Abstract
Conducting materials typically exhibit either diffusive or ballistic charge transport. When electron–electron interactions dominate, a hydrodynamic regime with viscous charge flow emerges^{1,2,3,4,5,6,7,8,9,10,11,12,13}. More stringent conditions eventually yield a quantumcritical Diracfluid regime, where electronic heat can flow more efficiently than charge^{14,15,16,17,18,19,20,21,22}. However, observing and controlling the flow of electronic heat in the hydrodynamic regime at room temperature has so far remained elusive. Here we observe heat transport in graphene in the diffusive and hydrodynamic regimes, and report a controllable transition to the Diracfluid regime at room temperature, using carrier temperature and carrier density as control knobs. We introduce the technique of spatiotemporal thermoelectric microscopy with femtosecond temporal and nanometre spatial resolution, which allows for tracking electronic heat spreading. In the diffusive regime, we find a thermal diffusivity of roughly 2,000 cm^{2} s^{−1}, consistent with charge transport. Moreover, within the hydrodynamic time window before momentum relaxation, we observe heat spreading corresponding to a giant diffusivity up to 70,000 cm^{2} s^{−1}, indicative of a Dirac fluid. Our results offer the possibility of further exploration of these interesting physical phenomena and their potential applications in nanoscale thermal management.
Main
During the last few years, signatures of viscous charge flow in the socalled Fermiliquid hydrodynamic regime were observed in twodimensional (2D) electron systems, especially graphene, using electrical device measurements^{7,8,9,11,12} and scanning probe microscopy^{10,13,22}. A second hydrodynamic regime, which has no analogue in classical fluids, can occur very close to the Dirac point. When the Fermi temperature (T_{F} = E_{F}/k_{B}, where E_{F} is the Fermi energy and k_{B} is the Boltzmann constant) becomes small compared to the electron temperature T_{e}, the system becomes a quantumcritical fluid^{3,6,14,15,17}. In this Diracfluid regime, the nonrelativistic description of the viscous fluid is replaced by its ultrarelativistic counterpart, which accounts for the presence of both particles and holes, as well as for their linear energy dispersion. In line with theoretical predictions in this regime^{15}, electrical measurements at cryogenic temperatures indicated a deviation from the Wiedemann–Franz law^{19} and from the Mott relation^{20}, and a terahertzprobe study revealed the quantumcritical carrier scattering rate^{21}.
Here, we follow electronic heat flow in the diffusive and hydrodynamic regimes at room temperature, and demonstrate a controlled Fermiliquid to Diracfluid crossover, with a strongly enhanced thermal diffusivity close to the Dirac point. These observations are enabled by ultrafast spatiotemporal thermoelectric microscopy, a technique inspired by alloptical spatiotemporal diffusivity measurements^{23,24,25}, with the crucial difference that the observable is the thermoelectric current, which is directly, and exclusively, sensitive to electronic heat^{26}. We use a hexagonal boron nitride (hBN)encapsulated graphene device that is both a Hall bar for electrical measurements and a splitgate thermoelectric detector (Fig. 1a). Since we use ultrashort laser pulses, with an approximate instrument response time (Δt_{IRF} where IRF means instrument response function) of 200 fs, to generate electronic heat, we are able to examine the system before momentum relaxation occurs, as we measure a momentum relaxation time, τ_{mr}, around 350 fs (Extended Data Fig. 1). In this temporal regime before momentum is relaxed, we enter the hydrodynamic window, because the electron–electron scattering time τ_{ee} is <100 fs (ref. ^{27}), that is τ_{ee} < Δτ_{IRF} < τ_{mr}. This is a different approach compared to most previous studies, where hydrodynamic effects were observed by using small system dimensions L to eliminate effects of momentum relaxation, that is v_{F}τ_{ee} < L < v_{F}τ_{mr} (refs. ^{7,8,9,10,11,12,13,19,22}) (v_{F} = 10^{6} m s^{−1} is the Fermi velocity). Our approach furthermore exploits elevated carrier temperatures, which greatly increases the accessibility of the Diracfluid regime, as for increasing carrier temperatures the crossover occurs increasingly far away from the Dirac point^{14,17} (Fig. 1b). As we will show, during the hydrodynamic window substantially more efficient heat spreading occurs in the Diracfluid regime than in the Fermiliquid regime and in the diffusive regime (Fig. 1c,d).
Our technique works by using two ultrafast laser pulses that produce localized spots of electronic heat within tens of femtoseconds^{27}. These spots are characterized by an increased carrier temperature T_{e} > T_{l}, with T_{l} the lattice temperature (300 K). The degree of spatial spreading of these electronic heat spots as a function of time is governed by the diffusivity D. We control the relative spatial and temporal displacement, Δx and Δt, of the two pulses with sub100nm spatial precision and roughly 200 fs temporal resolution. Each laser pulse is incident on opposite sides of a pn junction at a distance ∆x/2 from the junction. This pn junction is created by applying opposite voltages ±ΔU with respect to the Dirac point voltages to the two backgates that form a splitgate structure. The two photogenerated electronic heat spots spread out spatially and part of the heat can reach the pn junction after a certain amount of time, generating a thermoelectric current at the junction through the Seebeck gradient^{26}. The small region of the pn junction thus serves as a local probe of the electron temperature. While each of the heat spots can create thermoelectric current independently, we obtain spatiotemporal information by examining exclusively the signal that corresponds to heat generated by one of the pulses interacting with heat generated by the other pulse—the interacting heat current, ΔI_{TE}. Since the thermoelectric photocurrent scales sublinearly with incident power, we can isolate this interacting heat current ΔI_{TE} by modulating each laser beam at a different frequency, f_{1} and f_{2}, and demodulating the thermoelectric current at the difference frequency f_{1} − f_{2}. As illustrated in Fig. 1e,f, the higher the diffusivity D, the more interacting heat current ΔI_{TE} remains for increasing Δx and Δt.
Figure 2a shows the measured interacting heat current ∆I_{TE} as a function of Δx and Δt. As expected, the largest ∆I_{TE} occurs for the largest spatiotemporal overlap at the pn junction (Δx = Δt = 0). For increasing ∆t, we find that the normalized signal extends further spatially, indicating the occurrence of heat spreading (Fig. 2b). This spatial spread is quantified via the second moment <Δx^{2}>, which quantifies the width of the profile at different time delays (Methods). Similar to recent alloptical spatiotemporal microscopy^{24,25}, we obtain spatial information beyond the diffraction limit by precise spatial sampling of diffractionlimited profiles. The experimentally obtained spatial spread as a function of Δt (Fig. 2c) is very similar to the calculated results (Fig. 2d), obtained by simulating the experiment with a given diffusivity, D (Methods and Supplementary Note 1). The white lines indicate the values of the spatial spread <Δx^{2}> for different Δt. We also compare the simulated spatial spread <Δx^{2}> versus Δt (blue dashed line in Fig. 2e) with the theoretical expectation according to the heat diffusion equation, <Δx^{2}> = <Δx^{2}>_{focus} + 2DΔt (dashdotted line in Fig. 2e). Here, D is the same diffusivity that was used as input for the simulation, and <Δx^{2}>_{focus} is the minimum second moment from the two overlapping pulses (Supplementary Note 2 and Supplementary Figs. 1–4). The initial slope is the same for both the simulated heat spreading and the theoretical spreading following the heat diffusion equation.
We first discuss the experimental results in the diffusive regime, where Δt > τ_{mr}. For three different gate voltage combinations, corresponding to Fermi energies between 75 and 190 meV (T_{F} = 900 − 2,200 K), we extract the spatial spread as a function of time delay (symbols in Fig. 2e), and compare it with the results from simulations (solid lines). For these simulations, we have used the diffusivity values that we obtain directly from electrical measurements of charge mobility μ on the same device (Extended Data Fig. 1), and the relation between mobility and diffusivity: D = μE_{F}/2e (Methods). We find excellent agreement, if we account for shortlived ultrafast heat spreading around Δt = 0, which leads to a largerthanexpected initial spread at time zero <Δx^{2}>_{min}, as we will explain below. The agreement between the measured heat spread for Δt > τ_{mr} and the one calculated using the measured charge mobilities shows that electronic heat and charge flow together, as expected in the diffusive regime. Furthermore, it confirms that our technique is a reliable method for obtaining thermal diffusivities in a quantitative manner.
We now turn to the nondiffusive regime, by exploring the behaviour in the hydrodynamic window, where Δt < τ_{mr}. The experimentally obtained spatial spreads start at a minimum value <Δx^{2}>_{min} larger than 2 μm^{2}, rather than starting at an expected <Δx^{2}>_{focus} = 0.56 μm^{2}. A second device of hBNencapsulated graphene with similar mobility reproduces this largerthanexpected spatial spread at time zero (Supplementary Note 3 and Extended Data Fig. 2). We exclude the possibility of an experimental artefact such as an underestimation of the laser spot size, since we repeated the measurements while scanning through the laser focus, and measured the focus size (Supplementary Figs. 1–4). Furthermore, we observe that the offset depends on the Fermi energy, while keeping all other experimental parameters fixed. Finally, we measured a third device with a lower charge mobility and shorter hydrodynamic time window: τ_{mr} < 100 fs, which is smaller than Δt_{IRF}. This device exhibits systematically less heat spreading around time zero (Supplementary Note 4 and Extended Data Fig. 3), consistent with its smaller hydrodynamic time window. We therefore attribute the large experimentally observed minimum <Δx^{2}>_{min} in Fig. 2e to ultrafast initial heat spreading that occurs before momentum relaxation takes place, Δt ≲ 350 fs (see the schematic illustration of spatiotemporal heat spreading in Fig. 1d). The dynamics of this initial heat spreading are washed out by the finite time resolution Δt_{IRF}, and manifests as a large minimum <Δx^{2}>_{min} at time zero. The observed initial spatial spread suggests a thermal diffusivity of D = (<Δx^{2}>_{min} − <Δx^{2}>_{focus})/2Δt_{IRF} ≅ 70,000 cm^{2} s^{−1} for the lowest measured E_{F} of 75 meV. Simulations of heat spreading with an input diffusivity of 100,000 cm^{2} s^{−1} are indeed consistent with the experimentally observed spread in the hydrodynamic window (the red line in Fig. 2e).
We attribute this observation of highly efficient initial heat spreading to the presence of the quantumcritical electronhole plasma. We can exclude that the observed initial spreading is the result of ballistic transport, as we calculate that the ballistic contribution to initial heat spreading would give only <Δx^{2}>_{ball} = 0.68 μm^{2} (Supplementary Note 2 and Extended Data Fig. 4). Besides, ballistic transport has a very weak dependence (<10%) on carrier density in this range, as the Fermi velocity does not change appreciably for the Fermi energies considered here^{28}. The reason for the high diffusivity in the Diracfluid regime is that the hot electrons and hot holes that coexist in this regime move in the same direction under a thermal gradient, with interparticle scattering events conserving total momentum^{19}. We note that typical transport measurements probe the sum of the momentumconserving thermal resistivity (due to carrier–carrier scattering) and the momentumnonconserving thermal resistivity (due to carrier–impurity and carrier–phonon scattering), where the latter dominates at room temperature. The ability of our technique to interrogate the system during the 350 fs before any momentumnonconserving scattering occurs, means that this contribution to the overall resistivity is negligible. Therefore, we probe exclusively the momentumconserving thermal conductivity, which diverges towards the Dirac point and towards infinite electron temperature.
To provide further evidence of hydrodynamic heat transport, we demonstrate the ability to control the crossover between the Fermiliquid and quantumcritical Diracfluid regimes via the ratio T_{e}/T_{F}, by independently varying T_{e} via the incident laser power and T_{F} via the applied gate voltage. A larger ratio results in less Coulomb screening and correspondingly stronger hydrodynamic effects due to electron–electron interactions. If T_{e} is larger than T_{F}, electrons and holes coexist and the Diracfluid regime becomes accessible (Fig. 1b). We perform spatial scans in the hydrodynamic window at a temporal delay of Δt = 0, in a geometry with one laser pulse impinging on the junction, while scanning the other pulse across (x axis) and along (y axis) the junction region. Figure 3a–d shows four representative spatial ∆I_{TE} maps with varying T_{e}/T_{F}, yet similar signal magnitudes. Clearly, the signal is broader for larger T_{e}/T_{F}, indicating faster thermal transport. We repeat these measurements for a range of T_{e} and T_{F} values and quantify the initial heat spreading using Gaussian functions, with widths σ_{x} and σ_{y}, to describe ∆I_{TE} at Δt = 0 as a function of Δx or Δy (Fig. 3e,f and Supplementary Fig. 5). As expected for a crossover from the diffusive Fermiliquid regime to the hydrodynamic Diracfluid regime, both spatial spreads σ_{x} and σ_{y} increase substantially for increasing ratio T_{e}/T_{F}. These spreads correspond to a diffusivity up to 40,000 cm^{2} s^{−1} (Methods), similar to the 70,000 cm^{2} s^{−1} we found earlier.
We compare our experimental results to Boltzmann transport calculations following refs. ^{6,18}, including carrier interactions and longrange impurity scattering. We model impurities as Thomas–Fermi screened Coulomb scatterers of density 0.24 × 10^{12} cm^{−}^{2}. Figure 3g shows the calculated thermal diffusivity D as a function of T_{F} and T_{e}, when considering only the hydrodynamic term due to electron–electron interactions, relevant in the hydrodynamic window where Δt < τ_{mr}. A higher electron temperature, or lower Fermi temperature, leads to strongly increased diffusivity, signalling a crossover from the Fermiliquid to the Diracfluid regime. This is the same qualitative trend as for the experimental data taken at Δt = 0 in Fig. 3e–f, where a larger initial width originates from a larger diffusivity, thus supporting our interpretation of a hydrodynamic crossover.
A more quantitative comparison shows that the calculated D in the diffusive regime is around 2,000 cm^{2} s^{−1} (Fig. 3h), in quantitative agreement with the experiment in the diffusive regime. The obtained thermal diffusivity in the hydrodynamic window close to the Dirac point reaches values above 100,000 cm^{2} s^{−1}, even higher than our experimental estimates of 35,000–70,000 cm^{2} s^{−1}. Using the calculated diffusivities, we estimate the spatial spread at time zero σ_{calc} (Methods), as shown in Fig. 3g. These are similar to the experimentally obtained ones, thus confirming our conclusion of highly efficient heat spreading in the Diracfluid regime at room temperature, with a diffusivity that is almost two orders of magnitude larger than in the diffusive regime. We note that the theoretical calculations predict that even higher diffusivities are attainable.
Finally, we discuss the threedimensional (3D) thermal conductivity, to assess the ability to transport useful amounts of heat. We find roughly 100 W m^{−}^{1} K^{−1} in the diffusive regime (Methods), in agreement with ab initio calculations^{29}. In the Diracfluid regime, with an electron temperature of roughly 1,000 K, we obtain a thermal conductivity of 18,000–40,000 W m^{−}^{1} K^{−1}. This is in agreement with ref. ^{15}, where values up to 100,000 W m^{−}^{1} K^{−1} were predicted theoretically for large T_{e}/T_{F}. The thermal conductivity we obtain is about three orders of magnitude larger than the one obtained in the Diracfluid regime at cryogenic temperatures^{19}. Our results show that in the Diracfluid window the electronic contribution to heat transport can be much larger than the phononic contribution with a conductivity of >2,000 W m^{−}^{1} K^{−1} (ref. ^{30}), which is already exceptionally high and can also be enhanced hydrodynamically, as shown recently^{31}. Thus, the Dirac electronhole plasma can contribute strongly to thermal transport, extracting heat from hot spots much faster than predicted by classical limits.
In conclusion, our results show that the—until recently unreachable—physical phenomena associated with the Dirac fluid do not only offer an exciting playground for interesting physical phenomena, but also hold great promise for applications, for example in thermal management of nanoscale devices. We note that the quantumcritical behaviour can be switched on and off using a modest gate voltage and in systems prepared by standard fabrication techniques. Finally, we believe that the optoelectronic technique we have introduced—with the potential of increased spatial accuracy and temporal resolution—will be a valuable tool to reach a better understanding of the thermal behaviour of a broad range of quantum materials, with great promise for new technological applications.
Methods
Fabrication of splitgate thermoelectric device
The splitgate device with Hall geometry consists of exfoliated, single layer graphene encapsulated by hBN, prepared using standard exfoliation and dry transfer techniques. The hBNgraphenehBN stack is placed on a predefined splitgate structure made of graphene, grown by chemical vapour deposition, where the gap between the two gates is roughly 100 nm, created via electronbeam lithography and reactive ion etching (RIE). The top hBN and graphene are etched into a Hall bar shape with laser lithography and RIE, keeping the splitgate intact and not etching completely through the bottom hBN. Finally, the Ti/Au side contacts are created by a further step of lithography, RIE and metal evaporation. The fabrication steps are shown in Supplementary Fig. 6.
Spatiotemporal thermoelectric current microscopy setup
Our setup enables us to follow electronic heat spreading in space and time, because we use the thermoelectric signal generated by electronic heat interacting at a fixed location (the pn junction), while we vary the spatial displacement of our two laser pulses with respect to this junction and vary the temporal delay between the two ultrashort pulses. This means that we are following in space and time the diffusion of lightinduced electronic heat from the location of light incidence to the pn junction. It is the thermoelectric effect at the pn junction, governed by the Seebeck coefficient, that generates our observable signal, the thermoelectric current. We note that although the value of the Seebeck coefficient itself changes when changing E_{F}, and when entering the hydrodynamic regime^{20}, this only affects the magnitude of the thermoelectric current, rather than how electronic heat is diffusing outside the pn junction, which is what we are following with our spatiotemporal technique.
A sketch of the setup is shown in Supplementary Fig. 7. A Ti:sapphire oscillator (886nm centre wavelength, 76MHz repetition rate), is split into two beam paths. Both beams are modulated with optical choppers, at frequencies f_{1} = 741 and f_{2} = 529 Hz. The relative time delay between the two pulses is controlled by a mechanical delay line. The spatial offset of one beam with respect to the other is controlled with a mirror galvanometer, while the position of the sample with respect to the beams is controlled with a piezo scanning stage. The beams are focused onto the sample with a ×40 0.6 numerical aperture objective lens. We collect the thermoelectric (TE) photocurrent between the source and drain contacts through the graphene sheet on either side of the junction via lockin amplification. By demodulating the current signal at the difference frequency of the two modulation frequencies, f_{2} − f_{1} = 211.7 Hz, we isolate the signal caused by the interaction of both heating sources, which we call the interacting heat current ∆I_{TE}. The temporal resolution of the setup of 200 fs is determined in the sample plane of the microscope (Supplementary Fig. 8). The spatial resolution defined by our spot sizes is below 1 μm, whereas the accuracy with which we can observe electronic heat spreading is determined by the signaltonoise ratio, and is estimated to be below 100 nm.
Estimating Fermi temperature controlled by gate voltage
During photocurrent measurements, the gate voltage U_{x} is applied to one side (x is ‘A’) or the other side (x is ‘B’) side of the split gate. We always apply a symmetric voltage around the experimentally determined Dirac point voltage \(U_{{{\mathrm{x}}}}^{{{{\mathrm{DP}}}}}\): \(U_{{{\mathrm{A}}}} = U_{{{\mathrm{A}}}}^{{{{\mathrm{DP}}}}} + {\Delta} U\) and \(U_{{{\mathrm{B}}}} = U_{{{\mathrm{B}}}}^{{{{\mathrm{DP}}}}}  {\Delta} U\). The gate electrode and the graphene form a capacitor with the dielectric hexagonal boron nitride (hBN), with a thickness of t_{hBN} = 70 nm, and a relative permittivity of \({\it{\epsilon }}_{{{{\mathrm{hBN}}}}} = 3.56\). The carrier density n is calculated via \(n = \frac{{{\it{\epsilon }}_0{\it{\epsilon }}_{{{{\mathrm{hBN}}}}}}}{{e\,t_{{{{\mathrm{hBN}}}}}}}{\Delta} U\), where \(\epsilon_{0}\) is the vacuum permittivity. We calculate the Fermi energy E_{F} and the Fermi temperature T_{F} via \(E_{{{\mathrm{F}}}}^2\) = πħ^{2}\(v_{{{\mathrm{F}}}}^2 \cdot n\) and \(T_{{{\mathrm{F}}}} = \frac{{E_{{{\mathrm{F}}}}}}{{k_{{{\mathrm{B}}}}}},\) where k_{B} is the Boltzmann constant.
Estimating carrier temperature controlled by laser power
The thermoelectric photovoltage is assumed to be proportional to the timeaveraged increase of the electronic temperature T_{e} above the ambient temperature T_{0}, as in ref. ^{32}. The sublinear dependence of the thermoelectric current I_{TE} on optical power for the device under study here for illumination with a single pulsed laser (λ = 886 nm) is shown in Supplementary Fig. 9. With a linear temperature scaling of the electronic heat capacity for graphene away from the Dirac point, C_{e}(T) = γT, we integrate the heat energy per unit area dQ = C_{e}dT, that is, \(\mathop {\smallint }\limits_{Q_0}^{Q_0 + {\Delta} Q} {{{\mathrm{d}}}}Q = \mathop {\smallint }\limits_{T_0}^{T_e} \gamma T{{{\mathrm{d}}}}T\). With the incident power P proportional to the absorbed heat energy per unit area ∆Q, we find that the peak T_{e} as a function of the laser power P scales as in ref. ^{32}, \(T_e = \root {2} \of {{T_0^2 + bP}}\). Here, the parameter b is defined via bP = 2∆Q/γ, and is used to convert incident power to peak electron temperature (Supplementary Fig. 9).
Simulation of the experiment
A detailed description of the simulation can be found in Supplementary Note 1 and Supplementary Fig. 10. In brief, we calculate the spatiotemporal evolution of electronic heat generated by the two optical pulses in the graphene sheet via the heat equation with a finite difference method. We define Gaussian heating pulses and calculate their temperature rise via the experimentally measured nonlinear power scaling. We extract the differential TE current contribution as a function of ∆x and ∆t by the difference of the heating at the pn junction region in the presence of both pulses with respect to simulations with only one pulse at a time, analogous to the experimental differencefrequency demodulation.
Quantifying the spatial spread
The following analysis is conducted both on the experimental and the simulated data of ∆I_{TE}(∆x, ∆t) for ‘symmetric experiments’ with optical pulses incident at a distance ∆x on each side of the pnjunction (Figs. 1 and 2). For each ∆t of the datasets ∆I_{TE}(∆x, ∆t) we calculate the width of the signal via the second moment, which for an ideal Gaussian profile is equal to the squared Gaussian width σ^{2}. The second moment is calculated from the pixels ∆x_{i} (i = 1,…, N) via
We note that the minimum second moment at the focus <∆x^{2}>_{focus} of 0.56 µm^{2} comes from simulating the symmetric experiment, using as input the measured Gaussian beam width at the focus σ_{focus}^{2} = 0.14 µm^{2} (Supplementary Note 2). For the ‘asymmetric experiments’ with one optical pulse always incident on the pn junction (data for Fig. 3), we always consider the spatial profile only at time zero. Here we find that Gaussian fits with a background give the most reliable results. The entire set of data is shown in Supplementary Fig. 5. For each dataset ∆I_{TE}(∆x) or ∆I_{TE}(∆y) taken at ∆t = 0, we perform Gaussian fits using the function \({{{\mathrm{f}}}}({\Delta} x) = a\exp \left( {  \frac{{{\Delta} x^2}}{{2\sigma ^2}}} \right) + b\), where the Gaussian squared width σ^{2} indicates the thermal spreading. Here, the minimum simulated Gaussian widths are (σ_{x}^{2})_{focus} = 0.34 µm^{2} and (σ_{y}^{2})_{focus} = 0.44 µm^{2} (Supplementary Note 2). The experimentally obtained widths from this dataset as function of gate voltage and optical power are also shown in Supplementary Fig. 5, showing an increase with power, that is a larger T_{e}, and an increase towards the Dirac point, that is a smaller T_{F}. We estimate the theoretical Gaussian widths in Fig. 3g using \(\sigma _{{{{\mathrm{calc}}}}}^2\) = (σ_{x}^{2})_{focus} + 2D Δt_{IRF}, where D are the calculated diffusivities.
Electrical measurements
We characterize our device electrically with fourprobe measurements (Extended Data Fig. 1), finding a charge mobility μ of 30,000–50,000 cm^{2} Vs^{−1}, depending on carrier density. The measured mobilities correspond to a momentum relaxation time 𝜏_{mr} of 300–500 fs. These relaxation times are longer than the temporal resolution (the IRF) of our measurement technique, Δt_{IRF} ≅ 200 fs, thus allowing us to probe our system before and after momentum relaxation occurs, that is in the nondiffusive and diffusive regime. We use these measured charge mobilities to calculate the expected thermal diffusivity via the Einstein relation^{33,34} \(\mu _{{{{\mathrm{e}}}}/{{{\mathrm{h}}}}} = \frac{e}{{n_{{{{\mathrm{e}}}}/{{{\mathrm{h}}}}}}}\frac{{\partial n_{{{{\mathrm{e}}}}/{{{\mathrm{h}}}}}}}{{\partial E_{{{\mathrm{F}}}}}}D_{{{{\mathrm{e}}}}/{{{\mathrm{h}}}}}\), where e is the elementary charge, E_{F} is the Fermi energy and n_{e/h} is the electron/hole carrier density. For highly doped graphene (\(E_{{{\mathrm{F}}}} \gg k_{{{\mathrm{B}}}}T\)) the carrier density expression \(n_{{{{\mathrm{e}}}}/{{{\mathrm{h}}}}} = \frac{{E_{{{\mathrm{F}}}}^2}}{{\pi \hbar ^2v_{{{\mathrm{F}}}}^2}}\), leads to the simple relation:\(D_{{{{\mathrm{e}}}}/{{{\mathrm{h}}}}} = \frac{{E_{{{\mathrm{F}}}}}}{{2e}}\mu _{{{{\mathrm{e}}}}/{{{\mathrm{h}}}}}\). We note that we obtain the identical result by calculating D from the ratio of the 2D thermal conductivity κ_{e,2D} and the electronic heat capacity C_{e} and using the Wiedemann–Franz law: \(\kappa _{{{{\mathrm{e}}}},2{{{\mathrm{D}}}}}/\sigma = \pi ^2/3\cdot (k_{{{\mathrm{B}}}}/e)^2T_{{{\mathrm{e}}}}\), where k_{B} is the Boltzmann constant and e the elementary charge, together with the conductivity σ = neμ and the following heat capacity for graphene (valid for T_{e} < T_{F}): \(C_{{{\mathrm{e}}}} = \frac{{2\pi \varepsilon _{{{\mathrm{F}}}}k_{{{\mathrm{B}}}}^2T_{{{\mathrm{e}}}}}}{{3\hbar ^2v_{{{\mathrm{F}}}}^2}}\). Given the measured mobilities, we expect thermal diffusivities around 2,000 cm^{2} s^{−1} for our sample.
Thermal diffusivity and conductivity of the Dirac fluid
We estimate the enhanced thermal diffusivity of the Dirac fluid by comparing the measured width at time zero <Δx^{2}>_{min} to the expected width <Δx^{2}>_{focus} explained above, via D = (<Δx^{2}>_{min} − <Δx^{2}>_{focus})/2Δt_{IRF}. We find values up to 74,000 cm^{2} s^{−1} for the symmetric scan (Fig. 2), and 29,000 and 39,000 cm^{2} s^{−1} for the x and y directions of the asymmetric scan (Fig. 3), respectively, where <Δx^{2}> is replaced with (σ_{x}^{2}) and (σ_{y}^{2}), respectively. The same calculation for a second device (Supplementary Note 3 and Extended Data Fig. 2) gives a diffusivity of 100,000 cm^{2} s^{−1}. The 3D thermal conductivity κ_{3D} of the Dirac fluid is calculated from the diffusivity D and the electronic heat capacity C_{e}, via κ_{3D} = DC_{e}/d, where d is the thickness of graphene, 0.3 nm. For the Dirac fluid, we have T_{e} > T_{F}, and therefore use the ‘undoped’ electronic heat capacity^{35} \(\frac{{18\,\zeta (3)}}{{\pi (\hbar v_{{{\mathrm{F}}}})^2}}k_{{{\mathrm{B}}}}^3T_{{{\mathrm{e}}}}^2\), where \(\zeta \left( 3 \right) \approx 1.202\). With the above estimate D = 35,000–70,000 cm^{2} s^{−1} and T_{e} = 1,000 K, we obtain the 3D thermal conductivity κ_{3D} = 18,000–40,000 W mK^{−1}. This corresponds to a 2D κ_{2D} > 5 μW K^{−1}. This value is orders of magnitude larger than the value found in ref. ^{19}. The reason for this is that our electron temperature is more than ten times higher, and therefore the electronic heat capacity is >100× higher. Furthermore, we reach a T_{e}/T_{F} > 3, while their maximum T_{e}/T_{F} was around two, which means that we are further in the Diracfluid regime with its diverging thermal diffusivity.
Diracfluid crossover temperature
Following the treatment in ref. ^{14}, we find the crossover temperature from Fermi liquid to Dirac fluid, as a function of Fermi temperature as
where \(\lambda = {{{\mathrm{e}}}}^2/16\epsilon _0{\it{\epsilon }}_{{{\mathrm{r}}}}v_{{{\mathrm{F}}}}\hbar \approx 0.55/{\it{\epsilon }}_{{{\mathrm{r}}}}\) for graphene with the dielectric environment \({\it{\epsilon }}_{{{\mathrm{r}}}} \approx 3.56\) for hBN. The temperature \(T_0 = \frac{{2\hbar v_{{{\mathrm{F}}}}\sqrt \pi }}{{3^{3/4}k_{{{\mathrm{B}}}}a_0}} \approx 8.4\times 10^4\,{{{\mathrm{K}}}}\), with the interatomic distance \(a_0 = 1.42\times 10^{  10}\,{{{\mathrm{m}}}}\). The resulting crossover temperature is shown in Fig. 1b. We note that the relatively high refractive index of the hBN encapsulant makes the Dirac fluid more easily accessible, as it lowers the crossover temperature compared to vacuum, by a factor of about two for the range of T_{F} values studied here.
Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request.
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Acknowledgements
We thank M. Polini and P. Piskunow for fruitful discussions, and H. Agarwal and K. Soundarapandian for help with sample fabrication. We acknowledge the following funding sources: European Union’s Horizon 2020 research and innovation programme under grant nos. 804349 (K.J.T.), 873028 (A.P.), 785219 (F.H.L.K. and S.R.), 881603 (F.H.L.K. and S.R.) and 670949 (N.F.v.H.); Spanish MCIU/AEI under grant nos. RYC201722330 (K.J.T.), PID2019111673GBI00 (K.J.T.), BES2016078727 (M.L.), RTI2018099957JI00 (M.L.) and PGC2018096875BI00 (M.L. and N.F.v.H.); the Government of Catalonia under grant nos. SGR1656 (F.H.L.K.) and 2017SGR1369 (N.F.v.H.) and the CERCA program (ICN2 and ICFO); Spanish MINECO under grant nos. SEV20170706 (ICN2) and CEX2019000910S (ICFO); the International PhD fellowship program ‘la Caixa’ (A.B.); Leverhulme Trust grant no. RPG2019363 (A.P.); the Elemental Strategy Initiative conducted by the MEXT, Japan, grant no. JPMXP0112101001 (K.W. and T.T.), JSPS KAKENHI grant no. JP20H00354 (K.W. and T.T.) and the CREST (grant no. JPMJCR15F3) and JST (K.W. and T.T.); Fundació Privada Cellex (ICFO) and Fundació MirPuig (ICFO).
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K.J.T. conceived and supervised the project. A.B. developed the experimental setup under supervision of M.L. and N.F.v.H. and input from K.J.T. N.C.H.H. performed sample fabrication, with material input from K.W. and T.T., under the supervision of F.H.L.K. and K.J.T. A.B. performed the measurements under the supervision of K.J.T. A.P. performed the theoretical hydrodynamic transport calculations. A.B. and K.J.T. interpreted and analysed the data, with input from M.L., A.W.C. and N.F.v.H. A.B. developed the model that simulated the experiment with input from K.J.T. A.W.C. and S.R. developed the ballistic transport model. A.B. and K.J.T. wrote the paper, with input from all authors.
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Extended data
Extended Data Fig. 1 Electrical mobility measurement.
Momentum relaxation time from fourprobe measurements and corresponding (calculated) heat diffusivity (solid line). Fourprobe measurements were performed by applying 1 V to a MΩ series resistor, such that a current of 1 μA flows between the two outer contacts of the device (see inset). We then measure the voltage drop between two lateral contacts of the graphene device. This yields the sheet conductance σ as a function of gate voltage. We then use σ = neμ, in order to extract the mobility µ and use \(\tau _{mr} = \frac{{\mu E_{{{\mathrm{F}}}}}}{{ev_{{{\mathrm{F}}}}^2}}\) to obtain the momentum relaxation time. Here, n is the carrier density, E_{F} is the Fermi energy, v_{F} ≈ 10^{6} m/s is the Fermi velocity, and e is the elementary charge. We measured up to E_{F} = 150 meV, and extrapolated the data to higher Fermi energies. The three symbols indicate the Fermi energy and predicted diffusivity corresponding to the diffusivity measurements in Fig. 2e.
Extended Data Fig. 2 Spatiotemporal results from a second device.
a, Asymmetric spatiotemporal ∆I_{TE} maps for three different gate voltages. b, Extracted width as a function of ∆t. A lower Fermi level leads to a higher timezero width, in accordance with hydrodynamic transport, as presented for the main device in the manuscript. (c) ∆I_{TE} maps, taken at ∆t = 0, as a function of beam offset (∆x, ∆y), as well as sample height (z). (d) extracted line profiles for the two dimensions. e, Resulting signal width σ^{2} for both dimensions as extracted from Gaussian fits at each zposition. The same measurements are presented in Supplementary Fig. 4 for the main device. These experiments were performed with two beams of wavelength 443 nm and 886 nm, respectively.
Extended Data Fig. 3 Spatiotemporal results from a third device.
a, Microscope image of the device. This sample has splitgates made from graphite, with a gap that is 200 nm. It then has a 30 nm thick layer of SiO_{2}, and then we transferred a graphene flake grown by chemical vapour deposition (CVD) on top of the splitgate structure using an hBN flake. b, Gatedependent current measurement, which gives an estimated mobility of this halfencapsulated CVD graphene sample of around 8,500 cm^{2}/Vs (solid lines). cd, Asymmetric spatiotemporal ∆I_{TE} maps at time zero for different gate voltages and laser powers without (c) and with normalization (d). ef, Comparison between first and third device. Time zero Gaussian widths for spatial scans with one pulse on the junction and the second one scanning across (e) and along (f) the graphene pnjunction, as a function of power and gate voltage, for both the first device (hBNencapsulated with high mobility, presented in the main text, bluepurple colours, ‘hBN’) and the third device (on SiO_{2} with low mobility, yellowred colors, ‘SiO_{2}’). The lowmobility sample with shorter hydrodynamic time window shows systematically less heat spreading around time zero, in agreement with our picture of hydrodynamic heat spreading during the hydrodynamic time window.
Extended Data Fig. 4 Ballistic spreading simulation.
a, Timedependent output distributions for quantum mechanical calculations for a single electron (top row) and an ensemble of independent electrons (bottom row). b, Resulting width σ_{ball}^{2} for ballistic transport for the Monte Carlo method with varying Fermi velocities, as well as the quantum calculation for an ensemble of independent electrons. Both calculations essentially agree and the spread within 0.25 ps leads to final widths of below 0.25 µm^{2} for realistic values of the Fermi velocity.
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Supplementary Notes 1–4 and Figs. 1–10.
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Block, A., Principi, A., Hesp, N.C.H. et al. Observation of giant and tunable thermal diffusivity of a Dirac fluid at room temperature. Nat. Nanotechnol. 16, 1195–1200 (2021). https://doi.org/10.1038/s41565021009576
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DOI: https://doi.org/10.1038/s41565021009576
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