Observation of the Dirac fluidand the breakdown of the Wiedemann-Franz law in graphene

Jesse Crossno,1, 2 Jing K. Shi,1 Ke Wang,1 Xiaomeng Liu,1 Achim Harzheim,1 Andrew Lucas,1 Subir Sachdev,1, 3

Philip Kim,1, 2, Takashi Taniguchi,4 Kenji Watanabe,4 Thomas A. Ohki,5 and Kin Chung Fong5, †

1Department of Physics, Harvard University, Cambridge, MA 02138, USA2John A. Paulson School of Engineering and Applied Sciences,

Harvard University, Cambridge, MA 02138, USA3Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

4National Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan5Raytheon BBN Technologies, Quantum Information Processing Group, Cambridge, Massachusetts 02138, USA

(Dated: September 17, 2015)

Interactions between particles in quantum many-body systems can lead to collective behaviordescribed by hydrodynamics. One such system is the electron-hole plasma in graphene near thecharge neutrality point which can form a strongly coupled Dirac fluid. This charge neutral plasmaof quasi-relativistic fermions is expected to exhibit a substantial enhancement of the thermal con-ductivity, due to decoupling of charge and heat currents within hydrodynamics. Employing highsensitivity Johnson noise thermometry, we report the breakdown of the Wiedemann-Franz law ingraphene, with a thermal conductivity an order of magnitude larger than the value predicted byFermi liquid theory. This result is a signature of the Dirac fluid, and constitutes direct evidence ofcollective motion in a quantum electronic fluid.

Understanding the dynamics of many interacting parti-cles is a formidable task in physics, complicated by manycoupled degrees of freedom. For electronic transport inmatter, strong interactions can lead to a breakdown ofthe Fermi liquid (FL) paradigm of coherent quasiparti-cles scattering o↵ of impurities. In such situations, thecomplex microscopic dynamics can be coarse-grained toa hydrodynamic description of momentum, energy, andcharge transport on long length and time scales [1]. Hy-drodynamics has been successfully applied to a diversearray of interacting quantum systems, from high mobilityelectrons in conductors [2], to cold atoms [3] and quark-gluon plasmas [4]. As has been argued for strongly inter-acting massless Dirac fermions in graphene at the charge-neutrality point (CNP) [5–8], hydrodynamic e↵ects areexpected to greatly modify transport coecients as com-pared to their FL counterparts.

Many-body physics in graphene is interesting due toelectron-hole symmetry and a linear dispersion relationat the CNP [9, 10]. In particular, the Fermi surface van-ishes, leading to ine↵ective screening [11] and the forma-tion of a strongly-interacting quasi-relativistic electron-hole plasma, known as a Dirac fluid [12]. The Dirac fluidshares many features with quantum critical systems [13]:most importantly, the electron-electron scattering time isfast [14–17], and well suited to a hydrodynamic descrip-tion. A number of exotic properties have been predictedincluding nearly perfect (inviscid) flow [18] and a diverg-ing thermal conductivity resulting in the breakdown ofthe Wiedemann-Franz law [5, 6].

Away from the CNP, graphene has a sharp Fermi sur-face and the standard Fermi liquid (FL) phenomenologyholds. By tuning the chemical potential, we may mea-sure thermal and electrical conductivity in both the Dirac

fluid (DF) and the FL in the same sample. In a FL,the relaxation of heat and charge currents is closely re-lated as they are carried by the same quasiparticles. TheWiedemann-Franz (WF) law [19] states that the elec-tronic contribution to a metal’s thermal conductivity e

is proportional to its electrical conductivity and tem-perature T , such that the Lorenz ratio L satisfies

L e

T

=

2

3

kB

e

2

L0 (1)

where e is the electron charge, kB is the Boltzmann con-stant, and L0 is the Sommerfeld value derived from FLtheory. L0 depends only on fundamental constants, andnot on specific details of the system such as carrier den-sity or e↵ective mass. As a robust prediction of FL the-ory, the WF law has been verified in numerous metals[19]. However, in recent years, an increasing number ofnon-trivial violations of the WF law have been reportedin strongly interacting systems such as Luttinger liquids[20], metallic ferromagnets [21], heavy fermion metals[22], and underdoped cuprates [23], all related to theemergence of non-Fermi liquid behavior.The WF law is expected to be violated at the CNP

in a DF due to the strong Coulomb interactions betweenthermally excited charge carriers. An electric field driveselectrons and holes in opposite directions; collisions be-tween them introduce a frictional dissipation, resultingin a finite conductivity even in the absence of disorder[24]. In contrast, a temperature gradient causes electronsand holes to move in the same direction inducing an en-ergy current, which grows unimpeded by inter-particlecollisions (Fig. 3C inset). The thermal conductivity istherefore limited only by the rate at which momentum isrelaxed due to residual impurities.

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FIG. 1. Temperature and density dependent electrical and thermal conductivity. (A) Resistance versus gate voltageat various temperatures. (B) Electrical conductivity (blue) as a function of the charge density set by the back gate for di↵erentbath temperatures. The residual carrier density at the neutrality point (green) is estimated by the intersection of the minimumconductivity with a linear fit to log() away from neutrality (dashed grey lines). Curves have been o↵set vertically such thatthe minimum density (green) aligns with the temperature axis to the right. Solid black lines correspond to 4e2/h. At lowtemperature, the minimum density is limited by disorder (charge puddles). However, above Tdis 40 K, a crossover markedin the half-tone background, thermal excitations begin to dominate and the sample enters the non-degenerate regime nearthe neutrality point. (C-D) Thermal conductivity (red points) as a function of (C) gate voltage and (D) bath temperaturecompared to the Wiedemann-Franz law, TL0 (blue lines). At low temperature and/or high doping (|µ| kBT ), we find theWF law to hold. This is a non-trivial check on the quality of our measurement. In the non-degenerate regime (|µ| < kBT )the thermal conductivity is enhanced and the WF law is violated. Above Telph 80 K, electron-phonon coupling becomesappreciable and begins to dominate thermal transport at all measured gate voltages. All data from this figure is taken fromsample S2 (inset 1E).

Realization of the Dirac fluid in graphene requires thatthe thermal energy be larger than the local chemical po-tential µ(r), defined at position r: kBT & |µ(r)|. Impu-rities cause spatial variations in the local chemical po-tential, and even when the sample is globally neutral, itis locally doped to form electron-hole puddles with finiteµ(r) [25–28]. Formation of the DF is further complicatedby phonon scattering at high temperature which can re-lax momentum by creating additional inelastic scatteringchannels. This high temperature limit occurs when theelectron-phonon scattering rate becomes comparable tothe electron-electron scattering rate. These two temper-atures set the experimental window in which the DF andthe breakdown of the WF law can be observed.

To minimize disorder, the monolayer graphene samplesused in this report are encapsulated in hexagonal boronnitride (hBN) [29]. All devices used in this study aretwo-terminal to keep a well-defined temperature profile

[30] with contacts fabricated using the one-dimensionaledge technique [31] in order to minimize contact resis-tance. We employ a back gate voltage Vg applied tothe silicon substrate to tune the charge carrier densityn = ne nh, where ne and nh are the electron and holedensity, respectively (see supplementary materials (SM)).All measurements are performed in a cryostat controllingthe temperature Tbath. Fig. 1A shows the resistance R

versus Vg measured at various fixed temperatures for arepresentative device (see SM for all samples). From this,we estimate the electrical conductivity (Fig. 1B) usingthe known sample dimensions. At the CNP, the residualcharge carrier density nmin can be estimated by extrap-olating a linear fit of log() as a function of log(n) outto the minimum conductivity [32]. At the lowest tem-peratures we find nmin saturates to 8109 cm2. Wenote that the extraction of nmin by this method overesti-mates the charge puddle energy, consistent with previous

3

reports [29]. Above the disorder energy scale Tdis 40 K,nmin increases as Tbath is raised, suggesting thermal ex-citations begin to dominate and the sample enters thenon-degenerate regime near the CNP.

The electronic thermal conductivity is measured us-ing high sensitivity Johnson noise thermometry (JNT)[30, 33]. We apply a small bias current through the sam-ple that injects a joule heating power P directly into theelectronic system, inducing a small temperature di↵er-enceT TeTbath between the graphene electrons andthe bath. The electron temperature Te is monitored inde-pendent of the lattice temperature through the Johnsonnoise power emitted at 100 MHz with a 20 MHz band-width defined by an LC matching network. We designedour JNT to be operated over a wide temperature range3–300 K [33]. With a precision of 10 mK, we mea-sure small deviations of Te from Tbath, i.e. T Tbath.In this limit, the temperature of the graphene lattice iswell thermalized to the bath [30] and our JNT setup al-lows us to sensitively measure the electronic cooling path-ways in graphene. At low enough temperatures, elec-tron and lattice interactions are weak [33, 34], and mostof the Joule heat generated in graphene escapes via di-rect di↵usion to the contacts (SM). As temperature in-creases, electron-phonon scattering becomes appreciableand thermal transport becomes limited by the electron-phonon coupling strength [34–36]. The onset temper-ature of appreciable electron-phonon scattering, Telph,depends on the sample disorder and device geometry:Telph 80 K [33, 34, 37, 38] for our samples. Below thistemperature, the electronic contribution of the thermalconductivity can be obtained from P and T using thedevice dimensions (SM).

Fig. 1C plots e(Vg) alongside the simultaneously mea-sured (Vg) at various fixed bath temperatures. Here, fora direct quantitative comparison based on the WF law,we plot the scaled electrical conductivity as TL0 in thesame units as e. At low temperatures, T < Tdis 40 K,where the puddle induced density fluctuations dominates,we find e TL0, monotonically increasing as a func-tion of carrier density with a minimum at the neutralitypoint, confirming the WF law in the disordered regime.As T increases (T > Tdis), however, the measured e be-gins to deviate from the FL theory. We note that thisviolation of the WF law only appears close to the CNP,with the measured thermal conductivity maximized atn = 0 (Fig 1D). The deviation is the largest at 75 K,where e is over an order of magnitude larger than thevalue expected for a FL. This non-FL behavior quicklydisappears as |n| increases; e returns to the FL valueand restores the WF law. In fact, away from the CNP,the WF law holds for a wide temperature range, consis-tent with previous reports [33, 34, 37] (Fig. 1E). For thisFL regime, we verify the WF law up to Telph 80 K.Finally, in the high temperature regime T > Telph, theadditional electron-phonon cooling pathway causes the

0

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FIG. 2. Breakdown of the Wiedemann-Franz law inthe Dirac fluid regime. The Lorenz ratio is shown asa function of the charge carrier density and bath tempera-ture. Near the CNP and for temperatures above the disor-der (charge puddle) regime but below the onset of electron-phonon coupling, the Lorenz ratio is measured to be an orderof magnitude greater than the Fermi liquid value of 1 (blue).The WF law is observed to hold outside of the Dirac fluidregime. All data from this figure is taken from sample S1.

measured thermal conductivity to be larger than e. Wefind that near the CNP e tends to decrease just beforeTelph, restricting the maximal observable violation ofthe WF law.Our observation of the breakdown of the WF law in

graphene is consistent with the emergence of the DF.Fig 2 shows the full density and temperature depen-dence of the experimentally measured Lorenz ratio inorder to highlight the presence of the DF. The blue col-ored region denotes L L0, suggesting the carriers ingraphene exhibit FL behavior. The WF law is violatedin the DF (yellow-red) with a peak Lorenz ratio 22 timeslarger than L0. The green dotted line shows the corre-sponding nmin(T ) for this sample; the DF is found withinthis regime, indicating the coexistence of thermally pop-ulated electrons and holes. We find that disorder andphonon scattering bound the temperature range of theDirac fluid, Tdis < T < Telph.We investigate the e↵ect of impurities on hydrody-

namic transport by comparing the results obtained fromsamples with varying disorder. Fig. 3A shows nmin

as a function of temperature for three samples used inthis study. nmin(T = 0) is estimated as 5, 8, and10109 cm2 in samples S1, S2, and S3, respectively.All devices show qualitatively similar Dirac fluid behav-ior; the largest value of L/L0 measured in the Dirac fluidregime is 22, 12 and 3 in samples S1, S2, and S3, respec-tively (Fig 3B). For a direct comparison, we show L(n)for all three samples at the same temperature (60 K) inFig 3C. We find that cleaner samples not only have a

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H

(eV

/µm

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CHe

h

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h

∆Vg = 0

FIG. 3. Disorder in the Dirac fluid. (A) Minimum car-rier density as a function of temperature for all three sam-ples. At low temperature each sample is limited by disorder.At high temperature all samples become limited by thermalexcitations. Dashed lines are a guide to the eye. (B) TheLorentz ratio of all three samples as a function of bath tem-perature. The largest WF violation is seen in the cleanestsample. (C) The gate dependence of the Lorentz ratio is wellfit to hydrodynamic theory of Ref. [5, 6]. Fits of all threesamples are shown at 60 K. All samples return to the Fermiliquid value (black dashed line) at high density. Inset showsthe fitted enthalpy density as a function of temperature andthe theoretical value in clean graphene (black dashed line).Schematic inset illustrates the di↵erence between heat andcharge current in the neutral Dirac plasma.

more pronounced peak but also a narrower density de-pendence, as predicted [5, 6].

More quantitative analysis of L(n) in our experimentcan be done by employing a quasi-relativistic hydrody-namic theory of the DF incorporating the e↵ects of weakimpurity scattering [5, 6, 39].

L =LDF

(1 + (n/n0)2)2 (2)

where

LDF =HvFlm

T

2min

and n

20 =

Hmin

e

2vFlm

. (3)

Here vF is the Fermi velocity in graphene, min is the elec-trical conductivity at the CNP, H is the fluid enthalpydensity, and lm is the momentum relaxation length from

impurities. Two parameters in Eqn. (2) are undeter-mined for any given sample: lm and H. For simplic-ity, we assume we are well within the DF limit wherelm and H are approximately independent of n. We fitEqn. (2) to the experimentally measured L(n) for alltemperatures and densities in the Dirac fluid regime toobtain lm and H for each sample. Fig 3C shows threerepresentative fits to Eqn. (2) taken at 60 K. lm is esti-mated to be 1.5, 0.6, and 0.034 µm for samples S1, S2,and S3, respectively. For the system to be well describedby hydrodynamics, lm should be long compared to theelectron-electron scattering length of 0.1 µm expectedfor the Dirac fluid at 60 K [18]. This is consistent withthe pronounced signatures of hydrodynamics in S1 andS2, but not in S3, where only a glimpse of the DF appearsin this more disordered sample. Our analysis also allowsus to estimate the thermodynamic quantity H(T ) for theDF. The Fig. 3C inset shows the fitted enthalpy densityas a function of temperature compared to that expectedin clean graphene (dashed line) [18], excluding renormal-ization of the Fermi velocity. In the cleanest sample Hvaries from 1.1-2.3 eV/µm2 for Tdis < T < Telph. Thisenthalpy density corresponds to 20 meV or 4kBTper charge carrier — about a factor of 2 larger than themodel calculation without disorder [18].To fully incorporate the e↵ects of disorder, a hydrody-

namic theory treating inhomogeneity non-perturbativelyis necessary [40, 41]. The enthalpy densities reportedhere are larger than the theoretical estimation obtainedfor disorder free graphene, consistent with the picturethat chemical potential fluctuations prevent the samplefrom reaching the Dirac point. While we find thermalconductivity well described by Ref. [5, 6], electrical con-ductivity increases slower than expected away from theCNP, a result consistent with hydrodynamic transport ina viscous fluid with charge puddles [41].In a hydrodynamic system, the ratio of shear viscos-

ity to entropy density s is an indicator of the strengthof the interactions between constituent particles. It issuggested that the DF can behave as a nearly perfectfluid [18]: /s approaches a conjecture by Kovtun-Son-Starinets: (/s)/(~/kB) & 1/4 for a strongly inter-acting system [42]. A non-perturbative hydrodynamicframework can be employed to estimate , as we discusselsewhere [41]. A direct measurement of is of greatinterest.We have experimentally discovered the breakdown of

the WF law and provided evidence for the hydrodynamicbehavior of the Dirac fermions in graphene. This pro-vides an experimentally realizable Dirac fluid and opensthe way for future studies of strongly interacting rela-tivistic many-body systems. Beyond a diverging thermalconductivity and an ultra-low viscosity, other peculiarphenomena are expected to arise in this plasma. Themassless nature of the Dirac fermions is expected to re-sult in a large kinematic viscosity, despite a small shear

5

viscosity . Observable hydrodynamic e↵ects have alsobeen predicted to extend into the FL regime [43]. Thestudy of magnetotransport in the DF will lead to furthertests of hydrodynamics [5, 39].

Acknowledgements. We thank Matthew Fosterand Dmitri Efetov for helpful discussions. The ma-jor experimental work at Harvard is supported by DOE(DE-SC0012260) and at Raytheon BBN Technologies issupported by IRAD. J.C. thanks the support of theFAME Center, sponsored by SRC MARCO and DARPA.K.W. is supported by ARO MURI (W911NF-14-1-0247).J.K.S. is supported by ARO (W911NF-14-1-0638) andAStar. P.K. acknowledges partial support from theGordon and Betty Moore Foundation’s EPiQS Initia-tive through Grant GBMF4543 and Nano Material Tech-nology Development Program through the National Re-search Foundation of Korea (2012M3A7B4049966). A.L.and S.S. are supported by the NSF under Grant DMR-1360789, the Templeton foundation, and MURI grantW911NF-14-1-0003 from ARO. Research at PerimeterInstitute is supported by the Government of Canadathrough Industry Canada and by the Province of On-tario through the Ministry of Research and Innovation.K.W. and T.T. acknowledge support from the Elemen-tal Strategy Initiative conducted by the MEXT, Japan.T.T. acknowledges support from a Grant-in-Aid for Sci-entific Research on Grant 262480621 and on InnovativeAreas “Nano Informatics” (Grant 25106006) from JSPS.T.A.O. and K.C.F. acknowledge Raytheon BBN Tech-nologies’ support for this work. This work was performedin part at the Center for Nanoscale Systems (CNS), amember of the National Nanotechnology InfrastructureNetwork (NNIN), which is supported by the NationalScience Foundation under NSF award no. ECS-0335765.CNS is part of Harvard University.

SupplementaryMaterials

SAMPLE FABRICATION

Single layer graphene is encapsulated in hexagonalboron nitride on an n-doped silicon wafer with 285 nmSiO2 [31] and is subsequently annealed in vacuum for 15minutes at 350 C. It is then etched using reactive-ion-etching (RIE) to define the width of the device. A secondetch mask is then lithographically defined to overlap withthe sample edge, leaving the rest of the sample rectangu-lar shaped with the desired aspect ratio. After the RIEis performed, the same etch mask is used as the metaldeposition mask, upon which Cr/Pd/Au (1.5 nm / 5 nm/ 200 nm) is deposited. The resulting Ohmic contactsshow low contact resistances and small PN junction ef-

S1 S2 S3length (µm) 3 3 4width (µm) 9 9 10.5mobility (105 cm2 ·V1 · s1) 3 2.5 0.8nmin (109 cm2) 5 8 10

TABLE S.I. Basic properties of our three samples.

fects due to their minimum overlap with device edge.

OPTIMIZING SAMPLES FOR HIGHFREQUENCY THERMAL CONDUCTIVITY

MEASUREMENTS

To measure the electronic thermal conductivity e ofgraphene using high frequency Johnson noise the sampledesign should be made with three additional considera-tions: stray chip capacitance, resistance of the lead wires,and sample dimensions that enhance electron di↵usioncooling over phonon coupling.

Johnson noise thermometry (JNT) relies on measur-ing the total noise power emitted in a specified frequencyband and relating that to the electronic temperature onthe device; to maximize the sensitivity, high frequencyand wide bandwidth measurements should be made [33].In the temperature range discussed here, the upper fre-quency limit for JNT is typically set by the amount ofstray capacitance from the graphene, lead wires, and con-tact pads to the Si back gate. This is minimized by us-ing short, narrow lead wires and small (50 µm 50 µm)bonding pads resulting in an estimated 4 pF stray capac-itance.

The amount of Johnson noise emitted between any twoterminals is proportional to the mean electronic tempera-ture between them where each point in space is weightedby its local resistance. Therefore, to maximize the signalcoming from the graphene, contact resistance should bekept at a minimum. To compensate for the narrow leadwires, we deposit a thicker layer (200 nm) of gold result-ing in an estimated total contact resistance of < 80 .

Lastly, to e↵ectively extract e from the total elec-tronic thermal conductance Gth we want to enhance theelectron di↵usion cooling pathway with respect to theelectron-phonon cooling pathway (see below). This canbe accomplished by keeping the length of the sampleshort as the total power coupled into the lattice scalesas the area of the device while di↵usion cooling scales as1/R. In addition, the device should be made wide to min-imize the e↵ects of disordered edges. We find these highaspect ratio samples ( 3:1) are ideal for our measure-ments and serve the additional purpose of lowering thetotal sample resistance allowing us to impedance matchover a wider bandwidth.

6

−5 −4 −3 −2 −1 0 1 2 3 4 50

1

2

3

Vg (V)

R (

KΩ

)

S1

S2

S3

FIG. S1. 2-terminal resistance R vs. back gate voltage forthe 3 samples used in this report.

DEVICE CHARACTERIZATION

In this study we measure three graphene devices en-capsulated in hexagonal boron nitride (hBN), whose ba-sic properties are detailed in Table S.I. All devices aretwo-terminal with mobility estimated as

µ L

neRW

, (S.1)

where L and W are the sample length and width re-spectively, e is the electron charge, and n is the chargecarrier density. The gate capacitance per unit areaCg 0.11 fF/µm2 is estimated considering the 285 nmSiO2 and 20 nm hBN dielectrics. From this we esti-mate the charge density

n =Cg(Vg Vd)

e

(S.2)

where Vd is the gate voltage corresponding to the chargeneutrality point (CNP) estimated by the location of themaximum of the curve R(Vg). Fig. S1 shows the resis-tance of all samples as a function of gate voltage.

JOHNSON NOISE THERMOMETRY

The full Johnson noise thermometry (JNT) setup usedin this study is outlined in detail in [33]. Here, we onlygive a brief synopsis for completeness.

The electronic transport within a dissipative device canbe determined by the high frequency noise power col-lected by a low noise amplifier as

V

2↵= kBTe Re(Z)f

"1

Z Z0

Z + Z0

2#

(S.3)

where Z is the complex impedance of the device undertest, Z0 is the impedance of the measure circuit (typi-cally 50 ) and f is the bandwidth. From this formula,

50 75 100 125 150 175 200−90−80−70−60−50−40−30−20

Frequency (MHz)

R =

|S11

|2 (dB

)

LC matchingnetwork

5 mm

FIG. S2. Reflectance R = |S11|2 for a graphene deviceimpedance matched to 50 near 125 MHz.

we can see two critical components of JNT: impedancematching over a wide bandwidth and low noise amplifi-cation.

Graphene devices have a typical channel resistance onthe order of h/4e2 6 k near the CNP. To compensatefor this, we use an inductor-capacitor (LC) tank circuitmounted directly on the sample package to impedancematch the graphene to the 50 measurement network.Fig. S2 shows the reflectance coecient R = |S11|2 for atypical graphene device after impedance matching. Thebandwidth and measurement frequency of our JNT is setby the Q-factor and LC time of this matchingnetwork.

At 10 K, the power emitted by a resistor in a 1 Hzbandwidth is 1022 W or 190 dBm. To amplifythis signal we use a SiGe low noise amplifier (CaltechCITLF3) with a room temperature noise figure of about0.64 dB in our measurement bandwidth, correspondingto a noise temperature of about 46 K. We operate the am-plifier at room temperature, outside of the cryostat, toensure it is una↵ected by the 3–300 K temperature rampused for thermal conduction measurements. After am-plification, a homodyne mixer and low pass filter definethe measurement bandwidth and the power is found byan analog RF multiplier operating up to 2 GHz (AnalogDevices ADL5931). The result is a voltage proportionalto the Johnson noise power which – after calibration –measures the electron temperature in the graphene de-vice.

Calibration of our JNT device must be done on everysample as each device has a unique R(T, Vg) and there-fore couples di↵erently to the amplifier. The graphenedevice being measured is placed on a cold finger in acryostat with varying temperature Tbath. With no exci-tation current in the graphene, we collect the JNT signalVs(T, Vg) for all temperatures and gate voltage needed inthe study, as shown in Fig. S3. The linear temperatureslope at each point gives a gain factor g(T, Vg) = @Vs/@T .

7

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50

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150

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250

300

Gate Voltage [V]

Bat

h T

em

pe

ratu

re [

K]

100

140

180

220

260

µV

FIG. S3. Output voltage Vs from the JNT measuring agraphene device with no excitation current. This is used tocalibrate the JNT to a given sample.

MEASURING ELECTRONIC THERMALCONDUCTANCE

The procedure used to measure electronic thermal con-ductance is outlined in [30, 33, 44]. A small sinusoidalcurrent I(!) is run through the graphene sample, caus-ing a Joule heating power P (2!) to be injected directlyinto the electronic system. This causes a small tempera-ture di↵erence T (2!) between the electronic tempera-ture and the bath, described by Fourier’s law:

P = GthT. (S.4)

Here Gth is the total thermal conductance between theelectronic system and the bath. The component of John-son noise at frequency 2! is measured by a lock-in ampli-fier and then converted to a temperature di↵erence T

using the gain g(T, Vg) described in the previous section.Fig. S4 shows Te as a function of heating current I for agraphene device at three di↵erent bath temperatures: 3,30 and 300 K.

THERMAL MODEL OF GRAPHENEELECTRONS

In the regime presented here, Gth is dominated by twoelectronic cooling pathways. Hot electrons can di↵usedirectly out to the contacts (Gdi↵), or they can couple tophonons (Gelph):

G Gdi↵ +Gelph. (S.5)

In a typical metal, electron di↵usion is described by theWF law which is linear in Te. The electron-phonon cool-ing pathway has two components: first, the electronsmust transfer heat to the lattice via electron-phonon cou-pling, and then the lattice must conduct the heat to the

38.1

nW

/K

124 nW/K

995 nW/K

30

31

Ele

ctro

n T

em

pe

ratu

re (

K)

0 20 40 60 80

3

4

Heating Current (µA)

300

301

0

300 K

Bath Temperature

30 K3 K

FIG. S4. The electronic temperature in an encapsulatedgraphene device as a function of heating current for threedi↵erent bath temperatures. Te = Tbath + I2R/Gth. The to-tal thermal conductance between the electronic system andthe bath is found through Fourier’s Law (solid lines).

Diff

usi

on

Te

TBath

Ele

ctr

on

-Ph

on

on

P = I2 R

FIG. S5. Simplified thermal diagram of the electronic coolingpathways in graphene relevant for our experimental condi-tions. A current induces a heating power into the electronicsystem which conducts to the bath via two parallel pathways:di↵usion and coupling to phonons.

bath. Consistent with previous experimental studies ongraphene [33, 34, 37, 38], we find Gelph is bottleneckedby the weak electron-phonon coupling and hence the lat-tice is well thermalized to the bath. Fig. S5 shows thesimplified thermal diagram of the electronic cooling path-ways in graphene, relevant to our experiment.

At low temperature, Gth is dominated by Gdi↵ , whileat high temperature it is dominated by Gelph. Fig. S6shows an illustration of this e↵ect in one of our devices.

8

1 10 1001

10

100

1000

Bath Temperature (K)

Gth

(n

W/K

)

Data~T~T4

FIG. S6. The total thermal conductance of a graphene devicein the high density regime, illustrating how Gth is dominatedby either Gdi↵ T , as predicted by the WF law, or Gelph T 4.

MEASURING THERMAL CONDUCTIVITY

In the di↵usion-limited regime, we extract the elec-tronic thermal conductivity e as follows. This is detailedin [30] and we review the basic calculation for clarity. Thetotal power dissipated, P , is given by

P =J

2

LW = PLW (S.6)

where L is the length of the sample in the direction of cur-rent flow, W is the width in the perpendicular directionand P is the local power dissipated. Because this calcu-lation is done in linear response, and the external heatbaths on either side of the sample are at the same temper-ature, the contributions to power dissipated (T )2

do not enter so long as T J

2 is small. This is anappropriate assumption in the regime of linear response,where J is treated as a perturbatively small parameter.Fig. S4 shows our experiment is in this regime.

Let us now determine the change in the temperatureprofile. For simplicity we assume that the graphene sam-ple is homogeneous, that the approximately uniform elec-trical current is given by

J =

dV

dx ↵

dT

dx, (S.7)

and that the heat current is given by

Q = ↵T

dV

dx e

dT

dx, (S.8)

where

e e +T↵

2

= e(1 + ZT ). (S.9)

In the latter equation, ZT is the thermoelectric coe-cient of merit. As ↵ 0 at the CNP, we expect ZT 0,and that e e.

x

DTave

DT (x)

Is-d

FIG. S7. Cartoon illustrating the non-uniform temperatureprofile within the graphene-hBN stack during Joule heatingin the di↵usion-limited regime.

dT/dx is the temperature gradient in the sample, anddV/dx is the electric field in the sample. ↵/ is theSeebeck coecient. We assume that the response ofgraphene is dominated only by the changes in voltageV and temperature T to a uniform current density J ,which is applied externally. We also assume that devia-tions from constant V and T are small, so that the linearresponse theory is valid. Joule heating leads to the fol-lowing equations:

0 =dJ

dx, (S.10a)

P =J

2

=dQ

dx, (S.10b)

which can be combined to obtain

P = ed2T

dx2, (S.11)

assuming that e is approximately homogeneousthroughout the sample.The contacts in our experiment are held at the same

temperature T . Thus, writing

T (x) = Te +T (x), (S.12)

we find that

T (x) =P2e

x(L x). (S.13)

The average temperature change in the sample, which isdirectly measured through JNT, is

hT i =LZ

0

dx

L

T (x) =PL

2

12e. (S.14)

This non-uniform temperature profile is illustrated inFig. S7. Combining Eqs. (S.4), (S.6) and (S.14) weobtain

Gth =12L

W

e. (S.15)

As we have pointed out in the main text, our samplesare not perfectly homogeneous, but have local fluctua-tions in the charge density. Nevertheless, we do recover

9

the WF law in the FL regime, suggesting that our mea-surement of Gth – and thus e – using JNT, along withthe above formalism, is valid.

pkim@physics.harvard.edu† kc.fong@bbn.com

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