www.sciencemag.org/cgi/content/full/science.aad0343/DC1

Supplementary Materials for

Observation of the Dirac fluid and the breakdown of the Wiedemann-

Franz law in graphene

Jesse Crossno, Jing K. Shi, Ke Wang, Xiaomeng Liu, Achim Harzheim, Andrew Lucas,

Subir Sachdev, Philip Kim,* Takashi Taniguchi, Kenji Watanabe, Thomas A. Ohki, Kin

Chung Fong*

*Corresponding author. E-mail: pkim@physics.harvard.edu (P.K.); kc.fong@bbn.com (K.C.F.)

Published 11 February 2016 on Science First Release

DOI: 10.1126/science.aad0343

This PDF file includes:

Materials and Methods

Figs. S1 to S8

Table S1

Full Reference List

S-I. SAMPLE FABRICATION

Single layer graphene is encapsulated in hexagonal boron nitride on an n-doped silicon

wafer with 285 nm SiO2 [33] and is subsequently annealed in vacuum for 15 minutes at 350

C. It is then etched using reactive-ion-etching (RIE) to define the width of the device. A

second etch mask is then lithographically defined to overlap with the sample edge, leaving

the rest of the sample rectangular shaped with the desired aspect ratio. After the RIE is

performed, the same etch mask is used as the metal deposition mask, upon which Cr/Pd/Au

(1.5 nm / 5 nm / 200 nm) is deposited. The resulting Ohmic contacts show low contact

resistances and small PN junction e↵ects due to their minimum overlap with device edge.

S-II. OPTIMIZING SAMPLES FOR HIGH FREQUENCY THERMAL CONDUC-

TIVITY MEASUREMENTS

To measure the electronic thermal conductivity e of graphene using high frequency

Johnson noise the sample design should be made with three additional considerations: stray

chip capacitance, resistance of the lead wires, and sample dimensions that enhance electron

di↵usion cooling over phonon coupling.

Johnson noise thermometry (JNT) relies on measuring the total noise power emitted in

a specified frequency band and relating that to the electronic temperature on the device;

to maximize the sensitivity, high frequency and wide bandwidth measurements should be

made [35]. In the temperature range discussed here, the upper frequency limit for JNT is

typically set by the amount of stray capacitance from the graphene, lead wires, and contact

pads to the Si back gate. This is minimized by using short, narrow lead wires and small (50

µm 50 µm) bonding pads resulting in an estimated 4 pF stray capacitance.

The amount of Johnson noise emitted between any two terminals is proportional to the

mean electronic temperature between them where each point in space is weighted by its local

resistance. Therefore, to maximize the signal coming from the graphene, contact resistance

should be kept at a minimum. To compensate for the narrow lead wires, we deposit a thicker

layer (200 nm) of gold resulting in an estimated lead resistance of 50 . This, combined

with an estimated interfacial contact resistance of < 135 · µm results in a total measured

contact resistance of < 80 . The interfacial contact resistance may be density dependent

and we estimate it to be . 100–300 · µm. The most important e↵ect of this contact

resistance is likely the thermal resistance associated to the contacts, which may limit the

maximal observable Lorenz ratio. As such, the true value of L is likely slightly higher than

what we measure.

Lastly, to e↵ectively extract e from the total electronic thermal conductance Gth we

want to enhance the electron di↵usion cooling pathway with respect to the electron-phonon

cooling pathway (see below). This can be accomplished by keeping the length of the sample

short as the total power coupled into the lattice scales as the area of the device while

di↵usion cooling scales as 1/R. In addition, the device should be made wide to minimize

the e↵ects of disordered edges. We find these high aspect ratio samples ( 3:1) are ideal for

our measurements and serve the additional purpose of lowering the total sample resistance

allowing us to impedance match over a wider bandwidth.

S-III. DEVICE CHARACTERIZATION

In this study we measure three graphene devices encapsulated in hexagonal boron nitride

(hBN), whose basic properties are detailed in Table S.1. All devices are two-terminal with

mobility estimated as

µ L

neRW

, (S.1)

where L and W are the sample length and width respectively, e is the electron charge, and

n is the charge carrier density. The gate capacitance per unit area Cg 0.11 fF/µm2 is

estimated considering the 285 nm SiO2 and 20 nm hBN dielectrics. From this we estimate

the charge density

n =Cg(Vg Vd)

e

(S.2)

where Vd is the gate voltage corresponding to the charge neutrality point (CNP) estimated

by the location of the maximum of the curve R(Vg). Fig. S1 shows the resistance of all

samples as a function of gate voltage.

S-IV. JOHNSON NOISE THERMOMETRY

The full Johnson noise thermometry (JNT) setup used in this study is outlined in detail

in [35]. Here, we only give a brief synopsis for completeness.

The electronic transport within a dissipative device can be determined by the high fre-

quency noise power collected 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 under test, Z0 is the impedance of the

measure circuit (typically 50 ) and f is the bandwidth. From this formula, we can see

two critical components of JNT: impedance matching over a wide bandwidth and low noise

amplification.

Graphene devices have a typical channel resistance on the order of h/4e2 6 k near

the CNP. To compensate for this, we use an inductor-capacitor (LC) tank circuit mounted

directly on the sample package to impedance match the graphene to the 50 measurement

network. Fig. S2 shows the reflectance coecient R = |S11|2 for a typical graphene device

after impedance matching. The bandwidth and measurement frequency of our JNT is set

by the Q-factor and LC time of this matchingnetwork.

At 10 K, the power emitted by a resistor in a 1 Hz bandwidth is 1022 W or 190

dBm. To amplify this signal we use a SiGe low noise amplifier (Caltech CITLF3) with a room

temperature noise figure of about 0.64 dB in our measurement bandwidth, corresponding to

a noise temperature of about 46 K. We operate the amplifier at room temperature, outside

of the cryostat, to ensure it is una↵ected by the 3–300 K temperature ramp used for thermal

conduction measurements. After amplification, a homodyne mixer and low pass filter define

the measurement bandwidth and the power is found by an analog RF multiplier operating

up to 2 GHz (Analog Devices ADL5931). The result is a voltage proportional to the Johnson

noise power which – after calibration – measures the electron temperature in the graphene

device.

Calibration of our JNT device must be done on every sample as each device has a unique

R(T, Vg) and therefore couples di↵erently to the amplifier. The graphene device being mea-

sured is placed on a cold finger in a cryostat with varying temperature Tbath. With no

excitation current in the graphene, we collect the JNT signal Vs(T, Vg) for all temperatures

and gate voltage needed in the study, as shown in Fig. S3. The linear temperature slope at

each point gives a gain factor g(T, Vg) = @Vs/@T .

S-V. MEASURING ELECTRONIC THERMAL CONDUCTANCE

The procedure used to measure electronic thermal conductance is outlined in [32, 35,

56]. A small sinusoidal current I(!) is run through the graphene sample, causing a Joule

heating power P (2!) to be injected directly into the electronic system. This causes a small

temperature di↵erence T (2!) between the electronic temperature and the bath, described

by Fourier’s law:

P = GthT. (S.4)

Here Gth is the total thermal conductance between the electronic system and the bath. The

component of Johnson noise at frequency 2! is measured by a lock-in amplifier 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 a graphene device at three

di↵erent bath temperatures: 3, 30 and 300 K.

S-VI. THERMAL MODEL OF GRAPHENE ELECTRONS

In the regime presented here, Gth is dominated by two electronic cooling pathways. Hot

electrons can di↵use directly out to the contacts (Gdi↵), or they can couple to phonons

(Gelph):

G Gdi↵ +Gelph. (S.5)

In a typical metal, electron di↵usion is described by the WF law which is linear in Te. The

electron-phonon cooling pathway has two components: first, the electrons must transfer heat

to the lattice via electron-phonon coupling, and then the lattice must conduct the heat to

the bath. Fig. S5 shows the simplified thermal diagram of the electronic cooling pathways

in graphene, relevant to our experiment.

At low temperature, Gth is dominated by Gdi↵ , while at high temperature it is dominated

by Gelph. We plot in Fig. S6 our data compared to two di↵erent experimental reports. For

our device parameters, the heat transferred to the lattice by the supercollision mechansim is

much smaller (green solid line) as verified by three independent experiments in Ref. [51], [47],

and [40], based on the theories of Song-Reizer-Levitov [57] and Chen-Clerk [49]. Similarly,

the heat transferred to acoustic phonons is also smaller than the electronic di↵usion term

for our device geometry (purple solid curve) as verified by three sets of experiments in Ref.

[37], [32, 35, 36] and [54] based on the theories of Bistritzer-MacDonald [48], Tse-DasSarma

[58], Viljas-Heikkalla [59], and Kubakaddi [53].Yigen and Champagne reported in Ref. [39]

a similar graphene device dominated by the electronic thermal conductivity. Hence the

electronic contribution to the thermal conductivity and the Lorenz number can be directly

measured in Yigen-Champagne’s [39] and our experiments without the influence of acoustic

phonons below 100 K.

S-VII. MEASURING THERMAL CONDUCTIVITY

In the di↵usion-limited regime, we extract the electronic thermal conductivity e as fol-

lows. This is detailed in [32] and we review the basic calculation for clarity. The total 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 current flow, W is the width in

the perpendicular direction and P is the local power dissipated. Because this calculation

is done in linear response, and the external heat baths on either side of the sample are at

the same temperature, the contributions to power dissipated (T )2 do not enter so long

as T J

2 is small. This is an appropriate 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 temperature profile. For simplicity we assume

that the graphene sample is homogeneous, that the approximately uniform electrical 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 coecient of merit. As ↵ 0 at the CNP,

we expect ZT 0, and that e e.

dT/dx is the temperature gradient in the sample, and dV/dx is the electric field in

the sample. ↵/ is the Seebeck coecient. We assume that the response of graphene

is dominated only by the changes in voltage V and temperature T to a uniform current

density J , which is applied externally. We also assume that deviations from constant V and

T are small, so that the linear response theory is valid. Joule heating leads to the following

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 homogeneous throughout 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 is directly measured through JNT, is

hT i =LZ

0

dx

L

T (x) =PL

2

12e. (S.14)

This non-uniform temperature profile is illustrated in Fig. S7. Combining Eqs. (S.4), (S.6)

and (S.14) we obtain

Gth =12L

W

e. (S.15)

As we have pointed out in the main text, our samples are not perfectly homogeneous,

but have local fluctuations in the charge density. Nevertheless, we do recover the WF law

in the FL regime, suggesting that our measurement of Gth – and thus e – using JNT, along

with the above formalism, is valid.

BIPOLAR DIFFUSION

The bipolar di↵usion e↵ect occurs when di↵erent charge carriers of opposite sign move

in the same direction under an applied temperature gradient. The thermal conductivity ,

as defined in the main text in the absence of net electric current flow, is given in Ref. [55]:

e

T↵

2e

e

+

h

T↵

2h

h

+

Teh

e + h

↵e

e ↵h

h

2

(S.16)

The first two terms in the above equations are the thermal conductivity of electrons and holes

respectively. The third term is the bipolar di↵usion term, and accounts for the possibility

of electrons and holes flowing in the same direction.

Bipolar di↵usion has been used to explain the thermal conductivity of narrow gap semi-

conductors, such as bismuth telluride, when the chemical potential is close to the midgap.

Estimates of the Lorenz ratio in bismuth telluride have been reported as high as 7.2 L0

in Ref. [50]. In graphene, the two types of charge carriers correspond to above/below the

Dirac point. To use the formula (S.16) requires the assumption that interactions between

electrons and holes are negligible. If this is the case, it is reasonable to employ kinetic theory

to estimate e,h, ↵e,h and e,h. This was shown in Ref. [42] and we state the formalism here

for completeness. Employing the ultrarelativistic band structure of graphene near the CNP,

the transport coecients are given by:

e =

Zd2k

22e(k)v

2FF

~vFk ± µ

kBT

, (S.17a)

↵e = ±Z

d2k

22e(k)(~vFk ± µ)v2FF

~vFk ± µ

kBT

, (S.17b)

e =

Zd2k

22e(k)(~vFk ± µ)2v2FF

~vFk ± µ

kBT

, (S.17c)

where e,h are suitably defined energy relaxation times for electrons and holes, we use a ±

sign for electrons/holes, and

F(x) 1

kBT

ex

(1 + ex)2. (S.18)

Given a choice of , it is straightforward to numerically integrate these equations. This

was done in Ref. [42] for some choices of ; we have checked additional choices. We compare

the result of this two-band formalism at the CNP, to the same data for S1 used in Figure 3

of the main text. We can rule out the canonical bipolar di↵usion (BD) explanation of our

data for the following reasons:

(1) The BD theory predicts that L/LWF is independent of temperature at the CNP in a

clean sample. Adding disorder only adds a very weak temperature dependence [42]. This

is in stark contrast to our data – see Figure S8. Hydrodynamic models do predict a sharp

temperature dependence in L [41].

(2) Simple models of BD in the presence of charge puddles with local chemical poten-

tial fluctuations of 50 meV or higher predict factor of 2 violations of the WF law [42], and

a weak dependence of L on disorder, compared to hydrodynamics. In contrast we see a

sharp dependence on disorder, comparing our three samples (Figure 3 of the main text). In

samples where chemical potential fluctuations are comparable to 50 meV, the WF law is

obeyed to within 40% [32].

(3) Working under the theory of [52], (k) |k|, and hence the maximal Lorenz ratio

L/LWF 4 [42] – well below to our experimental observation of L/LWF 22.

(4) The BD theory of [42] predicts L/LWF & 0.8 under all conditions. Hydrodynamics

predicts that this ratio can become arbitrarily small in a clean sample, and we have indeed

observed L/LWF 1/3 at finite density in the DF in sample S1, only consistent with

hydrodynamics.

−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 for the 3 samples used in this report.

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 device impedance matched to 50 near 125 MHz.

−10 −5 0 50

50

100

150

200

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 a graphene device with no excitation current.

This is used to calibrate the JNT to a given sample.

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 encapsulated graphene device as a function of heating

current for three di↵erent bath temperatures. Te = Tbath+I2R/Gth. The total thermal conductance

between the electronic system and the 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 cooling pathways in graphene relevant for

our experimental conditions. A current induces a heating power into the electronic system which

conducts to the bath via two parallel pathways: di↵usion and coupling to phonons.

1 10 10010-10

10-9

10-8

10-7

10-6

Temperature (K)

Ther

mal

Con

duct

ance

(W/K

)

Wiedemann-Franzmeasured phonon coupling

supercollisionacoustic phonons

our data

FIG. S6. Comparison of the measured thermal conductance to the heat transfer to phonons due

to electron-phonon coupling. Data (blue diamonds) from Sample S1 at charge carrier density of

3.3 1011/cm2. Red solid and green dashed lines are the expected thermal conductance of heat

transfer from electrons to phonons.

x

DTave

DT (x)

Is-d

FIG. S7. Cartoon illustrating the non-uniform temperature profile within the graphene-hBN stack

during Joule heating in the di↵usion-limited regime.

Temperature [K]0

5

10

15

20

25

1 100

S1Yoshino-Murata ∆ε = 0 meV

Yoshino-Murata ∆ε = 50 meV

L / L

0

clean graphene BD

disordered graphene BD

FIG. S8. Comparison of the Lorenz ratio in graphene between bipolar di↵usion theory [42] and our

experimental data. The sharp temperature dependence of L is inconsistent with bipolar di↵usion

(see green dashed lines). This sharp behavior is predicted in hydrodynamics [41].

S1 S2 S3

length (µm) 3 3 4

width (µm) 9 9 10.5

mobility (105 cm2 ·V1 · s1) 3 2.5 0.8

nmin (109 cm2) 5 8 10

TABLE S.1. Basic properties of our three samples.

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