1. Eric Zhu1, Leo Liu1, Hong Guo1,2
2017/4/21
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
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2. Goal: simulate a transistor from atomic first principles
2017/4/21
Goal: simulate a transistor from atomic first principles
Picture from Taur and Ning, Fundamentals of Modern VLSI Devices
~100nm
~10nm
Doping & disorder,
Band gaps,
Large sizes,
Accuracy.
Other physics: phonons, magnons, photons, correlations…
Current: L=22nm Next: L=16nm
(10 nm)3 chunk of Si has ~64,000 atoms.
DFT: ~1,000 atoms
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3. … and many other systems with different materials
2017/4/21
… 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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4. Ex 1: Dopant fluctuation gives rise to device-to-device variability
2017/4/21
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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5. NEGF5, Jyvaskyla, Finland
2017/4/21
Ex 2: roughness scattering increases resistance of Cu interconnects
With Daniel Gall of RPI.
$: SRC
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6. NEGF5, Jyvaskyla, Finland
Ex. 3: disorder effect in topological insulator Bi2Se3
Experiment (Hasan etal)
Ab initio (Zhao etal)
Conductance:
Wang, Hu, H.G. PRB 85, (2012)
Calculated spin direction
Top surface
Bottom surface
Zhao, H.G. etal Nano Lett. 11, 2088 (2011).
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7. Can we calculate realistic device parameters?
2017/4/21
Can we calculate realistic device parameters?
Semi-empirical device modeling,
10,000 to 100,000 atoms
device parameters
TCAD
atomic simulations materials, chemistry, physics
quantum mechanics Physics
device modeling < 5nm (1000 atoms)
New
science
engineering
Quantitative prediction of quantum transport from atomic first principles without any parameter
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8. Two ingredients in theory: Quantum statistics and Hamiltonian
2017/4/21
Two ingredients in theory: Quantum statistics and Hamiltonian
H = Hleads + Hdevice + Hcoupling
Calculating electric current flow driven by a finite bias voltage is a non-equilibrium problem, it is important to correctly determine non-equilibrium statistics of the device scattering region that is embedded in an open environment.
A second consideration is the determination of device Hamiltonian H. H provides the energy levels of the device. How to fill these levels is given by the non-equilibrium statistics.
Keldysh non-equilibrium Green’s function (NEGF) is a natural approach to determine the non-equilibrium statistics. One may also evolve a non-equilibrium density matrix from some equilibrium initial condition.
What kind of H to use is an issue of taste and approximation: effective mass, kp, EH,TB, DFTB, HF, DFT, GW, higher-functionals, QMC, CI… In the end, one has to compare to experimental data without adjusting free parameters.
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9. Method - NEGF-DFT: non-equilibrium density matrix
2017/4/21
Method - NEGF-DFT: non-equilibrium density matrix
NEGF-DFT
‘DFT’: density functional theory
NEGF: Keldysh nonequilibrium Green’s function
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, (2001).
M. Brandbyge, J.-L. Mozos, P. Ordejon, J. Taylor, and K. Stokbro, PRB 65, (2002).
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10. 1. Within NEGF-DFT: solving the disorder averaging problem
2017/4/21
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
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. (
Youqi Ke, Ke Xia and Hong Guo PRL 100, (2008);
Youqi Ke, Ke Xia and Hong Guo, PRL 105, (2010);
Ferdows Zahid, Youqi Ke, Daniel Gall and Hong Guo, PRB 81, (2010);
Eric Zhu, Lei Liu and Hong Guo, preprint (2012).
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11. A tough problem of atomic calculations: disorder scattering
2017/4/21
A tough problem of atomic calculations: disorder scattering
Generating many configurations, compute each, and average result very time consuming
(Small x, large N)
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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12. To build intuition, let’s solve a toy problem exactly
2017/4/21
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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13. Self-energies for the leads:
2017/4/21
Self-energies for the leads: 1 2 SL SR
How to handle half-infinite chain ?
Self energy:
The problem is reduced to 3 sites plus self-energies
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14. Physical quantities and NEGF
2017/4/21
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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15. NEGF5, Jyvaskyla, Finland
2017/4/21 1 2 SL SR
Disorder average can be done exactly for the 3-site model
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16. Exact solution of the 3-site toy model:
2017/4/21
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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17. CPA - well established formalism
2017/4/21
CPA - well established formalism
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).
B. Velicky, Phys. Rev. 183 (1969).
Rev. Mod. Phys. 46, 466 (1974)
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18. NEGF5, Jyvaskyla, Finland
2017/4/21
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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19. Non-equilibrium density matrix: nonequilibrium vertex
2017/4/21
Non-equilibrium density matrix: nonequilibrium vertex
NEGF-DFT
Average over random disorder: X
specular part diffusive part
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.
Youqi Ke, Ke Xia and Hong Guo PRL 100, (2008)
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20. Essence of Nonequilibrium Vertex Correction (NVC)
2017/4/21
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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21. NVC Equation: some complicated technical details
2017/4/21
NVC Equation: some complicated technical details
Youqi Ke, Ke Xia and Hong Guo PRL 100, (2008).
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22. Consistency check: CPA-NVC identity
2017/4/21
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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23. NVC solution for the 3-site toy model
2017/4/21
NVC solution for the 3-site toy model
and are solved from NVC equation
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24. Comparison for the 3-site toy model:
2017/4/21
Comparison for the 3-site toy model:
specular part diffusive part
Excellent !
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25. NEGF5, Jyvaskyla, Finland
2017/4/21
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
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26. 2. Within NEGF-DFT: solving the band gap problem
2017/4/21
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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27. The band gap problem of local functionals in DFT
2017/4/21
The band gap problem of local functionals in DFT
DFT calculation of band gaps:
MBJ computation time is ~LDA
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28. Some relevant band gaps for transistors materials:
2017/4/21
Some relevant band gaps for transistors materials:
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29. Some relevant effective masses
2017/4/21
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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30. NEGF5, Jyvaskyla, Finland
2017/4/21
Experimental data
Calculated data
MBJ potential + CPA: works. Good agreement with experimental data.
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31. 3. Within NEGF-DFT: solving the large size problem
2017/4/21
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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32. Locality: the principle underlying all O(N) methods
2017/4/21
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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33. Density matrix computation:
2017/4/21
Density matrix computation:
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
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34. NEGF5, Jyvaskyla, Finland
2017/4/21
Roadmap for locality-less computation of density matrix of large systems
no preconditioner
with preconditioner
qmr
SOR
ILU
H-Matrix
Jacobi
gmres
bicgstab
algorithms
iterative method
direct method
MCS
Large: ~20,000 atoms
Do not depend on locality
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35. NEGF5, Jyvaskyla, Finland
2017/4/21
Roadmap (cont.)
single wall
double wall
thin & long
thick & short
algorithms
iterative method
direct method
nested dissection
principal layer
pardiso
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36. Performance: nano-device simulator (nanodsim)
2017/4/21
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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37. Solving ~20,000 atomic spheres for open devices at nonequilibrium
2017/4/21
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)
Two probe 1
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
( Orbitals)
195 NEGF-DFT steps,
14 min/step, total 46 hours
Charge 6.2 x 10-6 per atom
Two probe 3
1 x 20 x 60
19200 atomic spheres
( Orbitals)
267 NEGF-DFT steps,
30 min/step, total 134 hours
Same as above
Summary: NEGF-DFT modeling has reached realistic device sizes!
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38. If using tight binding model: huge systems can be done
2017/4/21
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 × 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 = nm, 1/3 of computation time is spent on surface Green’s function, and 2/3 spent on transmission calculation
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39. 4. How do we know all are well for real devices ?
2017/4/21
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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40. NEGF5, Jyvaskyla, Finland
2017/4/21
Commercial TCAD tool:
1328 pages of parameter and physics descriptions
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41. Hundreds of parameters are needed !
2017/4/21
Hundreds of parameters are needed !
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42. NEGF-DFT/CPA-NVC versus Sentaurus
2017/4/21
NEGF-DFT/CPA-NVC versus Sentaurus
Atomic model: parameter-free
Continuum model with external parameters
NEGF-DFT/CPA-NVC
Sentaurus: Drift-diffusion coupled with Poisson solver in real space grids
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43. Potential of Si TFET: p-i-n structure
2017/4/21
Potential of Si TFET: p-i-n structure
L. Zhang, F. Zahid, M. Chan, J. Wang, H.G. (2012).
p-i-n tunnel FET (TFET) potential: almost perfect agreement
Sentaurus (green) versus NEGF-DFT (red)
T=300K
Intrinsic channel
8nm
12nm
14nm
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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44. Verification for MOSFET channels
2017/4/21
Verification for MOSFET channels
Green: Sentaurus Red: Nanodsim
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45. New: non-uniform doping – delta doping
2017/4/21
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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46. Full double gate FET simulation:
2017/4/21
Full double gate FET simulation: LG LS LD
oxide sS sD
gate
Tox
TSi p n+
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47. Nanodsim (self-consistent atomic)
2017/4/21
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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48. Example 1: localized doping
2017/4/21
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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49. New idea: suppressing S-to-D off-state tunneling
2017/4/21
New idea: suppressing S-to-D off-state tunneling
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50. Example 2: MRAM simulations
2017/4/21
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
NEGF-DFT-CPA-NVC
Youqi Ke, Ke Xia and Hong Guo, PRL 105, (2010)
D.P. Liu, X.F. Han and Hong Guo, PRB 85, (2012).
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51. MTJ - magnetic tunneling junctions
2017/4/21
MTJ - magnetic tunneling junctions
Picture from W. Butler, Nature Mat., 3, 845 (2004)
Tunnel barrier is a few atomic layers thick.
TMR =
Spin transfer torque (STT)
Problem: for a given bias, STT is too small, or junction resistance too large.
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52. Solution: decreasing the junction resistance
2017/4/21
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, (2010)
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53. NEGF5, Jyvaskyla, Finland
2017/4/21
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, (2012).
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54. Newest: CPA to compute variance by evaluating <GGGG>
2017/4/21
Newest: CPA to compute variance by evaluating <GGGG>
Huge device to device variability.
Eric Zhu & H.G. (2012).
F.L. Yang et al., in VLSI Technol. Tech. Symp. Dig., pp. 208, June 2007.
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55. NEGF5, Jyvaskyla, Finland
2017/4/21
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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56. NEGF5, Jyvaskyla, Finland
2017/4/21
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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57. Today’s Circuit simulation:
2017/4/21
Today’s Circuit simulation:
How do we know device parameters?
Slide courtesy from Munsun Chan (HKUST)
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58. Device parameters: measured in foundry
2017/4/21
Device parameters: measured in foundry
capacitance
transconductances
diodes
Geometry scaling
Several hundred parameters are needed.
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59. Today: the Moore’s law of fitting parameters
2017/4/21
Today: the Moore’s law of fitting parameters
Too many parameters, very difficult to go on
Slide courtesy from Munsun Chan (HKUST)
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60. Why there are so many fitting parameters ?
2017/4/21
Why there are so many fitting parameters ?
Model is beautiful, real device is ugly.
Ex: parameters are added to take into account the fringing fields and shape irregularity.
More and more parameters are added to account for changes of device due to whatever reason.
Slide courtesy from Munsun Chan (HKUST)
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61. Computational nanoelectronics should have:
2017/4/21
Computational nanoelectronics should have:
discrete materials – atomic first principles
quantum transport in discrete materials – quantum equations
non-equilibrium quantum transport conditions - distributions
reasonably accurate results: band gap, disorder, phonon - methods
thousands and up to 70,000 atoms – large systems
wide scope of materials and device structures – complexity
reasonably fast computation – computer science, applied math …
Red – must have
Black – nice to have
Present status: certain level of every item has been developed, more at hands-on sessions.
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62. Can we simulate a FET from atomic first principles?
2017/4/21
Can we simulate a FET from atomic first principles?
Picture from Taur and Ning, Fundamentals of Modern VLSI Devices
~100nm
~10nm
2011: L=22nm
2013 (?): L=14nm
To model, needs ~10,000 to 70,000 Si atoms in the channel region. Can we calculate?
DFT: ~1000 atoms.
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63. Wide range of research are carried out by NEGF-DFT
2017/4/21
Wide range of research are carried out by NEGF-DFT
Leakage current is MOSFET;
Resistivity of Cu interconnects;
Conductance, I-V curves of transport junctions;
Computation of capacitance, diodes, inductance, current density;
TMR, spin currents, and spin injection in magnetic tunnel junctions;
Transport in nanowires, rods, films, clusters, nanotubes;
Resistance of surface, interface, grain boundary;
STM image simulations;
Strongly correlated issues in transport at the large U limit;
Transport in semiconductor devices;
Transport through molecules; ….
a progressing field...
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64. Direct method: principal layer (PL)
2017/4/21
Direct method: principal layer (PL)
After attempting many mathematical methods for solving the large matrix problem in NEGF, the principal layer approach for the LMTO implementation of NEGF-DFT wins for problems of roughly N < 20,000 atoms.
10 nm
6 nm
CHANNEL
SOURCE
DRAIN
Computation: O(N) along transport, O(M2) along transverse direction
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65. Verification on p-i-p MOSFET channels
2017/4/21
Verification on p-i-p MOSFET channels
Red – NEGF-DFT
Green – Sentaurus with Fermi statistics
Black – Sentaurus with Boltzman statistics
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