14-05-2013, 01:09 PM
Graphene Field Effect Transistors: Diffusion-Drift Theory
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Introduction
Recently discovered stable monoatomic carbon sheet (graphene) which is comprised of
field-effect structures has remarkable physical properties promising nanoelectronic
applications (Novoselov, 2004). Practical semiconductor device simulation is essentially
based on diffusion-drift approximation (Sze & Ng, 2007). This approximation remains valid
for graphene field-effect transistors (GFET) due to unavoidable presence of scattering
centers in the gate or the substrate insulators and intrinsic phonon scattering (Ancona, 2010).
Traditional approaches to field-effect transistors modeling suffer from neglect of the key and
indispensible point of transport description – solution of the continuity equation for
diffusion-drift current in the channels. This inevitably leads to multiple difficulties
connected with the diffusion current component and, consequently, with continuous
description of the I-V characteristics on borders of operation modes (linear and saturation,
subthreshold and above threshold regions). Many subtle and/or fundamental details
(difference of behaviour of electrostatic and chemical potentials, specific form of the Einstein
relation in charge-confined channels, compressibility of 2D electron system, etc.) are also
often omitted in device simulations. Graphene introduces new peculiar physical details
(specific electrostatics, crucial role of quantum capacitance etc.) demanding new insights for
correct modeling and simulation (Zebrev, 2007). The goal of this chapter is to develop a
consequent diffusion-drift description for the carrier transport in the graphene FETs based
on explicit solution of current continuity equation in the channels (Zebrev, 1990) which
contains specific and new aspects of the problem. Role of unavoidable charged defects near
or at the interface between graphene and insulated layers will be also discussed.
GFET electrostatics
Near-interfacial rechargeable oxide traps
It is widely known (particularly, from silicon-based CMOS practice) that the charged oxide
defects inevitably occur nearby the interface between the insulated layers and the device
channel. Near-interfacial traps (defects) are located exactly at the interface or in the oxide
typically within 1-3 nm from the interface. These defects can have generally different charge
states and capable to be recharged by exchanging carriers (electrons and holes) with device
channel. Due to tunneling exchange possibility the near-interfacial traps sense the Fermi
level position in graphene. These rechargeable traps tend to empty if their level ε
t are above
the Fermi level and capture electrons if their level are lower the Fermi level.
Electrostatics of graphene gated structures
Let us consider the simplest form of the gate-insulator-graphene (GIG) structure
representing the two-plate capacitor capable to accumulate charges of the opposite signs.
Without loss of generality we will reference the chemical potential in graphene from the
level of charge neutrality ENP.
High-field effects
As carriers are accelerated in an electric field their drift velocity tends to saturate at high
enough electric fields. Current saturation due to velocity saturation has been discussed in
recent electronic transport experiments on graphene transistors (Meric et al., 2008). The
validity of the diffusion-drift equations can be empirically extended by introduction of a
field-dependent mobility obtained from empirical models or detailed calculation to capture
effects such as velocity saturation at high electric fields due to hot carrier effects.