red curve; Fig. 2A, inset). (C) Time-dependent temperature 𝑇𝑇s(𝑑𝑑) used in the model (Eq. S6). The right vertical axis shows the corresponding scattering rates 𝜏𝜏s−1(𝑑𝑑). Probe pulses are chosen to arrive at four different delay times 𝑑𝑑delay, as indicated by the colored crosses. (D) Real and (E) imaginary parts of the optical conductivity extracted using Eq. S7. Different colors correspond to the different probe delays in (C). Solid curves are best fits using a Drude model with free parameters 𝜏𝜏−1,𝑇𝑇. (F) Values of 𝜏𝜏−1,𝑇𝑇 extracted from the fits (open circles) compared to exact values of 𝜏𝜏s−1,𝑇𝑇s (crosses) at each probe delay time 𝑑𝑑delay. The fit results are seen to very nearly reproduce the exact values. For reference, the dashed line shows 𝜏𝜏s−1 = 0.20 𝑘𝑘B𝑇𝑇s/ℏ.

19

Fig. S16. Photograph of Sample 2. Scale bar is 15 micron.

20

Fig. S17. Quantum-critical scattering at charge neutrality in Sample 2 (analogous to Fig. 3). (A) Real and (B) imaginary parts of the change in optical conductivity for four different fluences absorbed by graphene. For all fluences, the terahertz probe pulse was timed to arrive at a fixed delay ~3 ps after the optical pump. Solid curves show fits to the change in optical conductivity, with free parameters 𝜏𝜏−1,𝑇𝑇e. (C) Values of 𝜏𝜏−1,𝑇𝑇e extracted from the fits. The scattering rate is seen to be linear (𝜏𝜏ee−1 = 𝑖𝑖 𝑘𝑘B𝑇𝑇e/ℏ), with a similar slope 𝑖𝑖 = 0.16 to that of the sample described in the main text. (D) Real and imaginary parts (open and filled circles) of 𝜎𝜎 at different 𝑇𝑇e (i.e., different pump fluences), replotted as a function of ℏ𝑖𝑖/𝑘𝑘B𝑇𝑇e. Solid and dashed curves are fits to the real and imaginary parts of the universal function 𝜎𝜎U given in the main text; the fits yield 𝑖𝑖 =0.16.

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