doi: 10.1149/1.3569908
2011, Volume 35, Issue 3, Pages 161-172.
ECS Trans.
Boyd and Marc W. Bockrath
Nai-Chang Yeh, Marcus L. Teague, Renee T. Wu, Sinchul Yeom, Brian Standley, David
Charging Effects in CVD-Grown Graphene on Copper
Nano-Scale Strain-Induced Giant Pseudo-Magnetic Fields and
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Nano-Scale Strain-Induced Giant Pseudo-M
agnetic Fields and Charging Effects
in CVD-Grown Graphene on Copper
N.-C. Yeh
a
, M. L. Teague
a
, R. T.-P. Wu
a
, S. Yeom
b
, B. L. Standley
b
, D. A. Boyd
b
,
and M. W. Bockrath
b,c
a
Department of Physics, California Ins
titute of Technology, Pasadena, California 91125,
USA
b
Department of Applied Physics, Cali
fornia Institute of Technology, Pasadena,
California 91125, USA
c
Department of Physics, University of
California, Riverside,
California 92521, USA
Scanning tunneling microscopic
and spectroscopic (STM/STS)
studies of graphene grown by chem
ical vapor deposition (CVD) on
copper reveal that the monolayer
carbon structures remaining on
copper are strongly strained and
rippled, with different regions
exhibiting different lattice structures
and local electronic density of
states (LDOS). The large and non-uniform strain induces pseudo-
magnetic field up to
∼
50 Tesla, as manifested by the integer and
fractional pseudo-magnetic field
quantum Hall effects (IQHE and
FQHE) in the LDOS of graphe
ne. Additionally, ridges appear
along the boundaries of different lattice structures, which exhibit
excess charging effects. For graphe
ne transferred from copper to
SiO
2
substrates after the CVD growth, the average strain and the
corresponding charging effects and
pseudo-magnetic fields become
much reduced. Based on these findings, we consider realistic
designs of strain-engineered gr
aphene nano-transistors, which
appear promising for nano-e
lectronic applications.
The electronic properties of graphene exhi
bit significant depende
nce on the surrounding
environment and high susceptibility to diso
rder because of the single layer of carbon
atoms that behave like a soft membrane and
because of the fundamental nature of Dirac
fermions (1). In general, the sources of disord
er in graphene may be divided into intrinsic
and extrinsic disorder (1). Examples of in
trinsic disorder include surface ripples and
topological defects (1), whereas extrinsic diso
rder can come in many different forms,
such as adatoms, vacancies, extended defects including edges and cracks, and charge in
the substrate or on top
of graphene (1).
There are two primary effects associated with
disorder on the electr
onic properties of
graphene (1). The first effect is a local change
in the single site energy that leads to an
effective shift in the chemical potential for Di
rac fermions (1). One example of this type
of disorder stems from charge impurities. The
second type of disorder effect arises from
changes in the distance or angles between the
p
z
orbitals (1). In th
is case, the hopping
energies (and thus hopping amplitudes) betw
een different lattice sites are modified,
leading to the addition of a new term to the original Hamiltonian. The new term results in
the appearance of vector (gauge)
A
and scalar potentials
φ
in the Dirac Hamiltonian (1).
The presence of a vector potential in the pr
oblem indicates that an effective magnetic
field
B
S
= (
c
/
ev
F
)
∇
×
A
is also present, with opposite directions for the two inequivalent
Dirac cones at K and K
′
so that the global time-reversal symmetry is preserved. Here
v
F
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denotes the Fermi velocity of the Dirac fermi
ons. Similarly, the presence of a spatially
varying scalar potential can lead to local
charging effects known as self-doping (1). One
mechanism for inducing pseudo-magnetic fields
and charging effects is strain (2-10),
which is further elaborated below.
The strong susceptibility of graphene to
external influences can in fact provide
opportunities for engineering uni
que properties of graphene.
For instance, it has been
theoretically proposed that a
designed strain aligned along
three main crystallographic
directions induces str
ong gauge fields that effectively
act as a uniform pseudo-magnetic
field on the Dirac electrons (1
-3). This prediction was firs
t verified empirically via
scanning tunneling microscopy (STM) studi
es of graphene nano-bubbles grown on
Pt(111) substrates, with relatively uniform
pseudo-magnetic fields up to 600 Tesla over
each graphene nano-bubble (8). More specifically, the strain-induced gauge potential
A
= (A
x
, A
y
) may be related to the two-dimensional strain field
u
ij
(
x,y
) by the following
relation (with
x
-axis along the zigzag direction) (1,2):
,
[1]
where
t
denotes the nearest hopping constant,
a
represents the nearest carbon-carbon
distance,
β
is a constant ranging from 2 to 3 (1),
and for a two-dimensional displacement
field
u
= (
u
x
,
u
y
) we denote
u
xx
≡
(
∂
u
x
/
∂
x
),
u
xy
≡
(
∂
u
x
/
∂
y
) and
u
yy
≡
(
∂
u
y
/
∂
y
). It is clear from
Eq. [1] that any uniaxial strain
would lead to a uniform gaug
e potential and therefore no
pseudo-magnetic field.
In this work, we report our discovery of strained-induced giant pseudo-magnetic fields
and charging effects in graphene grown by means of chemical vapor deposition (CVD)
on copper (11). We attribute our findings to th
e result of large and non-uniform strain
induced by the expansion of graphene and th
e contraction of the copper substrate upon
cooling from the CVD-growth temperature
(~ 1000°C) to low temperatures (9,10). The
strain-induced pseudo-magnetic fields give rise
to discrete peaks in the electronic density
of states at quantized energies that are c
onsistent with the occurrence of quantum Hall
effects in nano-scales (1-3), and the pseudo-ma
gnetic fields in strongly strained regions
are found to exceed 50 Tesla (9). Additionally, the strain-induced charging effects
produce localized high conductance regions wh
erever maximum strain
s occur (9). Both
the pseudo-magnetic fields and charging effect
s diminish after the CVD-grown graphene
is transferred from the origin
al copper substrate to a SiO
2
substrate (9). Based on these
findings, we propose realistic designs of strain
-engineered graphene
nano-transistors and
examine the expected performance, and we
find the results promising for nano-electronic
applications.
Experimental
Our primary experimental tool for investig
ating the local electr
onic and structural
correlations of graphene is a homemade cr
yogenic STM, which is compatible with high
magnetic fields and also capable of variable
temperature control from room temperature
to 6 K, with a vacuum level of ~ 10
-10
Torr at the lowest te
mperatures. For studies
reported in this work, the measurements were made at 77 K and 300 K under high
vacuum (
<
10
-7
Torr) and in zero magnetic fields.
Both topographic and spectroscopic
()
ln
ln
22
xx
yy
xx
yy
xy
xy
uu
uu
ta
uu
aa
β
−−
−∂
∂
=≡
−−
⎛⎞⎛⎞
⎜⎟⎜⎟
⎝⎠⎝⎠
A
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measurements were performed simultaneously
at every location in a
(128×128) pixel grid.
At each pixel location, the tunnel junction wa
s independently established so that the
junction resistance of 1.5 G
Ω
was maintained across the sample. The differential
conductance, (
dI/dV
), was calculated from the best
polynomial fit of each current (
I
) vs.
bias voltage (
V
) curve.
Our procedures for CVD growth of graphene
on copper foils were largely consistent with
the literature (11). Specifically, gra
phene films were primarily grown on 25-
μ
m thick Cu
foils in a furnace consisting of a fused silica t
ube heated in a split tube furnace. The fused
silica tube loaded with Cu foils was first evacuated and then back filled with hydrogen,
heated to ~ 1000°C, and maintained under
partial hydrogen pressure. A flow of CH
4
was
subsequently introduced for a desired period
of time at a total pressure of 500 mTorr.
Finally, the furnace was cooled to room temp
erature, and the Cu foils coated with
graphene appeared shiner relative to the as-received Cu foils, consistent with previous
reports (11). The graphene sheets thus prep
ared were found to be
largely single- or
double-layered based on Raman spectroscopic st
udies. For the STM measurements of the
CVD-grown graphene on copper, the sample wa
s first cleaned and th
en loaded onto the
STM probe. The STM probe was subsequently evacuated to high vacuum condition (
<
10
-7
Torr), and the sample was studied at both ~ 77 K and room temperature. Both
topographic and spectroscopic measurements were carried out.
To transfer CVD-grown graphene samples to SiO
2
substrates, SiO
2
substrates were
prepared by first thermally
grown a 290 nm thick SiO
2
layer on the silicon wafer (p-type
h100i), followed by gentle sonication of the s
ubstrate in acetone and
then pure alcohol for
about two minutes. The substrate was then ba
ked at 115°C on a hotplate, nitrogen blown
dry while cooling down. Next, a layer of
PMMA was deposited on top of the CVD-
grown graphene on copper as scaffolding, and then the copper substrate was removed
with nitric acid. The PMMA/graphene sa
mple was subsequently placed on a SiO
2
substrate with the graphene side down. Fi
nally, the PMMA was removed with acetone.
The transferred graphene was cleaned and a
nnealed in an argon atmosphere at 400°C for
30 minutes. Finally, electrodes were created
by thermally evaporating 2.5 nm chromium
and 37.5nm gold through an aluminum foil sh
adow mask. Prior to STM measurements,
the sample was rinsed again gently with
acetone followed by pure alcohol to remove
possible organic surface contaminants. The ST
M/STS studies on the transferred graphene
sample on SiO
2
were all conducted on th
e single-layer region.
Results and Discussion
We observed significant differences in th
e strain-induced effects between the CVD-
grown graphene before and after transferred from Cu to SiO
2
, as detailed below.
CVD-Grown Graphene on Cu
The large ripples associated with the C
VD-grown graphene on copper (9,10) resulted
in significant lattice distorti
on to the arrangements of carbon atoms. As exemplified in the
atomically resolved topographic image in
Fig. 1a, different atomic arrangements are
apparent in different re
gions of the (3.0×3.0) nm
2
area in view. Specifi
cally, we find that
in a more relaxed region of the graphene sample (denoted as the
α
-region in the upper
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portion of the topographic map) the atomic
structure appeared to resemble the
honeycomb or hexagon-like latti
ce with slight distortion. On the other hand, for spectra
taken on the areas (
β
-region) slightly below
an apparent ridge, th
e atomic arrangements
deviate strongly from those
of graphene, showing either nearly square-lattice or
disordered atomic structure in the
β
-region. Additionally, we find certain areas that are
completely disordered without atomic resolution, as exemplified in Fig. 2a and denoted
as the
γ
-region.
Figure 1.
Strain-induced effects on the nano-scale st
ructural and electr
onic properties of
CVD-grown graphene on Cu, taken with a cr
yogenic STM at 77 K: (a) A representative
atomically resolved topography over a (3.0×3.0) nm
2
area, showing different atomic
arrangements of the graphene sample in
view, with slightly
skewed honeycomb (or
hexagonal) lattices in the
α
-region and a combination of nearly squared and disordered
atomic arrangements in the
β
-region. (b) Tunneling conductance
dI
/
dV
-vs.-
V
spectra
along the line-cut shown by the white dashed
line in (a). (c) Representative tunneling
conductance spectra after the subtraction of a smooth conductance background, showing
quantized conductance peaks th
at are associated with st
rain-induced pseudo-magnetic
fields (
B
S
). (d) (
E
−
E
Dirac
)-vs.-|
n
|
1/2
for a number of spectra taken in the
α
and
β
regions,
showing that the data for each point spectru
m follow a straight line whose slope it
proportional to the pseudo-magnetic field
B
S
. The slopes of all (
E
−
E
Dirac
)-vs.-|
n
|
1/2
curves
taken from different point spectra range from
B
S
~ 30 Tesla for most point spectra in the
α
-region to
B
S
~ 50 Tesla for the strongest
strained areas in the
β
-region.
To investigate the effect of lattice distortion on the local density of states (LDOS) of
graphene, we compare the normalized tunneling conductance (
dI
/
dV
)/(
I
/
V
)-vs.-
V
spectra
obtained from different regions of atomic a
rrangements. As shown in Fig. 1b for a series
of tunneling spectra along the
dashed line indicated in Fi
g. 1a, we observe apparent
spatial variations in the spectra that depe
nd sensitively on the local strain. Moreover, we
notice sharp conductance peaks in the tunneling
spectra. If we subtract off a parabolic
conductance background due to the copper contribution (indicated by the dashed lines in
Fig. 1b) and plot the resu
lting conductance against (
E
−
E
Dirac
) where
E
Dirac
is the Dirac
energy, we find distinct peaks occurri
ng at energies proportional to |
n
|
1/2
for
n
= 0,
±
1,
±
2,
±
3, as exemplified in Fig. 1c. Furthermore, for
β
-type spectra, sharp peaks at energies
proportional to sgn(
n
)|
n
|
1/2
, such as those proportional to
±
(1/3)
1/2
,
−
(2/3)
1/2
and (5/3)
1/2
,
are clearly visible. These peaks are in cont
rast to the weak “humps” occurring in the
α
-
type spectrum at energies proportional to (1/3)
1/2
and
−
(5/3)
1/2
. By plotting (
E
−
E
Dirac
) vs.
|
n
|
1/2
data for multiple spectra taken in both the
α
and
β
-regions in Fig. 1d, we find that
2
-2
-2
2
0
1/3
-1/3
2
-1
-2
-3
3
3
-0.4 -0.2 0 0.2 0.4
0.4
(B
S
~ 50 T)
(B
S
~ 30 T)
0 1 2
1/2
3
1/2
|n|
1/2
α
β
α
β
1
1
1
-1
-1
(c)
E-E
Dirac
(eV)
1
-1/3
1/3
-1
0
(a) (b)
(d)
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for every point spectrum, the energies at which the conductance peaks occur all fall
approximately on one linear curve in the (
E
−
E
Dirac
) vs. |
n
|
1/2
plot, although the slope
varies from point to point. In
general, the slopes for the
β
-type spectra are larger than
those for the
α
-type spectra.
The results manifested in Figs. 1c – 1d can
be understood in terms of the presence of
strain-induced pseudo-magnetic fields
B
S
, with larger strain giving rise to larger
B
S
values
(2,3,8,9). Specifically, the pseudo-Landau levels
E
n
of Dirac electrons under a given
B
S
satisfy the following relation:
,
[2]
where
n
denotes either integers or fractional numbers. Therefore, the slope of (
E
−
E
Dirac
)
vs. |
n
|
1/2
for each point spectrum is directly proportional to (
B
S
)
1/2
. Using
v
F
~ 10
6
m/s, we
find that the pseudo-magnetic fields range fr
om approximately 29 Tesla for most spectra
in the
α
-region to 50 Tesla for most strained areas in the
β
-region.
Although these
B
S
values are smaller than those
reported for graphene nano-bubbles
(8) and are also inhomogeneous, we note that
the conductance peaks associated with the
CVD-grown graphene are in fact
more distinct, being clearly
visible even at a relatively
high temperature of 77 K as opposed to the
weaker speaks associated with individual
graphene nano-bubbles at 4.2 K. Moreover, the
B
S
values for the CVD-grown graphene
on Cu are inhomogeneous in space due to comp
licated strain distributions, which is in
sharp contrast to the nearly homogeneous
B
S
values in graphene nano-bubbles. In
particular, pronounced fractional
n
-values in addition to the integer
n
-values are observed
for the first time in the strongly strained
β
-region, implying pseudo-magnetic field-
induced fractional quantum Hall effect (FQHE
) besides the integer quantum Hall effect
(IQHE). The occurrence of strain-induced FQ
HE in CVD-grown graphene may be due to
the need of including more complicated gauge po
tentials of the Chern-Simons type in the
Hamiltonian besides the simple gauge potential
given in Eq. [1] in order to fully account
for the behavior of Dirac fermions under ve
ry strong and inhomogeneous strain. This
situation is similar to the FQHE in two-di
mensional electron systems (2DES) induced by
strong external magnetic fields. In the latter
case the FQHE must be described in terms of
the effective Chern-Simons theory (12-15), wh
ich is in addition to the simple vector
potential associated with the
external magnetic field that
can only account
for the IQHE.
However, we note that to date there have not
been self-consistent th
eoretical calculations
for strain-induced FQHE.
The aforementioned strain-induced pseudo-
magnetic fields and IQHE reflect the
unique characteristics of the Dirac fermions
in graphene, namely, the massless linear
energy dispersion relation. This is in stark co
ntrast to the massive and parabolic energy
dispersion relation found in semiconducting 2D
ES. On the other hand, for the amorphous
regions of the CVD-grown graphene sample
shown in Fig. 2, we find that the spectral
characteristics differ fundamentally from those
of Dirac fermions. As exemplified in Figs.
2b and 2c, most point sp
ectra in the amorphous
γ
-region are largely smooth and parabolic
in energy, which suggests that
the characteristics of c
onduction carriers in amorphous
graphene differ fundamental from the behavi
or of Dirac fermions. However, further
theoretical studies would be necessa
ry to shed light on this finding.
()
2
sgn
2
FS
n
EnevBn
=
=
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(
a
)
(
b
)
(
c
)
a
b
c
Figure 2.
Strain-induced strong la
ttice distortion in CVD-
grown graphene on Cu,
leading to amorphous structures in certain
areas: (a) Topography of a representative
(3.0×3.0) nm
2
area, showing amorphous behavior in the
γ
-region. (b) (
dI
/
dV
)/(
I
/
V
)-vs.-
V
point spectra at three locations denoted by a, b, c in (a). All spectra exhibit a parabolic
background and are relatively smooth. (c) The three tunneli
ng spectra in
(b) after the
subtraction of a smooth conducta
nce background. Here the spectra are shifted vertically
for clarity. All data were taken at 77 K.
CVD-Grown Graphene Transferre
d from Cu substrate to SiO
2
We may further verify the eff
ect of strain on the electroni
c properties of graphene by
conducting a comparative study of CVD-grown
graphene transferred from Cu to SiO
2
. In
general we find that the height variations a
nd the corresponding strain in the transferred
graphene samples are much reduced (9), as exemplified in Fig. 3a. Hence, the
conductance spectra for most areas of the
sample are quite smooth without quantized
conductance peaks, as exemplified in Fig. 3b. However, occasional quantized LDOS
spectra are still present in regions that app
ear to be remnant “ridges” of the CVD-grown
graphene, as shown in Fig. 3c. These quantiz
ed peaks still obey the general relation (
E
−
E
Dirac
)
∝
|
n
|
1/2
except that the slopes are much sma
ller, implying much suppressed pseudo-
magnetic fields, (Fig. 3d). This is consistent
with the significant reduction in strain upon
transferring the graphene sample from Cu to SiO
2
.
Estimates of the Strain Fields
In contrast to the relatively weak latti
ce distortion in exfoliated graphene on SiO
2
substrates (4,10), the severe na
no-scale lattice distortion exemplified in Fig. 1a makes it
difficult to compute the resulting strain fields
directly. Nonetheless, we may estimate the
magnitude of strain in a severely distorte
d graphene sample by considering the strain
induced by corrugations along the
out of plane direction. In th
is case, the typical strain
〈
|
u
|
〉
is of order of the square of the height variation
l
over a strained region
L
,
〈
|
u
|
〉
~
(
l
/
L
)
2
(2,16,17). Hence, the
relative spatial varia
tions of the strain may be obtained from
the topography
z
(
x
,
y
) if we plot maps of [
∂
z
(
x
,
y
)/
∂
x
]
2
, [
∂
z
(
x
,
y
)/
∂
y
]
2
and [(
∂
z
/
∂
x
)
2
+(
∂
z
/
∂
y
)
2
].
As illustrated in Figs. 4a-4c for the maps of [
∂
z
(
x
,
y
)/
∂
y
]
2
, [
∂
z
(
x
,
y
)/
∂
x
]
2
and the quantity
[(
∂
z
/
∂
x
)
2
+(
∂
z
/
∂
y
)
2
] over the same area shown in Fig. 1a, the maximum strain occurs right
below the ridge that separates the
α
- and
β
-regions, consistent with our finding of
maximum pseudo-magnetic fields and most
pronounced FQHE in this area. Additional
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large strain regions also appear in the lo
wer section of the map, which correspond to
areas between two other lower ridge-like features.
Figure 3.
Topography and spectroscopy of a CVD-grown graphene sample transferred
from Cu to SiO
2
: (a) A representative t
opography over a (50×50) nm
2
area, showing
mostly flat regions (represented by the
α
-region) as well as regions containing ridges
(represented by the
β
-region) that are associated w
ith the rippling phenomena of the
CVD-grown graphene. (b) Representativ
e tunneling conductan
ce spectra of the
α
- and
β
-
regions, showing smooth graphene-like spectra in the
α
-regions and sharp peaks at the
β
-
region. (c) Representative tunneling conductance
spectra after the subtraction of a smooth
conductance background, showing strain-i
nduced quantized conductance peaks at
quantized energies with |
n
| = 1, 2, 3 only in the
β′
-region. (d) (
E
−
E
Dirac
)-vs.-|
n
|
1/2
for
spectra taken in the
β
-region, showing that the conductan
ce peaks fall on one linear curve
that corresponds to
B
S
= 8
±
1 Tesla.
Figure 4.
Estimate of the spatial variations of the typical strain
〈
|
u
|
〉
over the same
(3.0×3.0) nm
2
area shown in Fig. 1a: (a) [
∂
z
(
x
,
y
)/
∂
y
]
2
map, (b) [
∂
z
(
x
,
y
)/
∂
x
]
2
map, and (c)
[(
∂
z
/
∂
x
)
2
+(
∂
z
/
∂
y
)
2
] ~
〈
|
u
|
〉
map, showing maximum strain i
mmediately below the ridge in
the upper half of the scanned region and addition
al strained areas in the lower section of
the map that situates between
two lower ridge-like features.
Strain-Induced Charging Effects
The strain found in the CVD-grown graphene
on copper is not purely shear but also
contains dilation/compression components. The la
tter is theoretically predicted to gives
rise to an effective scalar potential
V
(
x
,
y
) and therefore a static
charging effect (3,16,17)
(a)
(b)
(c)
(d)
β
β
α
α
β
(a)
(b)
(c)
[
∂
z
(
x
,
y
)/
∂
y
]
2
[
∂
z
(
x
,
y
)/
∂
x
]
2
[(
∂
z
/
∂
x
)
2
+ (
∂
z
/
∂
x
)
2
]
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(a)
(b)
(c)
(d)
(e)
(f)
n
= 0
n
= 1/3
n
= 2/3
n
= 1
n
= 2
n
= 3
in addition to the aforementioned pseudo-magnetic field. The scalar potential
V
(
x
,
y
) is
given by (3,16,17):
,
[3]
where
V
0
~ 3 eV, and
〈
|
u
|
〉
is the dilation/compression strain. While the charging effect
may be largely screened if the height variation
l
is much smaller than the magnetic length
l
B
≡
[
Φ
0
/(2
π
B
S
)]
1/2
, where
Φ
0
is the flux quantum (1-3), we note that for
B
S
ranging from
30 to 50 Tesla, the corresponding
l
B
ranges from 5.5 nm to 3.5 nm. Given that the height
variation
l
over the sample area shown in Fig. 1a
is on the order of 1
nm, comparable to
the magnetic length, we expect significant ch
arging effect in the CVD-grown graphene
on Cu.
To visualize the charging effect, we show in Figs. 5a – 5f the conductance maps at
different constant bias voltages over the same (3.0×3.0) nm
2
area. The bias voltages are
chosen to represent the ps
eudo-Landau levels of the
β
-type spectra for
n
= 0, 1/3, 2/3, 1, 2,
3. A nearly one-dimensional high-conductance re
gion close to the most strained region
immediately below the ridge is clearly vi
sible for maps associated with smaller
n
values,
confirming the notion of strain-induced chargi
ng effects. For higher energies (larger
n
values), the high-conductance re
gion becomes less confined, which is reasonable because
of the higher confinement energies (
∝
|
n
|
1/2
) required for the Dirac electrons.
Figure 5.
Constant-voltage tunneli
ng conductance maps at quan
tized energies of the
pseudo-Landau levels over a (3.0
×
3.0) nm
2
area and for
T
= 77 K: (a)
n
= 0, (b)
n
= 1/3,
(c)
n
= 2/3, (d)
n
= 1, (e)
n
= 2, (f)
n
= 3, where the
n
values are referenced to the
β
-type
spectra with
B
S
~ 50 Tesla. For smaller
n
values, an approximat
ely one-dimensional high-
conductance “line” appears near
the topographical ridge wher
e the most abrupt changes
in height occur, suggesting significant ch
arging effects. The confinement of high-
conductance region diminishes with increasing
n
.
()
()
00
,
xx
yy
Vxy V u u
V
=+=
u
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The Fourier transformation (F
T) of the constant-energy conductance maps for integer
n
values are given in Figs. 6a – 6d. We find that for
n
equals to an even integer, the FT-
LDOS map exhibits sharper di
ffraction spots, implying better defined characteristic
wave-vectors for the Dirac fermions, which in co
ntrast to the broader distributions of the
spectral weights and therefore more di
ffusive motion of Dirac fermions for
n
equals to an
odd integer. The physical origin for the spec
tral differences in the FT-LDOS maps for
varying
n
values remains unknown and awaits furt
her theoretical inve
stigations.
Figure 6.
Fourier-transformation (FT) of the conduc
tance maps in Fig. 5 is shown in the
reciprocal space, with the white hexagon
representing the first
Brillouin zone of
graphene: (a)
n
= 0, (b)
n
= 1, (c)
n
= 2, (d)
n
= 3, where the
n
values are referenced to the
β
-type spectra with
B
S
~ 50 Tesla, and the units for the
k
x
and
k
y
axes are in (nm)
−
1
. For
n
equals to an even integer, the FT-LDOS
map exhibits sharper spots, implying better
defined wave-vectors for the Dirac fermions. In contrast, more extended momentum
distributions appear for
n
equals to an odd integer, suggesting more diffusive motion of
the Dirac fermions.
Discussion
Having established the existence of strain
-induced pseudo-magnetic fields at the nano-
scale, we compare the resulting electrical tr
ansport characteristics with those induced by
real magnetic fields. This quantitative compar
ison is essential to assessing the feasibility
of strain-engineered graphene nano-devices for electronic
applications. For realistic
strained graphene devices, we
may develop arrays of nano-
dots with optimized geometry
on the substrates for graphene so that the
strain thus induced corresponds to large and
uniform pseudo-magnetic fields on each nano-d
ot, similar to the situation demonstrated
for the graphene nano-bubbles (8).
In Figs. 7a – 7d we consider the electrical
transport properties (l
ongitudinal conductivity
σ
xx
, Hall conductivity
σ
xy
, longitudinal resistivity
ρ
xx
, and Hall resistivity
ρ
xy
) of graphene
under a uniform external magnetic field
B
= 14 Tesla and as a function of the gated
voltage, following the empirical results in
Ref. (18). For comparison, we consider a
uniform strain-induced pseudo-magnetic field of
B
S
= 50 Tesla and simulate the
corresponding
σ
xx
,
σ
xy
,
ρ
xx
, and
ρ
xy
vs. energy (
E
) in Figs. 7e-7h. Based on the simulated
results, we note that a sizable on/off resist
ance ratio can be achieved by engineering the
strained graphene into a transistor. Specifically, using the simulated results for
ρ
xx
-vs.-
E
(a)
n
= 0
(b)
n
= 1
(c)
n
= 2
(d)
n
= 3
0.07
π
⎛⎞
⎜⎟
⎝⎠
0
0.07
π
⎛⎞
−
⎜⎟
⎝⎠
0.07
π
⎛⎞
⎜⎟
⎝⎠
0
0.07
π
⎛⎞
−
⎜⎟
⎝⎠
0.07
π
⎛⎞
⎜⎟
⎝⎠
0
0.07
π
⎛⎞
−
⎜⎟
⎝⎠
0.07
π
⎛⎞
⎜⎟
⎝⎠
0
0.07
π
⎛⎞
−
⎜⎟
⎝⎠
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in Fig. 7g, we obtain a sizable on/off source-drain resistance ratio [(
ρ
xx
)
on
/(
ρ
xx
)
off
] > 10 for
E
on
= 0 and
E
1
<
E
off
<
E
2
, where
E
n
with
n
= 1, 2 follows the relation in Eq. [2]. The
advantages of such strain-engineered graphene nano-transistors include the smallness of
these nano-devices and the fact that the switching voltages and the magnitude of
[(
ρ
xx
)
on
/(
ρ
xx
)
off
] can be designed geometrically by controlling the surface texture of the
substrate for graphene.
Figure 7.
Comparison of the energy-dependent elec
trical transport properties of graphene
under a real magnetic field
B
((a) – (d)) and a strain-i
nduced pseudo-magnetic field
B
S
((e) – (h)): (a) longitudinal conductivity
σ
xx
, (b) Hall conductivity
σ
xy
, (c) longitudinal
resistivity
ρ
xx
and (d) Hall resistivity
ρ
xy
for
B
= 14 Tesla. (e) Long
itudinal conductivity
σ
xx
, (f) Hall conductivity
σ
xy
, (g) longitudinal resistivity
ρ
xx
and (h) Hall resistivity
ρ
xy
for
B
S
= 50 Tesla. We note that the sharp
ρ
xx
peak at the Dirac point of the strained graphene
may be employed to produce a significant on/o
ff ratio in a transistor. More specifically,
the source-drain resistance of a graphene nano
-transistor may be controlled by the gate
voltage, and a large resistance ratio [(
ρ
xx
)
on
/(
ρ
xx
)
off
] >~ 10 can be achieved for gate
voltages switching between
E
on
and
E
off
, with
E
on
= 0 and
E
1
<
E
off
<
E
2
, where
E
n
with
n
= 1, 2 follows the relation in Eq. [2]. For
B
S
= 50 Tesla and
v
F
= 10
5
m/s, we have
E
1
~
0.25 eV.
For realistic implementation of strained graphe
ne nano-transistors, we need to consider
an assembly of arrays of such transistors, which may be considered as a network of serial
and parallel connections of both
strained and unstrained graphene resistors. In Fig. 8a we
simulate the
ρ
xx
-vs.-
V
g
data (
V
g
being the gate voltage) for a serial connection of equally
weighted strained and unstrained graphene
resistors, and in Fig. 8b we consider the
ρ
xx
-
vs.-
V
g
data for the parallel connection of two se
ts of graphene resist
ors in Fig. 8a. We
find that the overall [(
ρ
xx
)
on
/(
ρ
xx
)
off
] ratio is better preserved in the serial connections,
whereas parallel connections tend to reduce the on/off ratios. Overall, the optimal strain-
induced effects can be achieved by maximizing
the weight associated with the strained
(a)
(b)
(c)
(d)
(e)
(f) (g) (h)
(
E
2
/
e
)
(
E
1
/
e
)
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graphene. Furthermore, the fabrication of
arrays of nano-dots on the substrates for
straining graphene can be achieved by means
of either standard na
no-lithography or self-
assembly techniques (19-21).
Figure 8.
Modeling the device performance of a
rrays of strain-eng
ineered graphene
nano-transistors embedded in
unstrained graphene: (a)
ρ
xx
-vs.-
V
g
characteristics for a
serial connection of two equall
y weighted strained and unstra
ined graphene resistors with
resistivity (
ρ
xx
)
S
and (
ρ
xx
)
0
, respectively, so that
ρ
xx
= 0.5(
ρ
xx
)
S
+ 0.5(
ρ
xx
)
0
, and the
resistivity ratio [(
ρ
xx
)
on
/(
ρ
xx
)
off
] ~ 7 for
V
on
= 0 and 4
eB
S
/(
α
0
h
) <
V
off
< 8
eB
S
/(
α
0
h
) where
h
is the Planck constant, and we have assume
B
S
= 50 Tesla, and the coefficient
α
0
= 7.3 ×
10
14
m
-2
V
-1
is defined by the linear relationshi
p between the gate voltage and the two-
dimensional carrier density
n
2D
=
α
0
V
g
. (b) Parallel conne
ctions of two sets of structures
shown in (a) so that
ρ
xx
= 0.25(
ρ
xx
)
S
+ 0.25(
ρ
xx
)
0
, and the resistivity ratio [(
ρ
xx
)
on
/(
ρ
xx
)
off
] ~
4 for
V
on
= 0 and 4
eB
S
/(
α
0
h
) <
V
off
< 8
eB
S
/(
α
0
h
) for
B
S
= 50 Tesla. (c) Schematics of a
graphene nano-transistor with graphene laid
over periodic arrays of
triangular nano-dots
to optimize the strength and uniformity of
the strain-induced pseudo-magnetic fields.
Conclusion
We have conducted spatially resolved top
ographic and spectroscopic studies of CVD-
grown graphene on copper and CVD-gr
own graphene transferred to SiO
2
. Our
investigation reveals the important influenc
e of the substrate and strain on the carbon
atomic arrangements and the electronic
DOS of graphene. For CVD-grown graphene
remaining on copper, the monolayer carbon stru
ctures exhibit strongly distorted, with
different regions exhibiting varying lattice st
ructures and electronic LDOS. In particular,
topographical ridges appear
along the boundaries between di
fferent lattice structures,
giving rise to excess chargi
ng effects. Additionally, the
large and non-uniform strain
induces pseudo-magnetic fields up to
∼
50 Tesla, as manifested by quantized conductance
peaks associated with the integer and frac
tional quantum Hall effects (IQHE/FQHE). In
contrast, for graphene tran
sferred from copper to SiO
2
after CVD growth, the average
strain on the whole is reduced, so are th
e corresponding charging effects and pseudo-
magnetic fields, except for limited regions
containing remnant t
opological defects. Our
findings suggest the feasibility of strain en
gineering of graphene
by proper design of the
surface textures of the substrates. Examples of
realistic nano-transistors based on strained
graphene nano-structures are examined and ar
e found to be promising for nano-electronic
applications.
(c)
(a)
(b)
(
E
1
/
e
)
(
E
2
/
e
)
(
E
1
/
e
)
(
E
2
/
e
)
k
Ω
ρ
xx
15
ρ
xx
15
0
0
k
Ω
0 100 200
-100 -200
Gate voltage (V
g
)
0 100 200
-100 -200
Gate voltage (V
g
)
0
4
S
eB
h
α
⎛⎞
⎜⎟
⎝⎠
0
8
S
eB
h
α
⎛⎞
⎜⎟
⎝⎠
0
4
S
eB
h
α
⎛⎞
⎜⎟
⎝⎠
0
8
S
eB
h
α
⎛⎞
⎜⎟
⎝⎠
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Acknowledgments
The work at Caltech was jointly supported by the National Science Foundation and
the Nano Research Initiatives (NRI) under the Center of Science and Engineering of
Materials (CSEM).
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ECS Transactions, 35 (3) 161-172 (2011)
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www.esltbd.org
address. Redistribution subject to ECS license or copyright; see
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