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Journal of Computational Electronics 2: 323–327, 2003
c 2003 Kluwer Academic Publishers. Manufactured in The Netherlands.
Quantum Aspects of Resolving Discrete Charges
in ‘Atomistic’ Device Simulations
GARETH ROY, ANDREW R. BROWN, ASEN ASENOV AND SCOTT ROY
Device Modelling Group, Dept. of Electronics and Electrical Engineering, University of Glasgow,
Glasgow G12 8LT, Scotland, UK
[email protected]
Abstract. In ‘atomistic’ device simulation the resolving of discrete charges onto a fine grained simulation mesh
can lead to problems. The sharply resolved coulomb potential can cause simulation artefacts to appear in classical
simulation environments using Boltzmann or Fermi-Dirac statistics. Various methods have been proposed in an
effort to reduce or eliminate artefacts such as the trapping of mobile carriers in sharply resolved Coulomb wells,
however they have met with limited success. In this paper we show an alternative approach for handling discrete
charges in drift diffusion ‘atomistic’ simulations by properly introducing the related quantum mechanical effects
using the Density Gradient formalism. This produces the desired effect of eliminating the trapping of mobile charge
in heavily doped regions of the device.
Keywords: MOSFET, density gradient, atomistic, charge assignment
1.
Introduction
The resolution of individual charges within an ‘atomistic’ simulation [1–3] using a fine mesh creates problems [4]. Due to the use of Boltzmann or FermiDirac statistics in classical drift-diffusion simulations
the electron concentration follows the electrostatic potential, gained from the solution of the Poisson equation. As a result a significant amount of mobile charge
can become trapped (localised) in the sharply resolved Coulomb potential wells created by discrete
dopant charges assigned to a fine mesh. Such trapping is physically impossible since, in quantum mechanical terms, confinement keeps the ground electron
state high in the well. Figure 1, which depicts a 1D
Poisson-Schrödinger solution, shows the Coulomb potential well and the bound energy states, illustrating
this point. It is clear that a large fraction of the potential in the well cannot be followed by the electron
concentration due to such quantisation. This results
in a quantum reduction of the electron concentration
in the well when compared to the classical electron
concentration.
Another effect of this charge trapping in classical
simulations is the strong sensitivity of the quantity of
trapped charge to the mesh size. If a finer mesh is used
then the Coulomb potential well is more sharply resolved, and therefore deeper, and the amount of charge
being trapped increases. The problem of charge trapping in classical ‘atomistic’ simulations of a MOSFET
is shown in Fig. 2. Figure 2(a) illustrates the case
where only the substrate is ‘atomistically’ doped. In
this case there is no charge trapping in the heavily
doped source/drain areas and the current flows around
the sharp potential peaks in the channel. Figure 2(b) illustrates the opposite scenario where the source/drain
regions are ‘atomistically’ doped. In this case electrons are trapped in the sharply resolved potential wells,
which can reduce the effective concentration of mobile
carriers and thus artificially increase the source/drain
access resistances in the simulation.
Attempts to correct these problems in ‘atomistic’
simulations have been made by charge smearing [5] or
by splitting the Coulomb potential into short and longrange components based on screening considerations
[6]. In the case of charge smearing a reduction in the
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Roy
Gradient [7] approximation applied to Drift Diffusion
simulations [8]. The Density Gradient formalism modifies the classical electron equation of state so it contains
an additional term which is dependent on the gradient
of the carrier density (1).
√
∇2 n
kB T
n
2bn √
= φn − ψ +
ln
q
ni
n
Figure 1. 1D Poisson-Schrödinger solution to the single particle
problem showing energy levels and electron concentration.
amount of trapping, and less mesh size sensitivity, can
be achieved. This approach is however purely empirical and can result in a loss of resolution in respect to
‘atomistic’ scale effects. The splitting of the Coulomb
potential into short and long range components also
suffers from drawbacks including the possible ‘double counting’ of mobile charge screening. This method
also relies upon a somewhat arbitrary cut-off point in
order to separate the long and short range components.
2.
Density Gradient Approach
In this paper we present a quantum mechanically consistent treatment of this problem using the Density
(1)
where bn = h 2 /12 qm ∗n
The impact of including quantum confinement effects in simulations by use of the Density Gradient
approximation is illustrated in Fig. 3. Quantum confinement in the sharp potential wells smoothes the overall electron concentration, reducing the sharper peaks
as well as raising the concentration in the surrounding area. This means that less of the mobile charge is
trapped within the coulomb potential wells and so the
overall resistance of the corresponding region in the
simulations is reduced.
3.
Simulation Study
To illustrate the overall increase in resistance associated
with charge trapping, we have simulated a resistor of
dimensions 30 × 20 × 20 nm. The resistor is modelled
as a slab of silicon, which has been doped to various
concentrations. For the simulations presented in this paper the doping concentration used was 1 × 1020 cm−3
Figure 2. 2D potential distribution at the SiSiO2 interface of MOSFET: (a) atomistic channel and continuous source/drain doping and (b)
continuous channel and atomistic source/drain doping.
Quantum Aspects of Resolving Discrete Charges
Figure 3. Electron concentration within an ‘atomistic’ resistor using both classical and DG approaches.
although the results were consistent at other doping
levels. This particular concentration is representative
of the doping concentrations in the source/drain region of current generation devices being simulated [9].
The potential distribution for one of the simulations is
shown in Fig. 4.
Figure 5 depicts the I -V characteristics obtained
from classical simulations of the resistor for both continuous doping and for two methods of charge assignment, Nearest Grid Point (NGP) and Cloud in Cell
(CIC) [10], for a mesh spacing of 1 nm. The continuously doped case is taken as the reference as it
matches the analytical conductivity of the resistor. In
Figure 4. 3D potential profile of a 30 × 20 × 20 nm ‘atomistic’
resistor used in this investigation.
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Figure 5. I -V characteristics of a 30 × 20 × 20 nm resistor comparing uniform continuous doping with atomistic averages when using
NGP and CIC charge assignment methods.
the NGP approach all of the charge associated with a
single dopant atom is assigned to a single mesh point,
which results in a large peak in electron concentration
at that node. The CIC approach is an improvement over
the NGP scheme as the charge associated with a single
dopant atom is spread over the surrounding eight mesh
points, smoothing the associated potential well, and
thus reducing the overall peak electron concentration
value. As can be seen from Fig. 5. the resistance of the
device increases dramatically in the ‘atomistic’ cases
and this increase is greater when the NGP approach is
used which is due to the fact that it does not smear the
charge and produces sharper potential wells.
Figure 6 shows the change in the I -V characteristics
of the resistor when DG is applied. As we have stated
Figure 6. I -V characteristics of a 30 × 20 × 20 nm resistor comparing uniform continuous doping with atomistic averages when using
NGP and CIC charge assignment methods with the addition of DG.
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Figure 7. I -V characteristics of a 30 × 20 × 20 nm resistor comparing uniform continuous doping with atomistic averages for classical
and DG simulations when varying the mesh spacing.
and shown in Fig. 3, when DG is applied to the simulation it acts to smooth the electron concentration. The
overall effect of this smoothing process is a reduction
in the amount of charge which is trapped by the associated dopant atoms. This reduction in trapped charge
leads to a decrease in the effective resistance of the device being simulated. This effect is shown in detail in
Fig. 6 where a significant improvement in the overall
resistance can be observed in the resistor simulation
which brings the average ‘atomistic’ current closer to
the continuously doped case.
Figure 7 shows the I -V characteristics of the resistor with various sizes of mesh spacing. The problem
of charge trapping is closely associated with the mesh
size. When the charge is assigned using NGP or CIC
techniques a smaller mesh size results in larger charge
density at that point. This larger localised charge density means that the potential wells are more sharply
resolved and the overall effect is that more charge
is trapped and localised and does not contribute to
the conduction. This problem is solved through the
DG approach which replicates quantum effects introducing a smooth effective quantum potential which
the electron concentration follows. The DG corrections are insensitive to the mesh spacing thus removing the mesh dependence present in classical DD
simulations. Figure 7 shows that the resistance dramatically depends on the mesh spacing in the classical resistor simulation case and that the inclusion
of DG quantum corrections completely removes this
sensitivity.
The improvements due to the addition of DG can
also be seen in MOSFET device simulations. Figure 8
Figure 8. I D -VG characteristics of a 50 nm MOSFET device
comparing uniform continuous doping with atomistic averages for
classical and DG simulations. Both linear and semi-log plots are
shown.
shows the I D -VG characteristics of a 50 nm MOSFET
illustrating the inclusion of the quantum corrections
in the source and drain regions. It shows the ‘atomistic’ average of 200 devices compared to the associated I D -VG curve corresponding to uniform continuous doping in both the classical and DG regime. It
can be seen on the linear plot that the ‘atomistic’ average current above threshold is lower compared to
the continuous case, which is due to the increased effective resistance associated with the charge trapped
in the source/drain regions. This effect is substantially
reduced in the DG case. Apart from this the proper
DG treatment in the source/drain regions results in
larger threshold voltage lowering in the ‘atomistic’
simulations.
4.
Conclusions
In this paper we have shown that the resolution of individual ‘atomistic’ dopants onto a fine mesh can lead
to substantial trapping of charge in heavily doped regions. This in turn artificially increases the effective
resistance of the corresponding regions which introduces errors into the simulation results. In MOSFET
simulation, for example, this artificially increases the
access source/drain resistance.
The use of the Density Gradient formalism offers a physical solution to this problem which reduces this charge trapping due to the inclusion
of quantum confinement into the simulation. We
have presented simulation results illustrating this
point in both simple resistor and complete MOSFET
simulations.
Quantum Aspects of Resolving Discrete Charges
Acknowledgments
This work was facilitated by IBM through a Shared
University Research grant. This work is also supported by EPSRC through grant GR/R47325/01 and
a platform grant GR/R89738/1.
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