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Towards a quantum resistance standard based on

epitaxial graphene

Alexander Tzalenchuk

1

*

, Samuel Lara-Avila

2

, Alexei Kalaboukhov

2

, Sara Paolillo

3

, Mikael Syva

¨

ja

¨

rvi

4

,

Rositza Yakimova

4

, Olga Kazakova

1

,T.J.B.M.Janssen

1

, Vladimir Fal’ko

5

and Sergey Kubatkin

2

The quantum Hall effect

1

allows the international standard for

resistance to be deﬁned in terms of the electron charge and

Planck’s constant alone. The effect comprises the quantization

of the Hall resistance in two-dimensional electron systems in

rational fractions of R

K

5 h/e

2

5 25 812.807 557(18) V, the

resistance quantum

2

. Despite 30 years of research into the

quantum Hall effect, the level of precision necessary for

metrology—a few parts per billion—has been achieved only in

silicon and

III–V heterostructure devices

3–5

. Graphene should,

in principle, be an ideal material for a quantum resistance stan-

dard

6

, because it is inherently two-dimensional and its discrete

electron energy levels in a magnetic ﬁeld (the Landau levels

7

)

are widely spaced. However, the precisions demonstrated

so far have been lower than one part per million

8

. Here, we

report a quantum Hall resistance quantization accuracy of

three parts per billion in monolayer epitaxial graphene at

300 mK, four orders of magnitude better than previously

reported. Moreover, by demonstrating the structural integrity

and uniformity of graphene over hundreds of micrometres, as

well as reproducible mobility and carrier concentrations

across a half-centimetre wafer, these results boost the pro-

spects of using epitaxial graphene in applications beyond

quantum metrology.

Graphene—a single layer of graphite—is a truly two-dimensional

gapless semiconductor, with electrons mimicking the behaviour of

relativistic (Dirac) electrons

9

. This last feature of charge carriers in

graphene is manifested most spectacularly through an unusual

sequence of the quantum Hall effect (QHE) plateaux

10

. The QHE

is a result of the Landau level quantization of the energy spectrum

of two-dimensional electrons. In the quantum Hall regime the

current is carried by a quantum state, spreading through the whole

sample, and the sequence of plateaux in the transverse resistance

R

xy

is determined by the topological (Berry) phase acquired by the

charge moving in the magnetic ﬁeld. This phase is zero in conven-

tional materials, where R

xy

¼+h/ne

2

(n-integer 1); it is equal to

2p in bilayer graphene

11,12

, leading to a sequence of QHE plateaux

at R

xy

¼+(h/4e

2

)/(n þ 1) (n 0), and it is p in the monolayers

13

,

which determines the QHE sequence R

xy

¼+(h/4e

2

)/(n þ 1/2)

(n 0), currently regarded as a smoking gun for the sample to

contain monolayer graphene

10

. The spacing between the n ¼ 0 and

n ¼ 1 Landau levels in graphene, DE

graphene

01

p

(B[T]) 36 meV

is large in comparison with conventional materials such as GaAs.

For example, at 15 T, DE

graphene

01

/DE

GaAs

5.4.

In reality, an impressive range of unconventional transport prop-

erties of electrons in graphene

14

, including QHE, have been seen

almost exclusively in ﬂakes mechanically exfoliated from bulk

graphite. Quantum Hall plateaux have been observed in such

material even at room temperature, albeit with an accuracy of

0.2% (ref. 15), while the highest experimentally achieved accuracy

8

of QHE measurements at 300 mK in exfoliated ﬂakes was 15 parts

in a million—still modest by metrological standards.

An alternative approach to producing graphene is to grow it epi-

taxially on silicon carbide (SiC)

16

. Although angle-resolved photoe-

mission studies of epitaxial graphene

17

have revealed an almost

linear dispersion of carriers around the corners of a hexagonal

Brillouin zone and scanning tunnelling microscopy (STM)

showed the sequence of Landau levels typical for graphene

18

, Hall

resistance quantization has not been observed in epitaxial graphene,

in contrast to the exfoliated material, despite numerous attempts.

The difﬁculty was related to the lack of atomically accurate thickness

control during ﬁlm growth on the C-terminated facet, and also a

strong variation of carrier density (doping) across the layers

grown on the silicon-terminated facet

19

.

Here, we demonstrate the viability of a quantum Hall resistance

standard based on large-area epitaxial graphene synthesized on the

silicon-terminated face of silicon carbide by observation of Hall

resistance quantization accurate to a few parts in a billion at

300 mK and a few tens in a billion at 4.2 K. The samples (Fig. 1)

studied in our experiments were produced on the silicon-terminated

face of a 4H-SiC(0001) substrate (Cree Inc.) using the protocol

described in the Methods. We concentrate on the transport charac-

teristics of the smallest and largest of the fabricated devices ident-

iﬁed by circles in Fig. 1c. Figure 2a shows the longitudinal

(dissipative) R

xx

and transverse (Hall) R

xy

resistance of a

11.6 mm 2 mm Hall bar at 4.2 K and 214 T , B , 14 T. The

absence of positive classical magnetoresistance at magnetic ﬁelds

jBj , 2 T indicates that carrier density in this material is quite

homogeneous over the length of at least several micrometres.

However, it was noticed that the magnetoresistance was slightly

asymmetric with respect to the reversal of the magnetic ﬁeld direc-

tion, which made it difﬁcult to determine the carrier density using

the Hall coefﬁcient (although the symmetry was preserved when the

voltage leads were swapped with the current leads as the ﬁeld was

reversed). In intermediate magnetic ﬁelds we observe Shubnikov–

de Haas (SdH) oscillations (as well as a weak localization peak at

jBj , 0.1 T characteristic of phase coherence electrons in disordered

graphene

20

). The analysis of SdH oscillations enabled us to deter-

mine the carrier density in this device n

s

¼ 6.5 10

11

cm

22

.

Using the density obtained from the SdH oscillations, we estimate

that the magnetic ﬁeld (B

n

¼ hn

s

/en) needed to reach the exact ﬁlling

factor n ¼ 2 in this device is 13.5 T (unfortunately we had no

means of controlling the carrier density in these experiments).

Two Hall resistance plateaux are clearly visible in Fig. 2a, at R

xy

(0)

¼

R

K

/2(n ¼ 0) and R

xy

(1)

¼ R

K

/6(n ¼ 1), corresponding to the ﬁlling

1

National Physical Laboratory, TW11 0LW Teddington, UK,

2

Department of Microtechnology and Nanoscience, Chalmers University of Technology, S-412 96

Go

¨

teborg, Sweden,

3

Department of Physics, Politecnico di Milano, 20133 Milano, Italy,

4

Department of Physics, Chemistry and Biology (IFM), Linko

¨

ping

University, S-581 83 Linko

¨

ping, Sweden,

5

Physics Department, Lancaster University, Lancaster LA1 4YB, UK.

*

e-mail: alexander.tzalenchuk@npl.co.uk

LETTERS

PUBLISHED ONLINE: 17 JANUARY 2010 | DOI: 10.1038/NNANO.2009.474

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factors v ¼ 2 and v ¼ 6, respectively. In graphene v ¼ 2 corresponds

to the fully occupied zero-energy Landau level characterized by the

largest separation v

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2h

eB=c

p

from other Landau levels in the spec-

trum, so that the Hall resistance quantization at R

xy

(0)

¼ R

K

/2 is par-

ticularly robust. This plateau appears already at jBj10 T and is

accompanied by a vanishing R

xx

. The n ¼ 1 plateau at v ¼ 6 is not

so ﬂat, with only a weak minimum in R

xx

, and there is also a

visible trace of a structure corresponding to v ¼ 10. Their presence

conﬁrms that the studied material is indeed monolayer graphene.

The magneto-transport measurements on a much bigger,

160 mm 35 mm Hall bar device (Fig. 1c) are presented in

Fig. 2b. A substantial positive magnetoresistance at low ﬁelds,

which was absent in the smaller sample, indicates that the carrier

concentration varies along the larger sample. Because of this, the

v ¼ 6 feature in R

xx

in the bigger sample is less prominent.

Nevertheless, despite the inhomogeneity of the carrier density, the

Hall resistance plateau at R

xy

(0)

¼ R

K

/2(n ¼ 0) is robust and is

accompanied by vanishing longitudinal resistance R

xx

.

Importantly, the large-area device has a low resistance R

c

1.5 V

of contacts to the graphene layer (determined at the plateau,

Fig. 2c) and, when compared to smaller devices, can sustain a

much higher current before QHE breaks down, as shown in the

I(V) characteristics in Fig. 2d. Because larger breakdown current

affords higher-precision measurements in the QHE regime, we

chose to perform such measurements in the larger sample. The

choice of the ﬁeld, 14 T, where the most accurate measurements

were performed, was determined simply by the limitation of

our superconducting magnet. This limit is below B ¼ 17.5 T,

where the ﬁlling factor would be exactly v ¼ 2 for this sample

(with n

s

¼ 8.5 10

11

cm

22

calculated from SdH oscillations).

Precision measurements of R

xy

were performed in the conditions

where R

xx

is very accurately zero: at B ¼ 14 T, I

sd

¼ 2.3 mA for 4.2 K

and I

sd

¼ 11.6 mA for 300 mK (see Supplementary Information).

Note that a higher probe current enables a higher precision of R

xy

measurements to be achieved, so we performed these studies at

300 mK. The accuracy of Hall resistance quantization in graphene

was established in measurements traceable to the GaAs quantum

Hall resistance standard using a cryogenic current comparator

bridge

21

. Figure 3a shows how the mean relative deviation of R

xy

(0)

from R

K

/2 depends on the measurement current through the gra-

phene device. The quantization accuracy þ0.4+3 parts in 10

9

(mean relative deviation of 129 measurements+standard error of

the mean) inferred from our measurements at 11.6 mA and

300 mK is a four orders of magnitude improvement on the previous

best result in exfoliated graphene. This readily puts epitaxial gra-

phene devices in the same league as their semiconductor counter-

parts. Note that our result was obtained on a sample that is

substantially smaller than the semiconductor devices used for cali-

bration and without any optimization. From Fig. 3a it can be seen

that graphene is still accurately quantized at 4.2 K; however, at

this temperature, the measurement current has to be reduced to

2.3 mA, which increases the uncertainty of the data accumulated

over a comparable time interval.

To demonstrate convergence of the measurement process and to

see what kind of noise limits the precision of our measurements, in

Fig. 3b we plot the Allan deviation

22

of R

xy

(0)

from R

K

/2 against the

measurement time

t

. These data follow a 1/

t

1/2

dependence—

behaviour typical of the predominantly white (uncorrelated

random) noise. This justiﬁes the use of the standard measures of

uncertainty and suggests that these (already very accurate) results

can be further improved if one is prepared to measure for longer

and at several smaller values of the ﬁlling factor within the R

xy

¼

R

K

/2 plateau. Further development should include methods to

control the carrier density either chemically or by electrostatic

y = 15 μm

z

= 14.2 nm

a

c

b

(μm)

(μm)

0 nm

8

nm

4

nm

0

0

5

5

10

10

Graphene

V

2

−

V

2

+

V

1

+

V

1

−

7 mm

I

source

I

drain

L

W

Graphene

SiC

I

drain

I

source

x = 15 μm

V

1

− V

2

− V

3

−

V

1

+ V

2

+

V

2

+

V

3

+

V

1

+

V

1

−

V

2

−

V

3

−

V

3

+

Figure 1 | Sample morphology and layout. a, AFM images of the sample: large ﬂat terraces on the surface of the Si-face of a 4H-SiC(0001) substrate with

graphene after high-temperatur e annealing in an argon atmospher e. b, Gr aphene patterned in the nominally 2-mm-wide Hall bar conﬁguration on top of the

terraced substrate. c, Lay out of a 7 7mm

2

wafer with 20 patterned devices. Encircled are two devices with dimensions L ¼ 11.6

m

mandW ¼ 2

m

m (wire

bonded) and L ¼ 160

m

mandW ¼ 35

m

m, for which the QHE data are pr esented in Fig. 2. The contact conﬁguration for the smaller device is shown in the

enlarged image. To visualize the Hall bar this optical micrograph was tak en after oxygen plasma trea tment, which formed the graphene pattern, but before

the removal of resist.

NATURE NANOTECHNOLOGY DOI: 10.1038/NNANO.2009.474

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© 2010 Macmillan Publishers Limited. All rights reserved.

gating, although we believe that the fastest route towards the

implementation of graphene in quantum metrology lies in increas-

ing the breakdown current by taking advantage of the ﬂexibility of

device design offered by the large area of graphene on a SiC

wafer: the optimization of contacts geometry and use of multiple

parallel Hall bar devices.

−15

−15

−10

−5

R

xy

(kΩ)

R

xx

(kΩ)

0

5

10

15

a

b

c

d

0

−13.0 −12.8 −12.6 −12.4 −12.2 −12.0 −11.8 −11.6 −11.4 −11.2 −11.0

2

Resistance (Ω)

4

6

8

10

–4

–2

Voltage (μV)

0

Magnetic ﬂux density (T)

2

4

300 mK

4.2 K

−15

−10

−5

R

xy

(kΩ)

0

5

10

15

−10 −5

R

1

R

xx

R

xx

+

R

l

+

R

c

0

Magnetic ﬂux density (T)

51015

−15 −10 −5 0

Magnetic ﬂux density (T)

51015

−20 −10 0

Current (mA)

10 20

R

xx

R

xx

R

xy

R

xy

R

xy

(1)

R

xy

(1)

R

xy

(0)

R

xy

(0)

20

16

12

8

4

0

R

xx

(kΩ)

20

16

12

8

4

0

Figure 2 | Quantum Hall effect in epitaxial graphene. a,Transverse(R

xy

) and longitudinal (R

xx

) resistance of the 11.6

m

m 2

m

mdevicemeasuredatT ¼ 4.2 K

with 1

m

Acurrent.R

xy

(0)

and R

xy

(1)

repr esent Hall resistance plateaux at ﬁlling factors n ¼ 2andn ¼ 6 respectively. The carrier density n

s

¼ 6.5 10

11

cm

22

was

obtained from SdH oscillations. b,Transverse(R

xy

) and longitudinal (R

xx

) resistance of the 160

m

m 35

m

m device measured at T ¼ 4.2 K with 1

m

Acurrent.

The carrier density n

s

¼ 8.5 10

11

cm

22

was obtained from SdH oscillations. c, Measur ements of the longitudinal resistance R

xx

performed in a four-point

conﬁguration (ﬁlled red circles), which excludes contact resistances, and in a three-point conﬁguration (open blue circles), which, as well as R

xx

, includes the

contact resistance R

c

and the resistance of the leads from room-tempera tur e electronics down to the sample R

l

¼ 2.5 V. On the plateau, R

xx

is very nearly

zero and R

c

is 1.5 V for all measured contacts. These measurements wer e performed while sweeping the magnetic ﬁeld; hence there is a relativ ely large

spread. d, Determination of the breakdo wn current I

max

of non-dissipativ e transport from measurement of the current–voltag e chara cteristic in the

longitudinal direction at 14 T: I

max

13

m

A at 300 mK and I

max

5

m

A at 4.2 K. The residual longitudinal resistance was conﬁrmed as R

xx

, 0.2 mV at

300 mK measured with I

sd

¼ 12

m

A, and R

xx

, 2.4 mV at 4.2 K measured with I

sd

¼ 2.5

m

A.

0

−60

−40

−20

0

20

2(R

xy

− R

K

/2)/R

K

(ppb)

2R

xx

/R

K

(ppb)

2,000

4,000

6,000

a

b

−60

−40

−20

0

20

Allan deviation

2,000

4,000

6,000

τ

–1/2

10

−7

10

−8

10

−9

2 4 6 8 10 12 14

Current (μA)

16 18 20 22 24

0.01 0.1 1 10

Time interval (h)

Figure 3 | Determination of Hall resistance quantization accuracy. a, Mean relative deviation of R

xy

(0)

from R

K

/2 at different bias currents (ppb, parts per

billion). The value at the smallest current was measured at 4.2 K (open blue squares), and all other values at 300 mK (ﬁlled red squares). The most accurate

measurement with an 11.6

m

A source–drain current at 14 T and 300 mK was performed over 11 h. The value of R

xx

/R

K

determined in the same conditions is

also shown (black star) together with the measurement uncertainty. b, Allan deviation of R

xy

(0)

from R

K

/2 versus measurement time

t

. The square root

dependence indicates purely white noise.

LETTERS

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© 2010 Macmillan Publishers Limited. All rights reserved.

To summarize, we report quantum Hall resistance quantization

accurate to a few parts in a billion at 300 mK in a large-area epitaxial

graphene sample. Several more devices have been studied at 4.2 K,

demonstrating quantization within an accuracy of some tens in

10

9

, conﬁrming the robustness of the QHE in graphene synthesized

on the silicon-terminated face of SiC. This remarkable precision

constitutes an improvement of four orders of magnitude over the

best previous results obtained in exfoliated graphene, and is similar

to the accuracy achieved in the established semiconductor resistance

standards. In the future, improvements in measurement precision

may advance the understanding of the QHE effect itself, by deter-

mining whether there exist systematic deviations of the quantized

Hall resistance in graphene from the fundamental values at the

rational fractions of h/e

2

. Even more importantly, the experiments

have demonstrated structural integrity over hundreds of micro-

metres and revealed relative uniformity of epitaxial graphene

across a half-centimetre SiC wafer (as well as from wafer to wafer).

This supports the potential of SiC technology for microelectronics

applications possibly extending far beyond quantum metrology.

Methods

The material used in the reported experiments was produced on the Si-face of SiC;

the reaction kinetics is slower there than on the C-face because of the higher surface

energy. This aids in the well-controlled formation of homogeneous graphene

23

.

Graphene was grown at 2,000 8C and 1 atm argon gas pressure, resulting in

monolayers of graphene atomically uniform over more than 50 mm

2

, as conﬁrmed

by low-energy electron microscopy. Twenty Hall bar devices of different sizes, from

160 mm 35 mm down to 11.6 mm 2 mm were produced on two 0.5-cm

2

wafers

using standard electron-beam lithography and oxygen plasma etching (Fig. 1c).

Atomic force microscopy (AFM) images reveal that the graphene layer covers the

substrate steps like a carpet, preserving its structural integrity (Fig. 1a). Contacts to

graphene were produced by straightforward deposition of 3 nm of titanium and

100 nm of gold through a lithographically deﬁned mask followed by lift-off, with a

typical area of graphene–metal interface of 1 10

4

mm

2

for each contact. This

process favourably compares with the laborious contact preparation to two-

dimensional electron gas in conventional semiconductor technology. The

manufactured devices were not cleaned in any way before measurements. Using low-

magnetic-ﬁeld measurements, we established that the manufactured material was

n-doped, owing to charge transfer from SiC, with the measured electron

concentration lying in the range 5 10

11

to 10 10

11

cm

22

, mobility of

2,400 cm

2

V

21

s

21

at room temperature and between 4,000 and

7,500 cm

2

V

21

s

21

at 4.2 K, almost independent of device dimensions and

orientation with respect to the substrate terraces. The scattering mechanisms in the

epitaxial graphene and the role of the substrate need further investigation. As seen in

the AFM images (Fig. 1b), the graphene Hall bars are patterned across many

substrate terraces; however, the measurements of the QHE reveal that the continuity

of graphene is preserved.

Received 14 September 2009; accepted 3 December 2009;

published online 17 January 2010; corrected online 23 F ebruary 2010

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Acknowledgements

The authors would like to thank L. Wallde

´

n, T. Lo¨fwander, F. Lombardi, J. Gallop and

T. Claeson for stimulating discussions and S. Giblin and J. Williams for help with

experiments. We are grateful to the NPL Strategic research programme, Swedish Research

Council and Foundation for Strategic Research, European Union FP7 SINGLE, UK

Engineering and Physical Sciences Research Council grant no. EP/G041954 and the

Science & Innovation Award EP/G014787 for ﬁnancial support.

Author contributions

S.K., A.T. and V.F. conceived and designed the experiments. A.T., S.L., A.K., S.P., O.K., T.J.

and S.K. performed the experiments. R.Y. and M.S. contributed materials. A.T., V.F. and

S.K. analysed the data and co-wrote the paper. All authors discussed the results and

commented on the manuscript.

Additional information

The authors declare no competing ﬁnancial interests. Supplementary information

accompanies this paper at www.nature.com/naturenanotechnology.

Reprints and

permission information is available online at http://npg.nature.com/reprintsandpermissions/.

Correspondence and requests for materials should be addressed to A.T.

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