WDS'14 Proceedings of Contributed Papers — Physics, 88–91, 2014.
ISBN 978-80-7378-276-4 © MATFYZPRESS
Ferroelectric Substrate Control of the Charge Modulation
in Multi-layered Graphene
A. S. Pusenkova,1 O. V. Varenyk,2 A. N. Morozovska,3 E. A. Eliseev,4 A. V. Ievlev,5
S. V. Kalinin,5 Ying-Hao Chu,6 V. Ya. Shur,7 and M. V. Strikha8
1
Taras Shevchenko Kyiv National University, Radiophysical Faculty, Kyiv, Ukraine.
Taras Shevchenko Kyiv National University, Physics Faculty, Kyiv, Ukraine.
3
Institute of Physics, National Academy of Sciences of Ukraine, Kyiv, Ukraine.
4
Institute for Problems of Materials Science, National Academy of Sciences of Ukraine, Kyiv, Ukraine.
5
The Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN, USA.
6
Department of Material Science and Engineering, National Chiao-Tung University, Hsinchu, Taiwan.
7
Ferroelectric Laboratory, Institute of Natural Sciences, Ural Federal University, Ekaterinburg, Russia.
8
Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Kyiv, Ukraine.
2
Abstract. Nanoscale heterostructures based on ferroelectrics have unique
electrophysical properties, and provide opportunities for creation of the next
generation of memory and nonlinear optical devices. In this article we further evolve
a continuum media theory of the ferroelectric domain structure influence on the spacecharge modulation in thin multi-layered graphene layers taking into account the
presence of buffer sapphire layer on the ferroelectric LiNbO3 surface. We consider
a graphene layer of finite thickness using a Tomas–Fermi approximation for the
description of its semiconductor properties in contrast with our previous studies,
where we assumed semi-infinite graphene (graphite) using Debye approximation.
We demonstrate how the space charge modulation in the graphene layers, induced
by the spontaneous polarization of the ferroelectric domain structure, depends
on the graphene layer thickness. Analytical results obtained for a single-domain case
provide an opportunity to explore the issue of the space charge accumulation control
in ultrafine graphene layers by ferroelectric substrates with a domain structure.
Introduction
Usage of ferroelectric substrates instead of traditional quartz substrates allows controlling the
space charge density distribution in multi-layered graphene by changing the spontaneous polarization
direction, value and domain structure properties of the ferroelectric [1–7]. Charge carriers in graphene
screen the depolarization electric field caused by spontaneous polarization at the ferroelectric surface,
and the sign of the screening carriers depends on the spontaneous polarization direction [7]. Allowing
for the screening mechanism, ferroelectric substrates make graphene highly sensitive to electric field,
elastic strain and temperature.
Below, we further expand a continuum media theory of the ferroelectric domain structure
influence on the space-charge modulation in thin multi-layered graphene layers allowing for the
presence of buffer layer of sapphire Al2O3 on the ferroelectric LiNbO3 surface. We consider a multilayer graphene layer of finite thickness using the Tomas–Fermi approximation for the description of
its semiconductor properties in contrast with our previous study [1], in which we used semi-infinite
graphene top layer (in fact graphite) using Debye approximation. These improvements are rather
important for application in realistic experiment, where graphene thickness is typically ultra-small (up
to 10–30 layers) [1–4]. The problem modification immediately allows to analyze the dependence of
the charge density stored by graphene on its thickness d (see Figure 1).
Problem statement
The geometry of the considered problem is shown in Figure 2. Multi-layered graphene (MLG) of
thickness d has permittivity ε G . The ultra-thin sapphire dielectric layer has thickness h and dielectric
permittivity ε g . The ferroelectric LiNbO3 film with 180-degree domain structure and spontaneous
polarization vector PS = (0,0, P3 ) has thickness l . The period of the 180-degree domain structure is a.
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PUSENKOVA ET AL.: FERROELECTRIC SUBSTRATE CONTROL IN M-L GRAPHENE
Figure 1. (a) Space charge redistribution in MLG modulated by LiNbO3 domain stripes, with the
presence of a sapphire layer on the ferroelectric surface. (b) Maximal total charge vs. graphene layer
thickness d.
MLG
Sapphire
d
h
x
l
z
LiNbO3
P3
P3
a
Figure 2. Geometry of the heterostructure MLG/dielectric sapphire layer/ ferroelectric LiNbO3 with
domain stripes.
The system of electrostatic equations for the given system is
∆ϕG −
ϕG
=0,
Rt2
for − d < z < 0 , (graphene),
(1a)
for 0 < z < h , (dielectric layer), (1b)
∆ϕ g = 0 ,
 f ∂2

for h < z < L . (ferroelectric film).(1c)
 ε 33
+ ε11f ∆ ⊥ ϕ f = 0 ,
2

∂
z


The Laplace operator is ∆, L = l + h , εijf are the components of linear dielectric permittivity tensor
components, Rt is a Tomas–Fermi screening radius whose value is given by expression [1]:
Rt =
e 0 e G (vF ) 2
.
4e 2 EF
Here EF is a Fermi energy, vF is a Fermi velocity, ε 0 is a universal dielectric constant.
89
(2)
PUSENKOVA ET AL.: FERROELECTRIC SUBSTRATE CONTROL IN M-L GRAPHENE
Eqs. (1) are supplemented by the boundary conditions of electric potential and displacement
continuity [8]:
ϕ G ( x, y , z = − d ) = 0 ,
ϕ G (x, y,0 ) = ϕ g (x, y,0 ) ,
(3a)
(3b)
∂ϕ g ( x, y,0) 

∂ϕ ( x, y,0)
 = 0 ,
DGn (x, y,0 ) − Dgn (x, y,0 ) ≡ ε 0  − ε G G
+ εg
∂z
∂z


ϕ g ( x, y , h ) = ϕ f ( x, y , h ) ,
D fn − Dgn ≡ −ε 33f
∂ϕ f ( x, y, h)
+
P3 ( x, y )
∂z
ε0
ϕ f ( x, y , z = L ) = 0 .
+εg
∂ϕ g ( x, y, h)
∂z
= 0,
(3c)
(3d)
(3e)
(3f)
For the case of periodic domain stripes with a period a, we can represent the spontaneous
∞
polarization of the ferroelectric in the form: P3 ( x) ≈ ∑ Pm sin (k m x ) , where Pm ≈ 4 PS π(2m + 1) and
m =0
k m = (2m + 1)(2π a ) . Results for the case of a single-domain ferroelectric can be obtained in the
limit k → 0 .
In Thomas–Fermi approximation [2] the space charge density ρ S ( x, y, z ) in the multi-layered
graphene and the total charge σ ( x, y ) are given by the expressions:
ρ S ( x, y , z ) = − N
σ ( x, y ) = − N
e e ϕ ( x, y , z )
4e 2 E F ϕ ( x, y )
≡ −N 0 G
,
2
πRt
π ( v F )
(4a)
0
ε 0 ε G ϕ ( x, y )
1
, ϕ ( x, y ) = ∫ ϕ ( x, y, z )dz.
d −d
πRt
(4b)
Here N is the number of graphene layers.
Potential and space charge distribution in a single-domain case
The system of electrostatic Eqs. (1) with boundary conditions (3) was solved analytically.
Solution has the simplest form for the single-domain case:
ϕG ( z ) =
ϕ g (z ) =
ϕ f (z ) =
− P3 Rt ε g l sinh[(d + z ) Rt ]
ε 0εG (ε h + ε g l ) cosh[d Rt ] + ε 0ε33f εG Rt sinh[d Rt ]
,
for − d < z < 0 ,
(5a)
ε 0εG (ε33f h + ε g l ) cosh[d Rt ] + ε 0ε33f εG Rt sinh[d Rt ]
,
for 0 < z < h ,
(5b)
f
33
− P3l (ε G z cosh[d Rt ] + ε g Rt sinh[d Rt ])
− P3 (l + h − z )(εG h cosh[d Rt ] + ε g Rt sinh[d Rt ])
ε 0ε G (ε33f h + ε g l ) cosh[d Rt ] + ε 0ε33f εG Rt sinh[d Rt ]
, for h < z < L .
The space charge density in multi-layered graphene and the total charge are:
− NP3ε g εG l sinh[(d + z ) Rt ]
,
ρG ( z ) =
f
πd εG (ε33h + ε g l ) cosh[d Rt ] + ε33f εG Rt sinh[d Rt ]
(
sG =
(
− NP3 Rt ε g εG l (cosh[d Rt ] − 1)
)
πd εG (ε h + ε g l ) cosh[d Rt ] + ε33f εG Rt sinh[d Rt ]
f
33
(5c)
)
(6a)
.
(6b)
Note that the main difference of Eqs. (5) and (6) from the case d → ∞ considered in Ref.[8] are
the hyperbolic functions cosh[d Rt ] and sinh[d Rt ] . One can see from Eqs. (6) that the concept of
effective gap cannot be introduced in a simple way even in a single-domain limit due to the hyperbolic
functions. Only in the limit d → ∞ it becomes possible.
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PUSENKOVA ET AL.: FERROELECTRIC SUBSTRATE CONTROL IN M-L GRAPHENE
Space charge redistribution caused by domain stripes
Below we study the space charge redistribution caused by domain stripes for the system multilayered graphene / dielectric sapphire Al2O3 / ferroelectric LiNbO3 with the following parameters:
Thomas–Fermi screening radius in graphene is Rt=0.5 nm (calculated using (2) with EF=0.2 eV,
vF=106 m/s), multi-layered graphene permittivity εG=15 (graphite), dielectric (sapphire) permittivity
εg=12.53, sapphire layer thickness h=5 nm, MLG thickness d=3.4 nm (N=10 graphene layers),
dielectric anisotropy of ferroelectric
γ = ε33f ε11f =0.58, ferroelectric permittivity ε33f=29,
ferroelectric polarization PS=0.75 C/m2 and ferroelectric thickness l=295 nm.
Figure 1a illustrates the space charge density distribution in the multi-layered graphene modulated
by ferroelectric LiNbO3 domain stripes. The density of charge along x has a harmonic function form
with maxima at the centre of domains and zero at the domain walls. Increasing the distance from the
dielectric, the absolute value of the maxima decreases exponentially and almost disappears beyond
2Rt. Figure 1b illustrates the monotonic decrease of the maximal total charge with graphene thickness
increase, i.e., finite size effect. Saturation appears with d increase in accordance with Eqs. (6).
Conclusion
We studied the induced charge in a graphene multilayer of arbitrary thickness under the influence
of a ferroelectric layer of a LiNbO3 with alternating domain structure placed underneath by modeling
graphene’s electrostatic response within a Thomas–Fermi approximation. Our analytical results
describe the dependence of the space charge modulation in the graphene layers, induced by the
spontaneous polarization of ferroelectric domain structure, on the graphene layer thickness. Those
results were compared with corresponding dependencies from our previous studies, which have dealt
with semi-infinite graphene using a Debye approximation. Analytical expressions obtained for a
single-domain case can be used for further exploration of mechanisms of controlling the space charge
accumulation in ultrafine graphene layers by a ferroelectric substrate with domain structure.
Acknowledgments. V.O.V., A.N.M. and E.A.E. acknowledge National Academy of Sciences of Ukraine,
Russian–Ukrainian grant 35-02-14. M.V.S. acknowledges State Fund of Fundamental Research of Ukraine, grant
53.2/006. S.V.K. acknowledges Office of Basic Energy Sciences, U.S. Department of Energy. Y.H.C
acknowledge the National Science Council, R.O.C. (NSC-101-2119-M-009-003-MY2), Ministry of Education
(MOE-ATU 101W961), and Center for Interdisciplinary Science of National Chiao Tung University. V.Y.S.
acknowledges the Russian Foundation of Basic Research (grant 14-02-92709 Ind-а).
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