Density of States and Zero Landau Level Probed through Capacitance of Graphene
L. A. Ponomarenko,
1
R. Yang,
1
R. V. Gorbachev,
1
P. Blake,
1
A. S. Mayorov,
1
K. S. Novoselov,
1
M. I. Katsnelson,
2
and A. K. Geim
1
1
Manchester Centre for Mesoscience & Nanotechnology, University of Manchester,
Manchester M13 9PL, United Kingdom
2
Theory of Condensed Matter, Institute for Molecules and Materials, Radboud University Nijmegen,
Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
(Received 27 May 2010; published 21 September 2010)
We report capacitors in which a finite electronic compressibility of graphene dominates the electro-
statics, resulting in pronounced changes in capacitance as a function of magnetic field and carrier
concentration. The capacitance measurements have allowed us to accurately map the density of states D,
and compare it against theoretical predictions. Landau oscillations in D are robust and zero Landau level
(LL) can easily be seen at room temperature in moderate fields. The broadening of LLs is strongly affected
by charge inhomogeneity that leads to zero LL being broader than other levels.
DOI: 10.1103/PhysRevLett.105.136801 PACS numbers: 73.22.Pr, 81.05.ue
One of the most celebrated consequences of the Dirac-
like electronic spectrum of graphene is its zero Landau
level (LL) centered at the neutrality point (NP) and shared
by hole- and electronlike carriers [1]. Although the elec-
tronic properties become particularly interesting near the
NP, it has proven difficult to probe this regime by transport
measurements. The problem is not only the potential fluc-
tuations that move the Dirac point spatially and average out
interesting features [2]. The situation is additionally
tangled because transport coefficients become nonmono-
tonic at the NP and sensitive to scattering details, even in
high magnetic fields B [3]. Capacitance measurements
provide an alternative. If graphene is incorporated in a
capacitor as one of its electrodes, there appears a signifi-
cant contribution into the total capacitance C due to the
electronic compressibility. This contribution is often
referred to as quantum capacitance C
q
¼ e
2
D and is a
direct measure of the density of state DðEÞ¼dn=dE at
the Fermi energy E
F
(e is the electron charge; n the carrier
concentration) [4,5].
As for experimental studies of C
q
, graphene is unique for
two reasons. First, it has an atomically thin body, which
allows capacitors in which the classical, geometrical con-
tribution plays a minor role so that C
q
can completely
dominate the device’s electrostatics. Second, D is a strong
function of E
F
and, therefore, C
q
can be changed by apply-
ing gate voltage V
g
. This distinguishes graphene from
conventional two-dimensional systems in which C
q
is usu-
ally a small and constant contribution that is difficult to
discern experimentally [5]. Thanks to the V
g
dependence,
several groups have already reported the observation of C
q
in graphene [6 –9]. Their measurements showed the ex-
pected V-shape dependence centered at the NP. However,
the electron and hole branches were often strongly asym-
metric [6,7], contrary to expectations, and the absolute
value of C
q
was either impossible to determine [8,9]orit
disagreed with theory [6]. Most recently, capacitance
measurements were also employed to prove the gap open-
ing in double-gated bilayer devices [10].
In this Letter we report large (100 100 m
2
) gra-
phene capacitors with a thin (10 nm) dielectric layer and
a high carrier mobility of 10 000 cm
2
=Vsmaintained
after the fabrication. In our devices, C
q
is no longer merely
a correction but reaches 30% of the measured capaci-
tance, so that its changes with varying V
g
and B are very
pronounced. This allowed us to compare the quantum
capacitance against theoretical predictions, the task proven
impossible in the previous studies. Second, we show that,
in capacitance experiments, zero LL is extremely robust
with respect to temperature T and disorder, and clearly
seen at room T in fields of 10 T (note that B as high as 30 T
were required to observe the quantum Hall effect at room T
[11]). Third, we have analyzed the broadening of LLs as a
function of B and T, and found that a charge inhomoge-
neity (electron-hole puddles) is as important as scattering
in defining LLs’ width. In particular, the inhomogeneity
results in zero LL being wider than other LLs, which is
directly visible from our experimental curves. This obser-
vation seems in contradiction to the suggestion that zero
LL is exceptionally narrow [12]. Fourth, we observe no
splitting at the NP in fields up to 16 T, reported in transport
experiments for devices of similar quality [13].
We developed the following technology for making
graphene capacitors (Fig. 1). Large crystals were obtained
by mechanical cleavage and deposited on top of an oxi-
dized Si wafer. High-resistivity (>1kcm) wafers were
essential to avoid a contribution from parasitic capacitan-
ces. 1 nm of Al was then deposited on top of graphene in
the presence of oxygen (thermal evaporation at a speed of
1
A= sec with O
2
pressure of 0:1 mbar), which
resulted in an initial (‘‘seed’’) layer of aluminum oxide.
A thick (100 nm) layer of Al was then evaporated on top of
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graphene in high vacuum, forming the second plate of a flat
capacitor. Although only 1 nm of Al oxide was deposited,
the resulting gate dielectric appeared to be much thicker
( 10 nm) as found by atomic force and electron micros-
copy (we used one of our devices and etched away the
metallic film in potassium hydroxide). This is attributed to
the oxidation of the Al gate from below by air that diffused
through pores in the initial oxide. Despite its simplicity, the
technology allowed us to avoid any noticeable reduction in
due to the gate dielectric and change E
F
by as much as
0:5eV. was measured by using multiterminal Hall bar
devices fabricated in parallel with the capacitors. The Hall
bars had some areas covered with the top gate and some
left open. The measured were statistically indistinguish-
able for the two areas. The only notable difference was
that, in the top-gated devices, the NP was always close to
zero V
g
(remnant doping n
D
< 10
11
cm
2
). In contrast,
nongated graphene typically exhibited n
D
10
12
cm
2
.
We attribute this behavior to the screening out of an
average electrostatic potential by the nearby Al gate.
A typical dependence of C on V
g
applied between
graphene and the Al gate is plotted in Fig. 2 (4 devices
were used in this study). We have found little contribution
from parasitic capacitances (<1pF) and, to illustrate this
fact, Fig. 2(a) also shows C measured in an alternative
approach (circles) by studying magnetocapacitance oscil-
lations (figure caption). C
q
contributes as a series capacitor
[4,5]: 1=C ¼ 1=C
ox
þ 1=C
q
where C
ox
is the gate inde-
pendent geometrical capacitance of the oxide layer. The
measured dependence in Fig. 2(a) is accurately described
by two parameters [4–10,14]: (1) C
ox
that gives the satu-
ration value in the limit of high n (where D becomes large
so that C
q
C
ox
) and (2) the Fermi velocity v
F
in gra-
phene that describes the rate of changes in C at low n. The
dashed curve in Fig. 2(a) shows the best fit that yields
C
ox
0:47 F=cm
2
and v
F
ð1:15 0:1Þ10
6
m=s.
The found v
F
is in excellent agreement with other experi-
ments (e.g., [15,16]) and theory [1,17]. The C
ox
value is the
only real fitting parameter and corresponds to the classical
capacitance expected for our oxide thickness of 10 nm.
By integrating the experimental curve in Fig. 2(a),we
can find n for a given V
g
and replot the data as a function
of n andE
F
¼ @v
F
ffiffiffiffiffiffiffiffiffiffi
jnj
p
(note that the standard equation
n / V
g
no longer holds for thin gate dielectrics).
The resulting curve in Fig. 2(b) is a textbook behavior
for the quantum capacitance of Dirac fermions. As ex-
pected for an ideal case, the curves are electron-hole
symmetric, and DðE
F
Þ exhibits equal slopes (within 3%)
for the valence and conduction bands, which are given by
the single material parameter v
F
. The prominent flat region
at the NP is attributed to electron-hole puddles [16] that
smear the sharp dip in C at the NP. The smearing in
Fig. 2(b) covers a region of n 4 10
11
cm
2
which
is close to n typically observed in transport experiments
[2]. By comparing Figs. 2(a) and 2(b), one can see that
electron-hole puddles dominate the spectral smearing near
the NP (because the energy smearing E / n=E
F
).
In perpendicular B, our capacitors exhibit pronounced
oscillations [Fig. 3(a)]. Zero LL is centered at the NP (zero
E
F
) as expected for graphene’s electronic spectrum [1,15].
First, let us understand the overall shape of the C
q
curves.
DðEÞ can be considered as a superposition of LLs having
the Lorentzian shape [18]. Figure 3(b) shows results of our
numerical analysis. For graphene with LLs equally broad-
ened by scattering and, therefore, having the same width ,
we could not reach any satisfactory agreement with our
experiment for any . Theory predicts stronger oscillations
in C
q
(dashed curve) whereas in reality the peaks become
progressively smaller at small n. To explain the latter, we
take into account the charge inhomogeneity inferred from
Fig. 2(b). This provides a natural explanation for zero LL
being broader than other LLs (in terms of E) and, thus,
-2 0 2
0.30
0.35
0.40
0.45
C (µF/cm
2
)
V
g
(V)
C
ox
10 K
a
-0.4 -0.2 0.0 0.2 0.4
0
2
4
6
C
q
(µF/cm
2
)
E
F
(eV)
n = 4·10
11
cm
-2
b
-2 0 2
0
2
4
D (10
17
eV
-1
m
-2
)
√n (10
6
cm
-1
)
FIG. 2 (color online). Graphene capacitance in zero B.
(a) Capacitance (per unit area) of the device in Fig. 1 (solid
curve). For consistency, all the plots presented in the report are
taken from the same device that exhibited slightly smaller n
that the others. The dashed curve is the best fit and the horizontal
line shows C
ox
. Open symbols are the alternative measurements
using the periodicity of magnetocapacitance oscillations. For a
given B, each period in gate voltage V
g
corresponds to the
filling of one LL, and this requires an increase in carrier
concentration n ¼ 4eB=h where h is the Planck constant.
C is then calculated as en=V
g
. Despite the lower accuracy
of this approach (due to oscillations in C
q
), it provides a valuable
crosscheck. (b) C
q
as function of E
F
(bottom axis) and n (top
axis). The right axis plots the density of states, D ¼ C
q
=e
2
.
These are the same measurements as in (a), and the flat region
near zero energy results from nonlinear stretching of the x axis.
The extrapolation lines (dashed) in Fig. 2(b) should cross at zero,
and the small vertical offset can be explained by parasitic
capacitances that we deliberately did not include in the analysis
to minimize the number of fitting parameters.
FIG. 1 (color online). Photograph of one of our devices (left)
and its schematics (right). Graphene is connected to the mea-
surement circuit by the contact deposited along the device’s
perimeter. The second capacitor plate is an Al electrode (central
rectangle in the left photo).
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having the smallest height in Fig. 3(a). It also allows us to
account for the overall shape of the C
q
ðV
g
Þ curve.
Furthermore, we have often observed an electron-hole
asymmetry in C
q
such that the oscillations for electrons
are more pronounced than those for holes [Fig. 3(a)]. The
degree of this asymmetry varied from sample to sample
and could change after thermal cycling, and we attribute it
to an asymmetric potential landscape. To this end, we have
modeled the charge inhomogeneity as a sum of two
Gaussians, which can be due to, for example, macroscopic
regions with different chemical doping. The resulting de-
pendence (solid curve) reproduces the essential features of
the observed asymmetry. In general, the analysis in
Fig. 3(b) shows that, at low T, the spectral broadening is
mostly due to electron-hole puddles and the finite plays
little role in the broadening of zero LL.
The T dependence in Fig. 3(a) has allowed us to find
cyclotron mass m
c
at different V
g
. The analysis is the same
as for the case of Shubnikov-de Haas oscillations [15],
and our results (not shown) are essentially identical,
yielding m
c
/ n
1=2
with a single fitting parameter v
F
ð1:1 0:1Þ10
6
m=s. Note that such analysis is not ap-
plicable at the Dirac point where the nominal value of m
c
is
zero. This emphasizes differences between transport and
capacitance measurements. The latter allow us to directly
assess DðEÞ whereas interpretation of transport experi-
ments is generally more complicated and sensitive to scat-
tering details. Also, note that under the same conditions
our reference multiterminal devices exhibited clear quan-
tum Hall effect plateaus and wide zeros in resistance (not
shown; for examples, see Refs. [1,15]). This difference can
be explained by the fact that the capacitance spectroscopy
probes both extended and localized states, whereas elec-
tron transport occurs over the former.
Because zero LL is hard to assess in transport experi-
ments, we focus below on this particular feature. Figure 4
details its behavior with changing T and B. We start with
pointing out two qualitative observations for zero LL. First,
we do not see any hint for zero LL’s splitting in fields up to
16 T and T down to 1 K [13 ]. We believe that, if present, this
splitting for the DðEÞ curves should be smeared by electron-
hole puddles. Second, zero LL can clearly be seen in C
q
even at room T in fields down to 10 T and an increase in
D
NP
Dð0Þ can be detected in B as low as 5 T [Figs. 3(a)
and 4(a)]. To analyze the observed T dependence, Fig. 4(b)
plots D
NP
as a function of T. As for theory, each of the
thermally broadened LLs contributes to D
NP
as [19]
D
i
ðTÞ¼
n
Z
1
0
e
0
t
cosðE
i
tÞ
Tt
sinhðTtÞ
dt
-2 -1 0 1 2
0
2
4
6
B = 0:
20 K
C
q
(µF/cm
2
)
V
g
(V)
a
B = 16 T:
20 K
60 K
100 K
150 K
250 K
-4 0 4
0
2
4
6
C
q
(µF/cm
2
)
n (10
12
cm
-2
)
b
FIG. 3 (color online). Magnetic oscillations in the density of
state. (a) Experiment: C
q
¼ e
2
D as a function of V
g
at various T.
(b) Theory: C
q
in 16 T at low T. The dashed curve shows the
oscillations if all LLs were equally broadened with the half
width at half maximum (HWHM), ¼ 15 meV [20]. The dotted
curve takes into account electron-hole puddles and was obtained
as a convolution of the above broadening with an added inho-
mogeneity in n which was modeled by a Gaussian distribution
[21] with standard deviation n ¼ 4 10
11
cm
2
, the value
found in Fig. 2. The solid line is a result of a similar convolution
but assuming an asymmetric doping, which was modeled as a
sum of two Gaussians with height ratios of 7:3; n ¼ 3:5
10
11
cm
2
for both; and the separation of 7 10
11
cm
2
. Note
that the device in Fig. 3(a) has changed with respect to its state in
Fig. 2. We find such changes to occur often after thermal cycling.
0 100 200
1.0
1.2
1.4
1.6
D
NP
(10
17
eV
-1
m
-2
)
T (K)
b
16 T
100 200
400
500
Γ
Σ
(K)
T (K)
-0.4 0.0 0.4
1.0
1.5
2.0
2.5
C
q
(
µF/cm
2
)
V
g
(V)
250 K
a
016
0.6
1.0
D
NP
B (T)
1.0
1.5
C
q
FIG. 4 (color online). Density of states near the NP. (a) The
emergence of zero LL with increasing B near room T. The
curves cover the range from 0 to 16 T in steps of 2 T. Zero
LL becomes clearly visible above 10 T. We avoided T above
250 K because the measurements become hysteretic due to an
increased mobility of adsorbates, similar to transport experi-
ments [22]. The inset shows C
q
and D at the NP as a function
of B in units of F=cm
2
and 10
17
eV
1
m
2
, respectively. The
solid curve is the best fit [19]. (b) T dependence of D
NP
at 16 T
(symbols). The curves are theory fits. Inset: same as in the main
figure but in terms of the width of zero LL.
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where
0
is the HWHM at T ¼ 0 and E
i
¼v
F
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2@eBjij
p
is
the energy of ith LL (i ¼ 0, 1; ...). The dashed theory
curve in Fig. 4(b) plots D
NP
if only zero LL is taken into
account, which is justified because at 16 T the distance to
the next LL is larger than
0
and T. If the next LL (i ¼1)
is included, this does not improve the fit (solid curve). We
find
0
30 meV that is twice larger than the one used in
the analysis in Fig. 3(b) because it now includes the smear-
ing E due to charge inhomogeneity. The inset in Fig. 4(b)
replots the data from the main figure in terms of the LL
broadening by using the standard expression
¼
n=D
NP
where n is the number of carriers at each LL.
This presentation is more intuitive because one can readily
see that zero LL has an intrinsic broadening
0
and broad-
ens approximately linearly with T as
0
þ T where
1:1 0:2, in agreement with general expectations.
Our model can also describe the field dependence of D.
The inset in Fig. 4(a) shows D
NP
as a function of B at
250 K, and the solid curve is the theory fit [19] with
broadening 50 meV found in Fig. 4(b). In general, our
analysis in Figs. 3 and 4 shows that the measured DðEÞ and
its quantization are in good agreement with theory.
In summary, changes in the density of states can com-
pletely dominate the behavior of graphene capacitors with
a thin dielectric layer. This provides a useful approach for
studies of quantization phenomena in graphene and, espe-
cially, its lowest Landau level that is difficult to assess in
transport experiments. In the capacitance measurements,
zero LL is easily observable at room T. This requires B of
only 10 T, which is probably the record for the Landau
quantization observed in any material. However, contrary
to previous experiments using similar-quality devices, we
find no indications that zero LL is narrower than other LLs
[12] and no splitting of zero LL is observed in fields up to
16 T [13]. This can be due to the difference in contributions
from extended and localized states in transport and capaci-
tance measurements but may also call for alternative inter-
pretations. Finally, we mention that our devices prove the
feasibility of bipolar variable capacitors that can have large
C 1 F=cm
2
per layer and be controlled by small volt-
ages ( 0:1V).
The work was supported by EPSRC (UK), the Ko
¨
rber
Foundation, the Office of Naval Research, the Air Force
Office of Scientific Research and the Royal Society.
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![FIG. 4 (color online). Density of states near the NP. (a) The emergence of zero LL with increasing B near room T. The curves cover the range from 0 to 16 T in steps of 2 T. Zero LL becomes clearly visible above 10 T. We avoided T above 250 K because the measurements become hysteretic due to an increased mobility of adsorbates, similar to transport experiments [22]. The inset shows Cq and D at the NP as a function of B in units of F=cm2 and 1017 eV 1 m 2, respectively. The solid curve is the best fit [19]. (b) T dependence of DNP at 16 T (symbols). The curves are theory fits. Inset: same as in the main figure but in terms of the width of zero LL.](/figures/fig-4-color-online-density-of-states-near-the-np-a-the-1bd7l707.webp)
![FIG. 3 (color online). Magnetic oscillations in the density of state. (a) Experiment: Cq ¼ e2D as a function of Vg at various T. (b) Theory: Cq in 16 T at low T. The dashed curve shows the oscillations if all LLs were equally broadened with the half width at half maximum (HWHM), ¼ 15 meV [20]. The dotted curve takes into account electron-hole puddles and was obtained as a convolution of the above broadening with an added inhomogeneity in n which was modeled by a Gaussian distribution [21] with standard deviation n ¼ 4 1011 cm 2, the value found in Fig. 2. The solid line is a result of a similar convolution but assuming an asymmetric doping, which was modeled as a sum of two Gaussians with height ratios of 7:3; n ¼ 3:5 1011 cm 2 for both; and the separation of 7 1011 cm 2. Note that the device in Fig. 3(a) has changed with respect to its state in Fig. 2. We find such changes to occur often after thermal cycling.](/figures/fig-3-color-online-magnetic-oscillations-in-the-density-of-fsoq183a.webp)
