Chemistry Reference
In-Depth Information
We note that electric fields of the magnitude considered here can in principle be
experimentally created, as recently obtained in the case of 3 L graphene (Zou et al.
2013
) which fields close to
0
.
6 V/Å were achieved taking into account HfO
2
gates.
In the case of MoS
2
, the high dielectric breakdown, due to the chemical character
of the Mo-S covalent bonds, allows the application of large electric bias as recently
reported in voltage-current measurements (Lembke et al.
2012
).
∼
14.3
Interlayer Electric Field: Spatial Dependence
Next we discuss the origin of the electric-field mediated tunable dielectric constant in
graphene and MoS
2
layered systems. Figure
14.2
shows the electric response in terms
of the effective electric field
E
eff
calculated from the Hartree potential
V
H
along the
supercell for bilayer structures. The application of the external field
E
ext
generates
an interlayer charge-transfer which partially cancels
E
ext
inducing the appearance
of
E
eff
in the region between the layers. At low
E
ext
, all the induced values of
E
eff
are approximately constant, within the numerical accuracy of our model, assuming
similar shapes as displayed in the dark regions of Fig.
14.2
a, c. At fields close to
those used to modify the band gap of 2 L graphene (Mak et al.
2009
; Zhang et al.
2009
; Castro et al.
2007
), or used in MoS
2
transistors (Radisavljevic et al.
2011
), that
is
E
ext
=
0
.
08 V/Å, the effective field E
eff
is already dependent on position z, with a
maximum at the mid-point between the layers. MoS
2
has the difference to be formed
by S-Mo-S bonds perpendicular to the external field, which induce a smaller but
finite contribution between the S atoms. Moreover, the effective field on2LMoS
2
assumes a narrower a shape relative to graphene with negligible values close to S.
The electric response can also be analyzed based on the induced charge densities,
Δρ
, at different fields as plotted in Fig.
14.2
b, d. Both layered systems show a charge
accumulation at the layer that is under positive potential
+
V
and a corresponding
depletion at the other one
−
V
. The integration of
Δρ
along z, utilizing the Poisson
2
V
(
z
)
equation
Δρ/ε
0
, where
ε
0
is the vacuum permittivity, results in a
response electric field
E
ρ
(solid black line in Fig.
14.2
b, d) that screens the external
electric field, that is,
E
eff
≈
∇
=−
E
ext
−
E
ρ
.
14.4
Electric Field Damping in Multilayer Systems
In the previous section, we have considered in detail the electric response due to
external fields on graphene and MoS
2
2 L structures. Although this is an impor-
tant system, other aspects are also crucial to understand and control the screening
associated to two-dimensional crystals. For example, one needs to explore the char-
acteristics of multilayer systems subjected to external bias, as well as the possibility
to compare structures with different electronic character. This kind of knowledge is
instrumental in possible applications in electronics and optoelectronics.
We address next the dependence of
ε
as a function of the number of graphene
and MoS
2
layers
N
as shown in Fig.
14.3
. Despite the electronic character of each
