Device Modeling from Atomistic First Principles: theory of
the nonequilibrium vertex correction
Eric Zhu1, Leo Liu1, Hong Guo1,2
1 Nanoacademic Technologies
2 Dept.
Inc. Brossard, QC J4Z 1A7, Canada
of Physics, McGill Univ., Montreal, Quebec, H3A 2T8 Canada
• Introduction: NEGF-DFT;
• 4 critical issues: disorder averaging, band gap, large sizes, verification;
• Two examples: localized doping in Si nanoFET; disorder scattering in MRAM;
• Summary.
Atomic model
Continuum model
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Goal: simulate a transistor from atomic first principles
~10nm
~100nm
1.
2.
3.
4.
Doping & disorder,
Band gaps,
Large sizes,
Accuracy.
Other physics: phonons,
magnons, photons,
correlations…
Picture from Taur and Ning, Fundamentals of Modern VLSI Devices
(10 nm)3 chunk of Si has
~64,000 atoms.
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Current: L=22nm
Next:
L=16nm
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DFT: ~1,000 atoms
page 2
… and many other systems with different materials
This talk:
In any real device made
of any real material, there
is a degree of disorder.
?
Such disorder impacts
device operation in
serious ways. How do we
compute these effects
from first principles?
Can we calculate ?
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Ex 1: Dopant fluctuation gives rise to device-to-device variability
Huge device to device variability.
If every transistor
behaves differently,
difficult to design a
circuit.
F.L. Yang et al., in VLSI Technol. Tech. Symp. Dig., pp. 208, June 2007.
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Ex 2: roughness scattering increases resistance of Cu interconnects
With Daniel Gall
of RPI.
$: SRC
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Ex. 3: disorder effect in topological insulator Bi2Se3
Conductance:
Experiment (Hasan etal)
Ab initio (Zhao etal)
Calculated spin direction
Top surface
Zhao, H.G. etal Nano Lett. 11, 2088 (2011).
Wang, Hu, H.G. PRB 85, 241402 (2012)
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Bottom surface
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Can we calculate realistic device parameters?
engineering
science
New
quantum mechanics atomic simulations
device modeling
Physics
materials, chemistry, < 5nm (1000 atoms)
physics
Semi-empirical
device modeling,
10,000 to 100,000
atoms
device parameters
TCAD
Quantitative prediction of quantum transport from
atomic first principles without any parameter
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Method - NEGF-DFT: non-equilibrium density matrix
NEGF-DFT
‘DFT’: density functional theory
DFT
Hˆ
 (r )
NEGF: Keldysh nonequilibrium Green’s function
 ~  G dE   GR GA dE
NEGF
DFT
NEGF-DFT
`DFT’ in NEGF-DFT is not
the usual ground state
DFT: density matrix of
NEGF-DFT is constructed
at non-equilibrium.
No variational solution.
Jeremy Taylor, Hong Guo and Jian Wang, Phys. Rev. B 63, 245407 (2001).
M. Brandbyge, J.-L. Mozos, P. Ordejon, J. Taylor, and K. Stokbro, PRB 65, 165401 (2002).
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1. Within NEGF-DFT: solving the disorder averaging problem
Doping and disorder scattering from atomic principles
Acknowledgements: Dr. Youqi Ke, Dr. Ke Xia, Dr. Ferdows Zahid, Dr. Eric Zhu,
Dr. Lei Liu, Dr. Yibin Hu
Drs. Eric Zhu, Leo Liu, and Yibin Hu: development of the
NEGF-DFT/CPA-NVC first principles package Nanodsim
(nano-device-simulator) – Nanoacademic Technologis Inc.
(www.nanoacademic.com).
NEGF-DFT/CPA-NVC
Youqi Ke, Ke Xia and Hong Guo PRL 100, 166805 (2008);
Youqi Ke, Ke Xia and Hong Guo, PRL 105, 236801 (2010);
Ferdows Zahid, Youqi Ke, Daniel Gall and Hong Guo, PRB 81, 045406 (2010);
Eric Zhu, Lei Liu and Hong Guo, preprint (2012).
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A tough problem of atomic calculations: disorder scattering
Generating many configurations, compute each, and average result
very time consuming
(Small x, large N)
Ax B1 x
T. Dejesus, Ph.D thesis,
McGill University, 2002.
For any theoretical calculation, disorder
averaging must be done.
How to do it in atomistic calculations at
non-equilibrium?
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To build intuition, let’s solve a toy problem exactly
1
2
1D tight binding chain
nearest neighbor coupling
on-site energy
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Self-energies for the leads:
L
1
2
R
How to handle half-infinite chain ?
Self energy:
The problem is reduced to 3 sites plus self-energies
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Physical quantities and NEGF
Express physical quantities in terms of NEGF:
average over disorder
configurations
L
R
The problem is reduced to calculate disorder averaged NEGF
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L
1
2
R
Disorder average can be done exactly for the 3-site model
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Exact solution of the 3-site toy model:
In general, the number of configuration is 2N (N is the number of disorder sites).
It is impossible to enumerate and compute all configurations for large N.
We need a better “statistical approach”
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Coherent potential Approx.
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CPA - well established formalism
Ax B1 x
When there are impurities, translational symmetry is broken.
Coherent Potential Approximation (CPA) is an effective medium
theory that averages over the disorder and restores the
translational symmetry. So, an atomic site has x% chance to
be occupied by A, and (1-x)% chance by B.
P. Soven, Phys. Rev. 156, 809 (1967).
T ( E, V )  Tr[G LG R ]
r
a
B. Velicky, Phys. Rev. 183 (1969).
Rev. Mod. Phys. 46, 466 (1974)
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CPA:
CPA picture: effective media
and
are solved
from CPA equation
Implementation:
needs a method that does
one atom at a time: LMTO,
KKR, etc..
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Non-equilibrium density matrix: nonequilibrium vertex

NEGF-DFT

 ~ G  dE  G R   G A dE
DFT
Average over random disorder:
Hˆ
 (r )
NEGF

X

 ~ G  dE  G R   G A dE
T  Tr  G R l G A r 


specular part
G R  G A  G R  G A
diffusive part
Take Home message #1: multiple disorder scattering at non-equilibrium is solved by the nonequilibrium vertex correction theory (NVC) and implemented in NEGF-DFT software Nanodsim.
Youqi Ke, Ke Xia and Hong Guo PRL 100, 166805 (2008)
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Essence of Nonequilibrium Vertex Correction (NVC)
X
Conventional vertex correction, i.e. that appears in computing Kubo formula in disordered
metal, is done at equilibrium.
NVC is done at non-equilibrium: it is related not only to multiple impurity scattering, but
also to the non-equilibrium statistics of the device scattering region.
Implementation: LMTO with atomic sphere approximation, plus CPA and NVC,
within NEGF-DFT.
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NVC Equation: some complicated technical details
Youqi Ke, Ke Xia and Hong Guo PRL 100, 166805 (2008).
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Consistency check: CPA-NVC identity
NVC
CPA
CPA
The CPA-NVC identity can also be proved analytically at non-equilibrium:
CPA and NVC are consistent approximations (Eric Zhu and H.G., 2012).
The identity is tested numerically: strong check of the code.
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NVC solution for the 3-site toy model
and
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are solved from NVC equation
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Comparison for the 3-site toy model:
Excellent !
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specular part
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diffusive part
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Non-toy system:
At equilibrium, fluctuation-dissipation theorem holds.
Left hand side has NVC; right hand side does not.
This gives a very strict check to the NVC formalism as well as to the numerical implementation.
no NVC
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NVC
exact
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2. Within NEGF-DFT: solving the band gap problem
The band gap problem …
Acknowledgements: Dr. Youqi Ke, Dr. Wei Ji, Mathieu Cesar, Dr. Eric Zhu, Dr. Lei Liu,
Dr. Zetian Mi, Dr. Ferdows Zahid
NEGF-DFT-CPA-NVC
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The band gap problem of local functionals in DFT
DFT calculation of
band gaps:
MBJ computation
time is ~LDA
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Some relevant band gaps for transistors materials:
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Some relevant effective masses
Take home message 2: the band gap problem is practically resolved by MBJ semi-local
exchange within LMTO implementation of NEGF-DFT.
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Experimental data
Calculated data
MBJ potential + CPA: works. Good agreement with experimental data.
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3. Within NEGF-DFT: solving the large size problem
Solving large problems from self-consistent first principles.
Acknowledgements:
Dr. Eric Zhu, Dr. Lei Liu (Nanoacademic Technologies Inc.)
Dr. Yibin Hu, Mohammed Harb, Vincent Michaud-Roux (McGill)
J. Maassan, E. Zhu, V. Michaud-Vioux, M. Harb and H.G., to appear in IEEE Proceedings (2012).
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Locality: the principle underlying all O(N) methods
Locality: the properties of a certain observation region comprising one or a
few atoms are only weakly influenced by factors that are spatially far away
from this observation region.
S. Geodecker Rev. Mod. Phys. (1999)
Equilibrium density matrix exhibits decaying property:
insulator
metal
LMTO (nanodsim)
LCAO (nanodcal)
Example: Si bulk
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Density matrix computation:
DFT – computes potential and energy levels of the device;
DFT
NEGF – non-equilibrium statistics that fills levels;
Hˆ
 (r )
The self-consistent loop – NEGF-DFT algorithm we use.
 ~  G dE   GR GA dE
NEGF
Practically, density matrix is divided into two parts: equilibrium
and non-equilibrium parts:
locality
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no locality
page 33
Roadmap for locality-less computation of density matrix of large systems
Large: ~20,000 atoms
algorithms
Do not depend on locality
iterative method
no preconditioner
gmres bicgstab
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qmr
direct method
with preconditioner
Jacobi
SOR
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ILU
MCS
H-Matrix
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Roadmap (cont.)
algorithms
iterative method
nested dissection
single wall
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double wall
direct method
principal layer
thin & long
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pardiso
thick & short
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Performance: nano-device simulator (nanodsim)
Nanodsim has been fully parallelized and optimized for both speed and memory costs.
The speed is made nearly O(N) along the transport direction.
speed performance
memory performance
Lx = periodic
Ly = 10 nm
Lz = 5, 10 ,15, 20 ,25, 30nm
160 cores
Benchmark: 160 cores, 3GB / core, 12,800 atomic sites, <30 min/step
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Solving ~20,000 atomic spheres for open devices at nonequilibrium
Open device structure of Si: parallel NEGF-DFT run on 480 cores.
Structure
Size
# of atoms
NEGF-DFT run
Convergence
Leads :
1 X 20 X 1
320 atomic spheres
(2880 Orbitals)
1 X 20 X 20
6400 atomic spheres
(57600 Orbitals)
107 NEGF-DFT steps,
5.5 min/step, total 9.8 hours
Potential 1.0 x 10-5
Charge 1.23 x 10-5 per atom
Two probe 2
1 X 20 X 40
12800 atomic spheres
(115200 Orbitals)
195 NEGF-DFT steps,
14 min/step, total 46 hours
Two probe 3
1 x 20 x 60
19200 atomic spheres
(172800 Orbitals)
267 NEGF-DFT steps,
30 min/step, total 134 hours
Two probe 1
Potential 1.0 x 10-5
Charge 6.2 x 10-6 per atom
Same as above
Summary: NEGF-DFT modeling has reached realistic device sizes!
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If using tight binding model: huge systems can be done
Run on a single
computing node
with 12 cores and
36 GB memory
1,024,000 Si atoms
10.9 nm ×10.9 nm
× 173.8 nm
time = 2 days
• Computation time scales linearly with the channel length
• Computation time increases 6~7 times if the cross section doubles
• For Lz = 173.8 nm, 1/3 of computation time is spent on surface Green’s function, and
2/3 spent on transmission calculation
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4. How do we know all are well for real devices ?
Bench-marking the NEGF-DFT atomistic model for
device simulations
Acknowledgements:
Lining Zhang (ECE, HKUST), Dr. Ferdows Zahid (Physics, HKU), Dr. Mansun Chan
(ECE, HKUST), Dr. Jian Wang (Physics, HKU),
Dr. Jesse Maassen (ECE, Purdue), Dr. Eric Zhu (Nanoacademic).
NEGF-DFT/CPA-NVC
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Commercial TCAD tool:
1328 pages of parameter
and physics descriptions
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Hundreds of parameters are needed !
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NEGF-DFT/CPA-NVC versus Sentaurus
Atomic model: parameter-free
Continuum model with external parameters
Sentaurus: Drift-diffusion
coupled with Poisson
solver in real space grids
NEGF-DFT/CPA-NVC
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Potential of Si TFET: p-i-n structure
L. Zhang, F. Zahid, M. Chan, J. Wang, H.G. (2012).
Sentaurus (green) versus NEGF-DFT (red)
Band gaps;
doping;
disorder;
large sizes;
computation;
…
Doping in the channel
does not affect the
potential profile due to
high doping at S/D
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Intrinsic channel
8nm
12nm
14nm
T=300K
p-i-n tunnel FET (TFET) potential: almost perfect agreement
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Verification for MOSFET channels
Green: Sentaurus.
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Red: Nanodsim
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New: non-uniform doping – delta doping
Atomistic treatment of doping (P-doped)
within CPA formalism
Red – NEGF-DFT
Green – Sentaurus with Fermi statistics
Black – Sentaurus with Boltzman statistics
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Full double gate FET simulation:
LG
gate
Tox
LS
oxide
LD
TSi
n+
p
n+
Tox
sS
oxide
sD
gate
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I-V characteristics
I-V characteristics calculated by atomic model are in good agreement with NanoMos
(effective mass model). Atomic model can go much further: surface roughness
scattering, inhomogeneous doping, new materials, etc.
Nanodsim (self-consistent atomic)
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NanoMos (Zhibin Ren’s thesis)
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Example 1: localized doping
NEGF-DFT-CPA-NVC
Localized doping suppresses off-state source-to-drain tunneling and
reduces performance variability.
Acknowledgement: Dr. Jesse Maassan (ECE, Purdue)
Jesse Maassan & H.G. preprint (2012).
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New idea: suppressing S-to-D off-state tunneling
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Example 2: MRAM simulations
Increasing spin transfer torque (STT) by impurity doping.
Acknowledgements:
Dr. Youqi Ke, Prof. Ke Xia, Dr. Eric Zhu, Dr. Dongping Liu, Prof. Xiu Feng Han
Youqi Ke, Ke Xia and Hong Guo, PRL 105, 236801 (2010)
D.P. Liu, X.F. Han and Hong Guo, PRB 85, 245436 (2012).
NEGF-DFT-CPA-NVC
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MTJ - magnetic tunneling junctions
Picture from W. Butler, Nature Mat., 3, 845 (2004)
Tunnel barrier is a few atomic layers thick.


I tot
 I tot
TMR =

I tot
Spin transfer torque (STT)
Problem: for a given bias, STT is too small, or junction resistance too large.
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Solution: decreasing the junction resistance
Why resistance is large? Because tunnel barrier is an insulator.
How to reduce resistance? Dope the insulator with metal atoms.
Why it does not work? Because impurity scattering destroys TMR.
Youqi Ke, Ke Xia and Hong Guo, PRL 105, 236801 (2010)
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New idea:
Can we find a dopant that exponentially decreases resistance, but only
linearly decreases TMR?
We thus predict that Zn doping into MgO barrier will solve our problem!
D.P. Liu, X.F. Han and Hong Guo, PRB 85, 245436 (2012).
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Newest: CPA to compute variance by evaluating <GGGG>
Eric Zhu & H.G. (2012).
Huge device to device variability.
F.L. Yang et al., in VLSI Technol. Tech. Symp. Dig., pp. 208, June 2007.
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Summary
 By solving 4 critical problems: disorder averaging, band gap, large size
and accuracy, NEGF-DFT method can begin to predict device
characteristics parameter-free for realistic nanoFET structures.
 Other details were included into NEGF-DFT as well: electron-phonon,
collinear and non-collinear spin, spin-orbit, photon, high frequency,
transient, etc.
 Endless application possibilities…
 Integration with industry TCAD tools possible.
 Further reduction of computation time underway …
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Thank you !
Acknowledgements to Canadian funding: NSERC, CIFAR, FQRNT, IRAP,
McGill University.
We are grateful to Hong Kong government which funded AoE at HKU where
the Sentaurus benchmark was done.
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