Post on 20-Mar-2016
description
Network for Computational Nanotechnology (NCN)UC Berkeley, Univ.of Illinois, Norfolk State, Northwestern, Purdue, UTEP
Generation and optimization of Tight Binding parameters using Genetic
Algorithms and their validation using NEMO-3D
Ganesh HegdeNetwork for Computational Nanotechnology (NCN)Electrical and Computer Engineering
Committee• Prof. Gerhard Klimeck (Major Prof.)• Dr. Michael McLennan• Prof. Supriyo Datta
Ganesh Hegde 2
Key points I wish to make in this presentation
• Need for optimization.
• Genetic Algorithm (GA) – general purpose technique.
• Tight Binding with GA
» InAs and GaAs at Low Temperature (4K)
• Validation Electronic Structure of InAs/GaAs Quantum Dots.
Ganesh Hegde 3
As the title suggests…
• Optimization • The Genetic Algorithm
• Tight Binding parameterization
• Other projects with GA + future work
…there are distinct topics tackled in this work.
Ganesh Hegde 5
The need for optimization
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The need for optimization
0 1 2 3 410-10
10-5
100
Energy (eV)
Abs
orpt
ion
(arb
itrar
y un
its)
Fig: Optical absorption plot obtained from Quantum Dot Lab tool on www.nanoHUB.org with parameters shown before.
• Forward procedure» Input Output
• Reverse procedure» Output Input
Ganesh Hegde 7
The need for optimization
• MOSFET tool on ww.nanoHUB.org
Ganesh Hegde 8
The need for optimization
0 0.5 1
10-5
Gate Voltage (V)
Sou
rce
- Dra
in C
urre
nt (A
)
Vd = 1.2 V
Vd = 0.05 V
Fig: Id-Vg plot obtained MOSFet tool on www.nanoHUB.org with parameters shown before.
0 0.5 1
10-5
Gate Voltage (V)
Sou
rce
- Dra
in C
urre
nt (A
)
Vd = 1.2 V
Vd = 0.05 V
Fig: Id-Vg plot obtained MOSFet tool on www.nanoHUB.org with parameters shown before.
Give me the input that gives me the output I want
Ganesh Hegde 9
Common features
• Input Output mapping.
• ‘N’ Input parameters» N-dimensional search space.
• Desired output(s)» Optimum solution(s) may exist
• Nature of Search space» Holes/Singularities/Discontinuities.
All of the above affect the choice of solution method!!
Linear/non-linear?
Time required
Constraints / Priorities
Gradient?
Ganesh Hegde 10
Broad comparison of commonly used optimization techniques
• Point - to - Point mathematical formulation
• E.g. Gradient-based, Gauss-Newton, Powell etc.
• Iterative» Y(i) = a.Y(i-1) + b.dY(i-1)/dx etx
• Local
• Depend on nature of search space
• Intuitive approach Analogy
• E.g. GA, SA, PSO, ACO. etc.
• Parallel • Global
• General purpose
Mathematical Techniques Heuristics
0 1 2 3 4-1
-0.5
0
0.5
1
x
y =
exp(
-x)s
in(8
x)
You need an optimum solution, not a mathematical way of getting from one point to another in search space!!
Ganesh Hegde 11
As the title suggests…
• Optimization • The Genetic Algorithm
• Tight Binding parameterization
• Other projects with GA + future work
…there are distinct topics tackled in this work.
Ganesh Hegde 12
The Genetic Algorithm – why choose it?
• Shares all +ve characteristics of heuristics
• PGAPack - Parallel Genetic Algorithm Package» David Levine, Argonne National Labs» Parallel (MPI)» Well documented, easy to interface.
• Previous experience with TB.» Klimeck et al. (1999)
General purpose, parallel, easy to interface your code
Scores over other optim. Tools!
Ganesh Hegde 13
GA – aim and analogy
• Heuristic
• Mimics biological genetic reproduction
• Survival of the fittest
Darwin Holland
Image Ref. [1] and [2]
Ganesh Hegde 14
Comparison -1• Gene
Biological AlgorithmicA unit of DNA.
Represents a physical characteristic.
A suitably encoded input parameter(binary, real, integer, exponential)
E.g.
Channel Length(nm) 23.2
Doping conc. 1e+18 /cm3
Image Ref. [3]
Ganesh Hegde 15
Comparison -2• Chromosome
Biological AlgorithmicBlueprint of an organism Collection of all inputs
E.g.
[23.2 1e+19 1e+18….]
[1101 1011 1111 0001 1110…]
[1 23 34 56 -9 -345 999 10247….]
Image Ref. [4]
Ganesh Hegde 16
GA - 1
• Input encoding» Binary» Real» Integer» Exponential» Combination of the above
0 5 10 15 200
50
100
150
200
x
y
y = (x-7)2+7
Choose an encoding suitable for your problem
Ganesh Hegde 17
GA-2
• Initialization (Playing God)» Population is created by ‘randomly’
sampling the search space
0010(2)
N individuals. N is usually large enough to accommodate memory constraints.
1111(15)
1010(10)
1101(13)
0 5 10 15 200
50
100
150
200
x
y
y = (x-7)2+7
Ganesh Hegde 18
GA-3
•Evaluation»Fitness – How ‘good’ is a potential solution?
0 5 10 15 200
50
100
150
200
x
y
y = (x-7)2+7 15
C2 is fitter than C1
1 1 1 1C1
03 0 0 1 1C2
Ganesh Hegde 19
GA-4: Selection and reproduction
n
N
C
C
CCCC
......
......4
3
2
1
)(
4
3
2
1
......Nnnew
new
new
new
new
C
CCCC
NC
CCCC
......4
3
2
1
n N
n-NUnfit to live
n
N
C
C
CCCC
......
......4
3
2
1
OLD NEWParents Mate Children are born
n (Parents+Children)
Ganesh Hegde 20
GA 5 - Crossover
0 5 10 15 200
50
100
150
200
x
y
y = (x-7)2+7
C1
C2
1 1 0 1
0 0 1 1
13
03
1 1 0 1
0 0 1 1
11
05
0 1 1
1 0 1
13
03
1 1 0 1
0 0 1 1
15
01
1 1 11 1
0 0 1 10 1
Crossover is an ‘exploitative’ operator!! It exploits the strengths of two chromosomes to form new chromosomes. Weaker children are discarded in
the next evaluation. Stronger ones improve fitness further.
Ganesh Hegde 21
GA 6 - Mutation
1 1 1 11 1 1 15
0 1 1 11 1 1 07
Mutation is an Explorative operator!!Prevents getting stuck in a local optima.Allows for exploration of search space.
0 5 10 15 200
50
100
150
200
x
y
y = (x-7)2+7
Standard GA In practice you can design your own operators
Ganesh Hegde 22
Summary
Physics CodeOutputsInputs
Optimization Algorithm
Modify
Evaluate
• Optimization Process
Genetic Algorithm
Initialization
Fitness Evaluation, SortSelection, Crossover,
Mutation, Replacement
Ganesh Hegde 23
As the title suggests…
• Optimization • The Genetic Algorithm
• Tight Binding parameterization
• Other projects with GA + future work
…there are distinct topics tackled in this work.
Ganesh Hegde 24
Tight Binding
• Electronic Structure Method
• LCAO
• Potential and material variation atomic scale
• Atomistic basis nearest neighbor sparse Hamiltonian
• sp3d5s*
(Image from http://cobweb.ecn.purdue.edu/~gekco)
Ganesh Hegde 25
TB as an optimization problem
nnn
n
HH
HH
......................................................
.........
1
111
atomson centered ions Wavefunct-
IntegralsCenter Two - ||Energies Onsite - ||
i
ijji
iiii
HHHH
nnH
HHHH
......
......
......22
21
12
11
Vhh1,Vhh2,Vhh3,….Vhhn
Ec1 ,Ec2 ,Ec3,….Ecn
Eg1 ,Eg2 ,Eg3,….Egn
P1 ,P2 ,P3,….Pn
m*1 ,m*2 ,m*3,….m*n
35 inputs/material, 100’s of outputs, unknown search space Genetic Algorithm
nnH
HHHH
......
......
......22
21
12
11m*1 ,m*2 ,m*3,….m*n
nnH
HHHH
......
......
......22
21
12
11m*1 ,m*2 ,m*3,….m*n
Eg1 ,Eg2 ,Eg3,….Egn
nnH
HHHH
......
......
......22
21
12
11m*1 ,m*2 ,m*3,….m*n
Eg1 ,Eg2 ,Eg3,….Egn
nnH
HHHH
......
......
......22
21
12
11m*1 ,m*2 ,m*3,….m*n
Ganesh Hegde 26
TB parameterization - methodology
Solve [H]{Ψ}= E{Ψ} Outputs (Band structure)Inputs (Hamiltonian Terms)
Optimization Algorithm
Modify
Evaluate
Masses, Band Edges, Gaps, etc (from experiment/theory)
Genetic Algorithm (PGAPACK)
Initialization (Random)
Fitness Extraction
Fitness Evaluation, SortSelection, Crossover,
Mutation, Replacement
Physics Code (NEMO-1D)
outputs
2
2
target)calculated-(target Fitness Weight
Ganesh Hegde 27
TB bulk results
Fig. Bulk band structure of (a) GaAs and (b) InAs at 4K
(a) (b)
-10
-5
0
5
10
Ene
rgy
(eV
)
k(L --> --> X)-10
-5
0
5
10
k(L --> --> X)E
nerg
y (e
V)
Ganesh Hegde 28
Quantity GaAs calculated GaAs Target % deviation InAs calculated InAs Target % deviation
Eg_gamma 1.5383092 1.5382 0.007099207 0.417766 0.418 0.055980861Ec_gamma 1.5383087 1.5382 0.007066701 0.6435875 0.6437 0.017477086Vhh -0.0000006 0 -- 0.2258215 0.2257 0.053832521Vlh -0.0000006 0 -- 0.2258215 0.2257 0.053832521Ec_X 1.899928 1.9 0.003789474 2.2799501 2.28 0.002188596Ec_L 1.7079764 1.708 0.001381733 1.5300286 1.53 0.001869281k_X 0.9 0.9 0 0.9 0.9 0 k_L 1 1 0 1 1 0
Electron effective Mass(Gamma)mstar_c_001 0.0658386 0.067 1.733432836 0.0229801 0.0239 3.848953975
Hole Effective masses(Gamma)mstar_lh_001 -0.0826915 -0.0871 5.061423651 -0.0281261 -0.0273 3.026007326mstar_lh_011 -0.0731901 -0.0804 8.967537313 -0.0270091 -0.0264 2.30719697mstar_lh_111 -0.0709123 -0.0786 9.780788804 -0.0266759 -0.0261 2.20651341mstar_hh_001 -0.3106723 -0.403 22.91009926 -0.3258846 -0.3448 5.485904872mstar_hh_011 -0.6059149 -0.66 8.194712121 -0.6236339 -0.6391 2.419981224mstar_hh_111 -0.8233913 -0.813 1.278142681 -0.8766644 -0.8764 0.030168873
Ganesh Hegde 29
InAs bulk variation with hydrostatic strain
εxx = εyy = εzz
Change lattice constant of material
to correspond to required strain
Fig. Gaps and edges at Gamma point for InAs at 4K versus hydrostatic strain
Solid Lines – TheoryCircles - calculated
Ganesh Hegde 30
GaAs bulk variation with hydrostatic strain
Fig. Gaps and edges at Gamma point for GaAs at 4K versus hydrostatic strain
Solid Lines – TheoryCircles - calculated
Ganesh Hegde 31
InAs bulk variation with uniaxial (001) strain
Fig. Gaps and edges at Gamma point for InAs at 4K versus uni-axial (001) strain.
εxx = εyy != εzz
Solid Lines – TheoryCircles - calculated
Ganesh Hegde 32
GaAs bulk variation with uniaxial (001) strain
Fig. Gaps and edges at Gamma point for GaAs at 4K versus uniaxial strain
Solid Lines – TheoryCircles - calculated
Ganesh Hegde 33
Numerical experiment in NEMO-3D
-8 -6 -4 -2 00.5
1
1.5
2
2.5
hydrostatic strain (percentage)
Ene
rgy
gaps
in e
V
LTRTLT
new• Free standing InAs box • 5nm X 5nm X 5 nm• Hydrostatic Strain• Energy gap measured
L, X valleys moved below Gamma valley calculated gap at Gamma NOT true gap!!
Ganesh Hegde 34
Validation of TB parameters – Electronic Structure of InAs/GaAs Dots
• Self – Assembly
• Experimental Uncertainties»GA diffusion (increases gap)»Size»Atomic Structure
• Previous theoretical studies +/- 10% error.
GaAs
GaAs
InAs
GaAs
InAs lattice constant > GaAs (7%)
Difficult to accurately model electronic structure of InAs/GaAs QD’s !!
Ganesh Hegde 35
Attempts at matching experiment
• Optical Gap = CBM - VBM• Coulombic correction not calculated (30-40 meV effect)• 2 Strain models in NEMO-3D (harmonic, Anharmonic)
Optical Gap (meV)
Dot dimensions Keating - Harmonic Keating - Anharmonic Experiment/Theory(Base nm X width nm) (3,4,5)
15 X 2.5 1238 1133 107820 X 6 1184 1000 1098
25.2 X 3.5 1164 1040 103225.2 X 2.5 1178 1059 113127.5 X 3.5 1154 1020 1016
Ganesh Hegde 36
Built in models in NEMO-3D for Atomic Structure
AsIn
const. distortion angle bond const. distortionlength bond
),(
fk
AsIn
)( , param. distortion angle bond param. distortionlength bond
),(
00 dxfkm(dx)g(dx)
fk
2
21 dxkE
dx
Harmonic
2)(21 dxdxkE
dx
Anharmonic
dxkF
Ganesh Hegde 37
Atomic Structure of QD’s – procedure and consequences
• Aim » To understand why the harmonic
model always gives a larger band gap than the anharmonic model
• Procedure» Lattice constant of GaAs entire
structure.» Minimize total strain energy.» Calculate bond length deviations
• Result» Both strain models InAs is only
compressively strained. (-1 to -5%)» Strain in Anharmonic model <
Strain in harmonic model.
-10 -5 0 5 100
0.05
0.1
0.15
0.2
0.25
Strain (deviation from unstrained bond length)
Stra
in E
nerg
y
AnharmonicHarmonic
-5 -4 -3 -2 -10
2
4
6
8
10
12x 104
Strain( % deviation from unstrained bond length)
Num
ber o
f rel
axed
bon
ds
AnharmonicHarmonic
Ganesh Hegde 38
The essential difference – an intuitive picture
-10 -5 0 5 100
0.05
0.1
0.15
0.2
0.25
Strain (deviation from unstrained bond length)
Stra
in E
nerg
y
AnharmonicHarmonic
|||| 1212 FFkk
In NEMO3D we initially set the lattice constant = lattice constant of GaAs
for both strain models!
AsIn
dxkF 11
Harmonic
In As
dxkF 22
Anharmonic
In As
dxkF 22
Anharmonic model minimizes its strain more effectively than Harmonic model.
Ganesh Hegde 39
Attempts at matching experiment
• Optical Gap = CBM - VBM• Coulombic correction not calculated (30-40 meV effect)• 2 Strain models in NEMO-3D (harmonic, Anharmonic)
Atomic Structure effects are extremely important in validation!!!
Optical Gap (meV)
Dot dimensions Keating - Harmonic Keating - Anharmonic Experiment/Theory(Base nm X width nm) (3,4,5)
15 X 2.5 1238 1133 107820 X 6 1184 1000 1098
25.2 X 3.5 1164 1040 103225.2 X 2.5 1178 1059 113127.5 X 3.5 1154 1020 1016
Ganesh Hegde 40
Summary
• Genetic Algorithm» General purpose» Parallel» Easy to implement and interface
• TB is a non-trivial optimization problem» TB parameterization and results» Effect of strain on bulk electronic structure
• Matching to experiment for InAs/GaAs dot system is non-trivial» Experimental uncertainties» Atomic structure effects
Ganesh Hegde 41
As the title suggests…
• Optimization • The Genetic Algorithm
• Tight Binding parameterization
• Other projects with GA + future work
…there are distinct topics tackled in this work.
Ganesh Hegde 42
Additional projects with the GA
• Tight Binding Parameters » Si (4K)» AlAs (4K and 300K)» InSb, AlSb and GaSb at 300K. (Intend to publish Sb parameters)
• Force Field Optimization (collaboration with Strachan group)» Energy, Force and Stress minimization (Ni,Ti)» Force Field parameters» Replace ab-initio calculations
Ganesh Hegde 43
General purpose optimization engine for nanoHUB
GUI
Rappture – <Language>API
Tool
Rappture – <Language>API
GUI
Rappture Optimization API
Rappture Optimization API
Tool Tool Tool Tool
Analyze
Launch
Ganesh Hegde 44
Future Work
• Arbitrariness of TB parameters
• Parameters for Surfaces/Interfaces scope for work in this area.
• Fitness = single number.
• Alternate optimization techniques.
• Atomic Structure effects greater accuracy required!
Ganesh Hegde 45
Acknowledgments
• Committee Members» Prof Klimeck for guidance, constant encouragement (+ve and -ve) and
funding support.» Dr. McLennan for his initial guidance with the optimization API and for funding
support.» Prof. Datta for agreeing to be a part of my committee in spite of the confusion
and for ECE 495 and 659, both excellent courses from which I’ve learned a lot.
• George Howlett for helping me out whenever I needed it. (If I have problems with my code, I’m coming back for more help!!)
• All EE-350 lab-mates – in particular Sunhee, Usman and Sebastian. Everyone else for the long hours of discussion – technical and non-technical. (…and for tolerating me!!)
• Cheryl Haines, Vicki Johnson – Mother Hens of EE-350!!
Ganesh Hegde 46
• Images
1. http://user.uni-frankfurt.de/~scherers/blogging/AdventsKalenderPlots/GaAs/BandStructureGaAs_s_mark.jpg
2. http://www.mun.ca/computerscience/news/distinguished_lect.php 3. http://en.wikipedia.org/wiki/File:ADN_animation.gif