Towards Utilizing Spontaneous Coherence in Bilayer Graphene for Ultra-Low Power Switches

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Towards Utilizing Spontaneous Coherence in
Bilayer Graphene for Ultra-Low Power
Switches M.J. Gilbert Micro and Nanotechnology
Laboratory Department of Electrical and Computer
Engineering University of Illinois
Urbana-Champaign Urbana, IL 61801 Email
matthewg_at_illinois.edu
Understanding the initial Conditions
Introduction
Device Structure
Harnessing collective motion of electrons is a
promising approach to creating ultra-low power
logic devices . In a switch that is based on a
set of independently-moving electrons, switching
requires a change in gate voltage large enough to
shift each electrons energy by more than room
temperature thermal energy. With collective
motion of electrons, one may hope to achieve
switching with smaller changes in gate voltage
the energy of an entire ensemble of electrons can
be shifted by more than room temperature thermal
energy, turning on transport, even though each
electrons energy is shifted by only a small
fraction of that amount. One of the most
spectacular demonstrations of collective behavior
in the last 10 years is the giant enhancement in
tunnel current seen in semiconductor bilayer
devices in the quantum Hall regime. The enhanced
tunnel current between the two layers occurs when
electrons in the top layer bind with vacancies in
the bottom layer to form indirect excitons,
which in turn organize into a Bose-Einstein
condensate. This type of behavior may be viewed
as spontaneous coherence, or pseudospin
ferromagnetism selection of a particular
superposition of states in the two layers (a
particular direction of pseudospin, where the
layer degree of freedom is treated as if it were
a spin) for the entire system.  In order for
this behavior to be useful in the context of a
device, pseudospin ferromagnetism must survive at
room temperature, rather than being restricted to
cryogenic temperatures as in the GaAs
heterostructures in which this effect was
discovered. Room temperature operation is
possible if the carrier densities in the two
layers are equal but with opposite polarity
electrons in one layer, holes in the other. This
condition is very difficult to engineer this in a
semiconductor system, but could be feasible in
graphene. Graphene, a recently discovered form of
carbon consisting of a single atomic sheet of
graphite, is closely related to graphite,
fullerenes and nanotubes, all well-known to both
industry and academics. Carbon nanotubes are
being explored for use in FETs to extend scaling
of conventional charge-based (CMOS) switches.
The recent achievement of high quality monolayer
graphene with mobilities in excess of 104 cm2 V-1
s-1 suggests that graphene too may be useful for
advanced electronics.4 Beyond its remarkable
room-temperature mobility, graphene is predicted
to have exotic properties due to its unique
bandstructure. Most importantly for the present
proposal, the carrier density may be tuned
through zero with the same velocity and
mobility for electrons and holes simply by
applying a gate voltage5. We propose to develop
the theory of spontaneous coherence in bilayer
graphene, all with a view toward applications to
ultra-low power switching devices. Specifically,
we hope to use pseudospin torque and spontaneous
interlayer coherence to create a scalable
switching device that requires far less energy to
switch between low and high conductivity, with
equal or better drive current, than optimum
MOSFET switching energy predicted by the
International Technology Roadmap.  
ABOVE Schematic of the proposed device geometry
which consists of two monolayers of graphene
separated by a tunnel oxide with local gates to
control the electron density in the layers
separately. Our proposed device consists of two
monolayers of graphene separated by a thin
insulating tunnel oxide (Fig. 1). The electron
densities in the top and bottom monolayers of
graphene may be individually tuned by top and
bottom gate voltages, VTG and VBG, respectively.
By applying a bias Vinterlayer between the top
and bottom layers, we will drive a current
across the tunnel oxide of thickness d.
Experimentally, we would like d 1-5 nm, to
achieve a uniform barrier with large tunnel
coupling ( mV). BELOW Description of the
bilayer system in the pseudospin language where
the top layer is described as pseudospin up and
the bottom layer as pseudospin down both
quantized along the z-direction. Due to the small
distance between the layers, the electrons in the
top layer are intimately aware of the locations
of the vacancies   in the bottom layer. The
effect that we exploit here is an
exchange-correlation (interaction) enhancement of
the quasiparticle interlayer tunneling amplitude
?t, by a factor S. The enhancement of the bare
tunneling amplitude by interactions is very
closely related to the quantum well exchange
enhancement  effect that has been studied
previously using electron gas theory. More
specifically, the bare tunneling amplitude
combined with the Pauli exclusion principle
results in an increased occupation of states with
a common inter-layer phase relation.
TOP The initial electron and hole density for
VTG -VTB 0.1 V. LEFT The calculate
bandstructure for a 125 nm wide nanoribbon of
graphene.
Fig. 5 (a) Schematic of the bilayer with the top
layer lightly doped with electrons and the bottom
layer heavily doped with holes. (b) Schematic
of the bilayer with equal populations of
electrons in the top layer and holes in the
bottom. This situation allows spontaneous
coherence to form leading to large interlayer
currents.
The Pseudospin Torque Effect
eV
Electron and Hole Density
Pseudospin Field (Longitudinal Only)
0
0
Finding a Bose-Einstein Condensate at Room
Temperature??
Switching Operation
Hongki Min et al. arXiv0802.3462 (2008)
y (nm)
y (nm)
125
250
0
625
x (nm)
0
625
x (nm)
Here, we show the evolution of the system in
equilibrium (TOP) and with an interlayer bias
(BOTTOM). In equilibrium, (TOP), since we have
composed our wavefunction of z and z oriented
pseudospin states we may state that the
pseudospin polarization is along the x direction
in the plane of the bilayer system. When
interlayer bias is smaller than the splitting
between the symmetric and anti-symmetric states,
the net effective pseudospin field is aligned
with the x-direction and this allows the
electrons to tunnel more easily from the top
layer to the bottom layer. In other words, the
transport electrons have yet to produce a torque
sufficient to cause the net pseudospin phase
angle to anti-align with the transport direction.
Therefore, states from the top layer continue to
easily precess to the bottom layer, leading to a
large tunneling current in the bilayer. As the
bias is increased, the Fermi velocity will
correspondingly increase in one layer relative to
the other layer, resulting in an increase in the
net pseudospin phase angle from 0º to 90º, which
tells us that the pseudospin phase angle is no
longer aligned with the x, or transport,
direction, (BOTTOM). At this point, the system
switches from a state of high conductivity to a
state of low conductivity as the effective
pseudospin field produced by the motion of the
carriers prevents the carriers from tunneling.
This type of operation is analogous to the
operation of a Datta-Das transistor9 where the
role of the gate voltage used to control the spin
precession in narrow gap III-V materials is now
played by the interlayer bias. We call this the
pseudospin torque effect.  
We envision that our bilayer graphene switch will
operate in much the same way as a typical MOSFET,
but with far better subthreshold slope. In our
bilayer system, we would set VGB so that the
bottom graphene is very heavily hole doped. We
then may use VTG to tune the sign of the
quasiparticle in the top layer. In Fig. 4, we
demonstrate the envisioned Iinterlayer-VTG
characteristics for our bilayer switch at an
interlayer voltage Vinterlayer lt ?t. We begin
with VTG 0, which sets up the situation
depicted. In (a). Here there are not many
electrons in the top layer and the resultant
exchange enhancement between the two layers is
small. As VTG is swept to increasingly positive
values, more electrons enter the top graphene
layer. This increases the interlayer exchange
enhancement between the two layers which results
in increasingly interlayer currents. We can
continue to increase VTG until we reach an
electron density in the top layer equal to the
hole density in the bottom layer. At this point,
the system should undergo spontaneous coherence,
shown in (b), and the interlayer current will
reach a maximum with a subthreshold slope for our
switch S2, potentially with a value far less
than the typical 60 mV/decade. e plot the
hypothetical Iinterlayer-Vinterlayer
characteristics for several different values of
VTG. We expect the transfer characteristics to
have a very fast rise for applied voltages less
than the bare tunnel coupling ?t.  When the
applied voltage exceeds ?t then the pesudospin
torque effect causes the interlayer current to go
into saturation resulting in I-V curves very
reminiscent of those of a MOSFET.
Pseudospin Field (Longitudinal-Top,
Transverse-Bottom)
eV
ABOVE The critical temperature for coupling
between electrons and holes as a function of the
electric field applied across the bilayer and the
physical separation between the layers. For
certain separations and fields, we see that the
critical temperature is above room
temperature. BELOW Electron (blue) and hole
(red) paths with identical dispersion relations
simulated with quantum Monte Carlo. On the left,
the density is too low and condensation does not
occur. (right) When we cut the domain or
equivalently double the density of electrons and
holes we see that the paths permute around the
domain which is a signature of Bose condensation.
1013 cm-2
Electron and Hole Density
0
eV
y (nm)
250
0
625
x (nm)