of 5
Supporting Materials
Electronic-mechanical coupling in gra
phene from in-sit
u nanoindentation
experiments and multiscale
atomistic simulations
Mingyuan
Huang
†,
*
,
Tod
A.
Pascal
§,
,
Hyungjun
Kim
§,
,
William
A.
Goddard
III
§,
,
Julia
R.
Greer
Division of Engineering and Applie
d Science, California In
stitute of Technology, 1200 E. Cali
fornia Blvd, MC 309-81, Pasadena,
California 91125-8100, USA.
§
Center for Materials Simulations and Design, Graduate School
of EEWS, Korea Advanced Institu
te of Science and Technology,
Daejeon, Korea
Materials and Process Simulation Center, Cali
fornia Institute of Technology, 1200 E.
California Blvd, MC 309-81, Pasadena,
California 91125-8100, USA.
*
To
whom
correspondence
should
be
addressed.
E
mail:
mingyuan@caltech.edu.
Tel:
626
395
2243.
Sample fabrication:
Figure S1, Raman spectra of the 2D mode of single layer and bi-layer graphene. The Raman spectrum
(blue curve) was taken from the thinner area part of the graphene flake (an optical image was shown
in the inset). The sharp symmetric peak at about 2680 cm
-1
proves that the thinner part of the flake is
single layer graphene. The broad and asymmetric nature of the Raman spectrum (red curve) taken
from the thicker part of the flake shows th
at the thicker part is bi-layer graphene.
The first step in our fabrication process is mechanical exfoliation of the graphene flakes onto
SiO
2
/Si wafer by the well-known ‘Scotch tape’ method
1
. Single layer graphene flakes are identified
based on their optical contrast and confirmed by Raman spectroscopy
2
, shown in Fig. S1. To ensure
that strain is uniform during the nanoindentation
experiment, graphene flakes were shaped into
graphene ribbons with the widths between 1.5 to 4
μ
m by E-beam lithography and subsequent argon
plasma etching. The electrodes were patterned by E-beam lithography with the separation from 0.8 to
1.2
μ
m. Thin films comprising the electrodes, 3 nm-thick Cr under 100~200 nm-thick Au, were
deposited by e-beam evaporation, followed by aceton
e “lift-off” process. The large thickness of the
electrodes was chosen to increase their stiffness. Finally, Buffered Oxide Etchant (BOE, 50:1) was
used to etch the underlying SiO
2
and critical point dryer was used to release the suspended graphene
(Fig. 1b). Because BOE can diffuse freely under the graphene ribbons
3
, the SiO
2
under graphene
ribbons, including the part under the electrodes, was etched away at the same rate and left a constant
distance (~200 nm) between the graphene ribbons and the substrate. Thus, the part of the electrodes
with the graphene is also suspended, as shown in Fig. 1c.
The
8 μ
m
-wide wedge tip with ~15
o
angle was fabricated from a standard Berkovich indenter
tip by focused ion beam (FIB) and coated with 30 nm Al
2
O
3
by a magnetron sputtering system to
electrically isolate the graphene from the indenter
tip. The nanoindentation tests were performed at a
constant displacement rate of 2 nm/s controlled
by a custom-written feedback-loop algorithm. As
shown in Fig. 1b, the graphene devices were made
in two-terminal configuration to ensure uniform
strain during the nanoindentation experiment, and th
e highly doped Si substrate was taken as the gate.
Graphene devices were mounted in the standard chip
carriers (the up-right inset in Fig. 2a) and the
electrical measurements were carried out by a Keithley source meter. To incorporate the electrical
measurements into the nanoindentation tests, several intentional load holds were added during loading
and the electrical tests were performed during these holds.
Stiffness of gold electrodes:
Figure S2. The simulated stiffness of the electrodes wa
s plotted as a function of width of graphene for
three different thicknesses of electrodes.
As we can observe in the nanoindentation video,
the supporting electrodes aren’t infinitely
stiff and they bend when graphene is stretched. To figure out how much the electrodes deflected, the
stiffness of the electrodes was simulated by using finite element method (ABAQUS). As shown in
Fig. 1c, the part of electrodes with graphene
underneath was suspended due to the BOE freely
diffusing along the graphene ribbon. Also, 200 nm
undercut was taken into account for all edges.
Because the bending is mainly perpendicular to the electrodes and no slippage between the electrodes
and graphene occurs, only the perpendicular stretching force was considered and the average
disp
of t
h
fun
c
calc
u
Gat
e
As
w
abo
u
Al
2
O
ind
e
gra
p
Dur
i
was
volt
a
corr
e
this
Fig
u
ind
e
sho
w
lacement of
t
h
e electrodes.
c
tion of widt
h
u
lated based
e
capacitan
c
The bac
k
w
e described
u
t 100 nm)
w
O
3
was then
e
nter head.
W
p
hene and th
e
i
ng our expe
r
grounded. S
o
a
ge. In our
m
e
sponding to
voltage is ne
g
u
re S3, Illust
r
Now, w
e
e
nter tip push
e
w
n schemati
c
t
he interface
b
The stiffnes
s
h
of graphene
on this stiffn
c
e
k
gate capaci
t
before, the
s
w
as shaped
b
coated on th
W
hen the in
d
e
indenter tip
r
iment, the ti
p
o
, the voltag
e
m
easurement,
35 mV back
g
ligible.
r
ation of inde
n
e
turn to calc
u
e
s the graph
e
c
ally in Fig. s
3
ܥ
௚௥௔௣
b
etween the
g
s
of the elect
r
and thickne
s
ess, was the
n
t
ance of susp
ܥ
௕௔௖௞
ܣ
s
tandard B-d
o
b
y the focus
i
e tip surface
d
enter tip t
o
can be calcu
l
ܥ
௧௜௣
ܣ
௚௥௔௣
p
was contac
t
e
between th
e
the bias volt
a
gating from
n
ting a susp
e
u
late the gat
e
e
ne toward th
3
. The capac
i
௘௡௘ିௌ௜ைଶ
2
g
raphene an
d
r
odes was th
e
s
s of electro
d
n
taken out fr
o
ended devic
e
௥௔௣௛௘௡௘
o
ped diamon
d
i
on beam int
o
to create a
n
o
uches the g
l
ated:
௘௡௘
ಲ೗మೀయ
t
with the ce
n
e
graphene a
n
a
ge is 10 mV
the Si substr
a
e
nded graphe
n
e
capacitance
e Si substrat
e
i
tance betwe
e
2
߳
׬
ଶ௛௫
d
the electrod
e
e
n extracted
o
d
es. The defle
o
m the total
d
e
can be esti
m
ାఢ
ൌ3.9
d
Berkovich
i
o a wedge-s
h
n
insulating l
a
raphene dev
೟೔೛
೒ೝೌ೛೓೐೙೐
2
n
ter of the gr
a
n
d the indent
e
, so the gate
e
a
te. Compari
n
n
e device.
change duri
n
e
, thereby inc
e
n SiO
2
and
g
ௗ௫
ି௛ି௛
e
s was taken
o
ut and plott
e
ction of the
e
d
isplacement
m
ated as:
(nF/cm
-2
)
i
ndenter tip
(
h
aped tip. A
a
yer betwee
n
ice, the cap
a
2
6.6
(nF/cm
-
a
phene ribbo
n
e
r tip was ab
o
e
ffect from t
h
n
g to the gat
e
n
g the indent
a
reasing the g
g
raphene can
௪௟
ሺ݊ܮ
ି௛
ି௛
ି
as the displa
c
e
d in Figure
S
e
lectrodes as
of the inden
t
(
the width of
layer of 30
n
the grahen
e
a
citance bet
w
-
2
)
n
and the in
d
o
ut half of th
e
h
e tip is
e
voltage we
a
a
tion process
ate capacita
n
be expresse
d
ି
c
ement
S
2 as a
t
er tip.
(S1)
the tip is
nm thick
e
and the
w
een the
(S2)
d
enter tip
e
bias
a
dded,
.
The
n
ce, as
d
as:
(S3)
where h and h
1
is the deflection of the center of the graphene ribbon and the bending deflection of the
electrodes. The gate capacitance can then be derived as:
ܥ
௕௔௖௞
ܣ
௚௥௔௣௛௘௡௘
ାఢ
௛/௅௡ሺ
ష೓
ష೓
ష೓
(S4)
Based on this equation, the relative change of gate capacitance can be calculated and is plotted in the
inset of Fig. 3C. As we can see, the calculated values match the results extracted from experimental
data very well.
MD simulation details
We constructed a 25 x 37 nm graphene nanoribbon (Fig. S4) by replicating the orthorhombic
graphene crystal structure (a0 = 4.275
Å
, b0 = 4.93670
Å
). We then subjected the replicated cell to 500
steps of conjugate gradient minimization, with a force tolerance of 10E-4. We equilibrated the
structure at room temperature for 1ns of molecula
r dynamics in the canonical (NVT) ensemble. The
temperature of the nanoribbon was controlled with a Nose-Hoover thermostat and a temperature
coupling constant of 100fs. The cell was then
indented the center of the nanoribbon with a
cylindrically repulsive (1/r
9
) potential along z dimension in the isobaric (NPT) ensemble with a piston
coupling constant of 2.0ps. The Nose-Hoover baro
stat was only applied along the y-direction,
allowing the cell to shrink in response to the applied stress. The system was indented to a maximum
depth of 10nm (14% strain) over 10 ns (a tip velo
city of 1 m/s). All simulations were performed with
the LAMMPS
4
2001 atomistic simulator.
Obtaining the relative resistivity from DFT
All DFT calculations are performed using genera
lized gradient approximation by Perdew-Burke-
Ernzerhof (PBE)
5
with the projector-augmented wave (PAW) method
6, 7
as implemented in VASP
8
.
The simulation cell is orthorhombic cell containing 4 ca
rbon atoms; x-axis is chosen as an armchair
direction and y-axis is chosen as a zigzag direction. The cut-off value of the plane-wave basis set is
given by 500 eV and 24
×
24 k-points are sampled from reciprocal space using Monkhorst-Pack
method. We first obtained a stress-free state by op
timizing the atomic positions and cell parameters,
then, deformed the simulation cell by 1) stretc
hing the armchair direction up to 2% and 2)
compressing the zigzag direction up to 0.333%, simu
ltaneously. This leads us to maintain the Poison
ration value as 0.17. Local band structure near the
Dirac point is obtained from each simulation set.
Then, we linearly fitted the band structure to obtain the Fermi velocity.
Fig
u
fini
t
at t
h
alo
n
Re
f
1.
Gri
g
2.
Jian
g
3.
M.;
G.
N
4.
5.
6.
7.
8.
u
re S4: Sche
m
e in the x dir
e
h
e boundary.
T
n
g the z-axis
a
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