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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
NATURE NANOTECHNOLOGY | VOL 5 | MARCH 2010 | www.nature.com/naturenanotechnology186
© 2010 Macmillan Publishers Limited. All rights reserved.
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
LETTERS
NATURE NANOTECHNOLOGY | VOL 5 | MARCH 2010 | www.nature.com/naturenanotechnology 187
© 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
NATURE NANOTECHNOLOGY DOI: 10.1038/NNANO.2009.474
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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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