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.

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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.

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The need for optimization

• Quantum Dot Lab www.nanoHUB.org

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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

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The need for optimization

• MOSFET tool on ww.nanoHUB.org

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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

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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?

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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!!

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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.

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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!

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GA – aim and analogy

• Heuristic

• Mimics biological genetic reproduction

• Survival of the fittest

Darwin Holland

Image Ref. [1] and [2]

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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]

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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]

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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

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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

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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

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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)

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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.

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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

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Summary

Physics CodeOutputsInputs

Optimization Algorithm

Modify

Evaluate

• Optimization Process

Genetic Algorithm

Initialization

Fitness Evaluation, SortSelection, Crossover,

Mutation, Replacement

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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.

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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)

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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

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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

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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

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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

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GaAs bulk variation with hydrostatic strain

Fig. Gaps and edges at Gamma point for GaAs at 4K versus hydrostatic strain

Solid Lines – TheoryCircles - calculated

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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

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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

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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!!

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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 !!

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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

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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

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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