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Slide 1
Apology for not being here with you.
I had my tooth extracted last week and a nasty infection developed over the
weekend. I have to undergo a surgery this week.
This talk is focused on specific aspects related to the TCAD simulation of
statistical variability.
This is a large area and therefore I was very selective in choosing few of the most
relevant topics.
Slide 2
This is the summary of my talk.
After a brief introduction I will discuss selected topics of drift-diffusion, Monte
Carlo and Non-equilibrium Green’s function simulation of statistical variability
and will end up with brief conclusions.
Slide 3
Let start with the background.
Slide 4
The different components of variability are summarised in this slide courtesy of
Takeuchi-san from Renesas (NEC)
My talk is entirely focused on the purely statistical variability, which dominates
the on-chip variability particularly of small SRAM transistors.
Slide 5
There are results from a SELETE project on characterisation of statistical
variability led by Hiramoto-san in Japan.
Measurements of 1,000,000 million transistors on a chip show that for SRAM
size transistors practically all on-chip variability is statistical variability.
Slide 6
The main sources of statistical variability include:
Random discrete dopants both in the channel and in the source/drain regions of
the transistors
Granularity of materials including the poly-silicon or the metal gate and the highk dielectric
The line edge roughness of the gate in bulk MOSFETS or the Fin and the gate in
FinFETs.
Slide 7
A full spectrum of drift diffusion (DD), Monte-Carlo (MC) and quantum transport
simulation tools (QT) are needed to capture all aspect of statistical variability.
DD accurately captures the electrostatic effects associated with the variability
sources and deliver accurate results in the subthreshold region.
MC is needed to capture the impact of variability sources on transport variations
and on current.
For sub 10 nm transistors when source-to drain tunnelling plays important role
full QT simulations become a necessity.
Slide 8
Full 3D simulations of large statistical samples are needed for characterizing the
statistical variability of particular device design.
We have developed fully automated cluster engine for running statistical
variability simulation.
The engine submits automatically thousands of jobs, monitors their progress,
resubmits jobs fallen due to hardware faults, and provides computational
steering for jobs that have fail to converge.
It also collects the results in a database for statistical analysis and statistical
compact model extraction.
Slide 9
Let us focus first on important aspects of DD simulations.
Slide 10
Resolving the Coulomb potential of individual dopants results in artificial charge
trapping, acute mesh sensitivity and increased resistance.
In reality this is prevented by the quantum confinement effects in the Coulomb
potential of the individual dopants.
Slide 11
To avoid the artificial charge trapping Sano proposed to use only the long range
Coulomb potential contribution of the individual dopant. However the screening
parameter involved depends on the doping concentration (right-down picture)
The approach also cannot take into account the change in the screening in the
channel with the change of the gate bias and the inversion carrier concentration.
Slide 12
The most elegant and economic solution is to use density gradient (DG) quantum
corrections for both electrons and holes.
The use of DG corrections makes the charge distribution around the dopant
independent of the mesh spacing (bottom-left).
Also in the ‘atomistic’ simulation of a resistor (top-right) the use of DG
corrections removes the mesh space dependence of the simulated resistance.
Slide 13
The figures on the top illustrate the sharp coulomb potential wells for holes in
the substrate and for electrons in the source/drain region of a MOSFET.
The figures at the bottom illustrate the smooth effective quantum potential
obtained from the simultaneous application of DG quantum correction for both
electrons and holes.
Slide 14
A simple example of the simulations of 10x10 nm double gate MOSFET illustrates
that the Sano approach can lead to wrong result.
Two dopants in the middle of the channel are simulated on the left using DG
approach and on the right using the Sano approach.
The doping concentration representing the two dopants in the Sano approach
result in doping concentration maximum not at the position of the dopants but in
the middle of the channel.
Slide 15
This picture shows the current density in a cross-section of the channel trough
the dopants.
In the DG simulations on the left the current flows both in-between and on the
two sides of the dopants. This is confirmed also through NEGF simulations.
In the Sano approach simulations on the right the current is blocked by the
unphysically high dopant concentration in the middle of the channel and flows
only on the two sides of the dopants.
Slide 16
Here is an example of the simulation of threshold voltage variability in 45 nm LP
technology transistors developed by ST Microelectronics.
The simulations are carried out with the GSS ‘atomistic’ simulator GARAND.
Slide 17
The atomistic simulations allow to quantify the contribution of each source of
statistical variability including random discrete dopants (RDD), line edge
roughness (LER) and poly-Si granularity PSG.
The simulations when all the relevant sources of statistical variability are
combined together yield excellent agreement with the measurements for the
standard deviation of the threshold voltage.
Slide 18
The NBTI/PBTI degradation associated with random charge trapping in the
oxide results not only in a threshold voltage drift but also in an increase of the
threshold voltage variability.
Rare combination of underlying random dopant distribution and trapped
charges may result in ‘gigantic’ shift in the threshold voltage.
Slide 19
This picture presents on a normal probability plot the change of the threshold
voltage distribution as a result of low, medium and high degree of degradation
for the 45 nm PMOS. The higher degradation results in an increase of the
standard deviation and tailing of the distribution. The simulation result are in
excellent agreement with measurements conducted at ST Microelectronics.
Slide 20
On the left in this slide there is a fresh transistor where a narrow current
percolation path on the right side of the channel determines the threshold
voltage.
The trapping by chance of two holes in this percolation path result in a ‘gigantic’
change in the threshold voltage.
Slide 21
A set of well scaled bulk MOSFETs are used to illustrated the role of the metal
gate granularity on the variation of the threshold voltage. The 35nm transistor is
designed to meet the performance of the 45 nm technology Intel HP MOSFETs.
Slide 22
As expected with the scaling from 35 to 18 nm the statistical threshold voltage
variability increases.
The increase in the metal grain size result in larger variability.
Metal last technology (Intel) that allow better control of the metal granularity
results in lower variability compared to the metal first technology.
Slide 23
FD SOI and DG transistors can tolerate very low channel doping due to superior
electrostatic integrity.
They offer excellent potential for reducing RDD and LER variability.
Theoretically matching (Av) factors as low as 0.5 are possible.
Slide 24
How ever the presence of metal gate variability (top-left) can very rapidly erode
the advantage of the FDSOI transistors.
Also trapped charge density (bottom left) due to NBTI/PBTI can contribute to
the increase of the Vt variability.
Please note that the metal gate granularity (right) results in non Gaussian
distribution of the threshold voltage.
Slide 25
In order to capture accurately the on-current variability we need MC
simulations.
Slide 26
In order to capture the transport variability associated with ionised impurity
scattering from random dopants in the channel we have developed 3D MC
simulation technology that includes ‘ab-initio’ scattering through the real space
trajectories of the carriers interacting with the Coulomb potential of the
individual impurities.
The approach is verified by reproducing the doping concentration dependence of
the mobility in Si (top-left) running ‘atomistic’ MC simulations of ensembles of
resistors (bottom-left) with different average doping concentrations.
Slide 27
This picture illustrate results from the MC simulation of two microscopically
different 35 nm MOSFETs. The electron concentration distribution on the left
illustrate the electrostatic impact of the individual dopants. The current density
distribution illustrates their impact on the transport and current variation.
Single dopants has very localised effect on the carrier concentration but
delocalised effect on velocity and current distribution.
Slide 28
There is some correlation between the current variability obtained from DD and
MC simulations but the MC simulations yield higher current variability (left).
We have developed technology to transfer the current variability obtained from
MC simulation into DD simulations and to simulate corresponding target current
voltage characteristics for statistical compact model extraction (right).
Slide 29
The top two figures illustrate current voltage characteristics obtained from DD
simulations only showing little or no dispersion in the transconductance.
In the bottom two figures the current voltage characteristics simulated based on
MC data for the on current variability show significant transconductance
dispersion.
Slide 30
The transconductance dispersion obtained from the combined DD=MC
simulations (top-right) is in good agreement with experimental measuremets
reported by Hiramoto-san (bottom-left).
His analysis also shows that the DD simulation only cannot reproduced the
measured dispersion in the transconductance (bottom-right).
Slide 31
In sub 10 nm transistors when source-to-drain variability plays important role
full QM simulations are needed.
Slide 32
We have also developed the first fully 3D quantum transport simulator
incorporating atomic scale variability and phonon scattering.
The main difficulty was to achieve convergence in the presence of quasi-resonant
states in the attractive Coulomb potential of individual impurities (top-right).
Slide 33
This slide illustrate the impact of random discrete dopants in the source/drain
region of a 6 nm channel length nanowire MOSFET.
Corresponding variations in the access resistance result in significant variation
in the on current (top-right).
Both resonant states and secreting from individual dopants are clearly
distinguishable in the transmision coefficients of individual devices (bottomright)
Slide 34
Interface roughness and corresponding random body thickness variation result
in both threshold voltage variation and on current variation (top-right).
Slide 35
Here we report for the first time statistical variability simulations in a 6nm
channel length nanowire MOSFET in the presence of RDD and LER and including
phonon scattering. The phonon scattering result in additional suppression of the
average on current (top-right).
The phonon scattering results in states intermixing and broadening (bottomright).
Slide 36
It is time now to draw the conclusions
Slide 37
The simulation of statistical variability in nano CMOS transistor requires a
portfolio of DD, MC and NEGF simulation tools.
DD captures well the electrostatic effect of the variability sources and provides
accurate results for the threshold voltage variability.
MC simulation with ‘ab-initio’ impurity scattering are needed to capture
accurately the on current variability.
For sub 10 nm transistors full quantum transport simulations are needed to
capture effects associated with source-to-drain tunnelling.