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Device Modeling from Atomistic First Principles: theory of the nonequilibrium vertex correction
Eric Zhu1, Leo Liu1, Hong Guo1,2
1 Nanoacademic Technologies Inc. Brossard, QC J4Z 1A7, Canada
2 Dept. 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.
Continuum model Atomic model
Goal: simulate a transistor from atomic first principles
(10 nm)3 chunk of Si has ~64,000 atoms.
Picture from Taur and Ning, Fundamentals of Modern VLSI Devices
~100nm~10nm
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Current: L=22nm Next: L=16nm DFT: ~1,000 atoms
1. Doping & disorder,2. Band gaps,3. Large sizes,4. Accuracy.
Other physics: phonons, magnons, photons, correlations…
… and many other systems with different materials
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?Can we calculate ?
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?
This talk:
Ex 1: Dopant fluctuation gives rise to device-to-device variability
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Huge device to device variability.
F.L. Yang et al., in VLSI Technol. Tech. Symp. Dig., pp. 208, June 2007.
If every transistor behaves differently, difficult to design a circuit.
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Ex 2: roughness scattering increases resistance of Cu interconnects
With Daniel Gall of RPI.
$: SRC
Ex. 3: disorder effect in topological insulator Bi2Se3
Calculated spin direction
Top surface Bottom surface
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Experiment (Hasan etal) Ab initio (Zhao etal)
Conductance:
Wang, Hu, H.G. PRB 85, 241402 (2012)Zhao, H.G. etal Nano Lett. 11, 2088 (2011).
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Quantitative prediction of quantum transport from atomic first principles without any parameter
Semi-empirical device modeling,
10,000 to 100,000 atoms
device parameters
TCADatomic simulations
materials, chemistry, physics
quantum mechanics Physics
device modeling < 5nm (1000 atoms)
New
science engineering
Can we calculate realistic device parameters?
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Method - NEGF-DFT: non-equilibrium density matrix
( )r H
DFT
NEGF
NEGF-DFT
R A ~ G dE G G dE
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.
‘DFT’: density functional theory
NEGF: Keldysh nonequilibrium Green’s function
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).
1. Within NEGF-DFT: solving the disorder averaging problem
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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
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).
NEGF-DFT/CPA-NVC
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).
A tough problem of atomic calculations: disorder scattering
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xxBA 1
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?
Generating many configurations, compute each, and average result very time consuming(Small x, large N)
To build intuition, let’s solve a toy problem exactly
1D tight binding chain
1 2
nearest neighbor coupling
on-site energy
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Self-energies for the leads:
How to handle half-infinite chain ?
Self energy:
The problem is reduced to 3 sites plus self-energies
1 2
L R
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Physical quantities and NEGF
Express physical quantities in terms of NEGF:
The problem is reduced to calculate disorder averaged NEGF
average over disorder configurations
L
R
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Disorder average can be done exactly for the 3-site model
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1 2
L R
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” Coherent potential Approx.
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CPA - well established formalism
xxBA 1 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.
Rev. Mod. Phys. 46, 466 (1974)
][),( Ra
Lr GGTrVET B. Velicky, Phys. Rev. 183 (1969).
P. Soven, Phys. Rev. 156, 809 (1967).
CPA:
CPA picture: effective media
and are solved from CPA equation
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Implementation:
needs a method that does one atom at a time: LMTO, KKR, etc..
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Non-equilibrium density matrix: nonequilibrium vertex
( )r H
DFT
NEGF
NEGF-DFT
rlAR
AR
AR
GGTr T
dE GG dE G ~
dE GG dE G ~
Average over random disorder: X
Take Home message #1: multiple disorder scattering at non-equilibrium is solved by the non-equilibrium vertex correction theory (NVC) and implemented in NEGF-DFT software Nanodsim.
ARAR G GGG
specular part diffusive part
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).
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 are solved from NVC equation
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Comparison for the 3-site toy model:
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Excellent ! specular part 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 NVC exact
2. Within NEGF-DFT: solving the band gap problem
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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
The band gap problem of local functionals in DFT
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MBJ computation time is ~LDA
DFT calculation of band gaps:
Some relevant band gaps for transistors materials:
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Some relevant effective masses
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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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MBJ potential + CPA: works. Good agreement with experimental data.
Experimental data
Calculated data
3. Within NEGF-DFT: solving the large size problem
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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).
Locality: the principle underlying all O(N) methods
Equilibrium density matrix exhibits decaying property:
insulator
metal
Example: Si bulk
LMTO (nanodsim) LCAO (nanodcal)
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)
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Density matrix computation:
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( )r H
DFT
NEGF
DFT – computes potential and energy levels of the device;
NEGF – non-equilibrium statistics that fills levels;
The self-consistent loop – NEGF-DFT algorithm we use.
Practically, density matrix is divided into two parts: equilibrium and non-equilibrium parts:
no locality locality
R A ~ G dE G G dE
Roadmap for locality-less computation of density matrix of large systems
no preconditioner with preconditioner
qmr SOR ILU H-MatrixJacobigmres bicgstab
algorithms
iterative method direct method
MCS
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Large: ~20,000 atoms
Do not depend on locality
Roadmap (cont.)
single wall double wall thin & long thick & short
algorithms
iterative method direct method
nested dissection principal layer pardiso
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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.
Benchmark: 160 cores, 3GB / core, 12,800 atomic sites, <30 min/step
speed performance memory performance
Lx = periodic
Ly = 10 nm
Lz = 5, 10 ,15, 20 ,25, 30nm
160 cores
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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 1320 atomic spheres
(2880 Orbitals)
Two probe 1 1 X 20 X 206400 atomic spheres
(57600 Orbitals)107 NEGF-DFT steps,
5.5 min/step, total 9.8 hoursPotential 1.0 x 10-5
Charge 1.23 x 10-5 per atom
Two probe 2 1 X 20 X 4012800 atomic spheres
(115200 Orbitals)195 NEGF-DFT steps,
14 min/step, total 46 hours
Potential 1.0 x 10-5
Charge 6.2 x 10-6 per atom
Two probe 3 1 x 20 x 6019200 atomic spheres
(172800 Orbitals)267 NEGF-DFT steps,
30 min/step, total 134 hoursSame 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
• 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
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
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4. How do we know all are well for real devices ?
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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
Commercial TCAD tool:
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1328 pages of parameter and physics descriptions
Hundreds of parameters are needed !
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NEGF-DFT/CPA-NVC versus Sentaurus
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Continuum model with external parameters
Sentaurus: Drift-diffusion coupled with Poisson solver in real space grids
Atomic model: parameter-free
NEGF-DFT/CPA-NVC
Potential of Si TFET: p-i-n structure
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Band gaps; doping; disorder; large sizes; computation; …
L. Zhang, F. Zahid, M. Chan, J. Wang, H.G. (2012).
Doping in the channel does not affect the potential profile due to high doping at S/D p-i-n tunnel FET (TFET) potential: almost perfect agreement
Sentaurus (green) versus NEGF-DFT (red)
T=300K
Intrinsic channel8nm
12nm
14nm
Verification for MOSFET channels
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Green: Sentaurus. Red: Nanodsim
New: non-uniform doping – delta doping
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Red – NEGF-DFT
Green – Sentaurus with Fermi statistics
Black – Sentaurus with Boltzman statistics
Atomistic treatment of doping (P-doped)
within CPA formalism
Full double gate FET simulation:
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LG
LS LDoxide
oxideS D
gate
gate
Tox
TSi
Tox
pn+ n+
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) NanoMos (Zhibin Ren’s thesis)
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Example 1: localized doping
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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).
NEGF-DFT-CPA-NVC
New idea: suppressing S-to-D off-state tunneling
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Example 2: MRAM simulations
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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)
TMR =
tot
tottot
I
II
Tunnel barrier is a few atomic layers thick.
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Problem: for a given bias, STT is too small, or junction resistance too large.
Spin transfer torque (STT)
Solution: decreasing the junction resistance
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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)
New idea:
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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).
Newest: CPA to compute variance by evaluating <GGGG>
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Huge device to device variability.
F.L. Yang et al., in VLSI Technol. Tech. Symp. Dig., pp. 208, June 2007.
Eric Zhu & H.G. (2012).
Summary
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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.