resumo aula passada

Resumo aula passada

• Diferentes processos litográficos, por projeção, por contato, por mergulho, por escrita direta

• Evolução tamanho linha escrita• Litografia soft. Litografia nanoimpressa• SAW, dispositivos integrados• Sala limpa• Materiais fotônicos, MEMS – MOEMS

20110620

Cristais Fotônicos

Elétrons de um lado e fótons do outro lado,

junção de fóton + eletrônico (fotônico)

Temos elétrons em sólidos e fótons

em......materiais fotônicos

Referencias

• Fundamentals of Photonics (SALEH AND TEICH · Fundamentals of Photonics, Second Edition. ISBN: 978-0-471-35832-9 )– Ch. 9: Fiber Optics

• Photonic Crystals: Molding the Flow of Light(ISBN: 978-0-691-12456-8)– J. D. Joannopoulos

• Photonic crystal tutorials– Steven G. Johnson– http://ab-initio.mit.edu/photons/tutorial/

• MIT Photonic Bands Software– Free program for computing photonic crystal band structures– http://ab-initio.mit.edu/wiki/index.php/MIT_Photonic_Bands

Cristais fotônicos

Em Cristal Sólido

• elétrons • potencial periódico• banda de energia• defeitos: estados

dentro da banda proibida

Em Cristal Fotônico• fótons• modulação da constante

dielétrica• Banda de energia

fotônica = photonic band gap (PBG)

• defeitos: estados dentro da banda com direcionalidade bem definida

Yablonovitch, PRL 58 (1987) 2059; John, PRL 58 (1987) 2486

Analogia entre cristal sólido e cristal fotônico.

Analogias • portadores • estrutura• bandas• defeitos

Solid of N atomsTwo atoms Six atoms

Band Theory: “Bound” Electron Approach• For the total number N of atoms in a solid (1023 cm–3), N energy

levels split apart within a width E.– Leads to a band of energies for each initial atomic energy level (e.g.

1s energy band for 1s energy level).

Electrons must occupy different energies due to

Pauli Exclusion principle.

Filtro de Fabry-Perot C_MEMS

http://www.npphotonics.com/files/article/OEG20030324S0088.htm

O seguinte é um seminário dado porSteven G. Johnson, MIT Applied

Mathematics

From electrons to photons: Quantum-inspired modeling in nanophotonics

Steven G. Johnson, MIT Applied Mathematics

Nano-photonic media (l-scale)

synthetic materials

strange waveguides

3d structures

hollow-core fibersoptical phenomena

& microcavities[B. Norris, UMN] [Assefa & Kolodziejski,

[Mangan, Corning]

1887 1987

Photonic Crystals

periodic electromagnetic media

periodic intwo directions

periodic inthree directions

periodic inone direction

can have a band gap: optical “insulators”

Cristal fotônico 1D1887

periodic inone direction

e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2

e(x) = e(x+a)a

Elétron numa rede periódica 1D

Elétron num ambiente livre

− ℏ2

2𝑚𝜕2 (𝑥 )𝜕 𝑥2 =𝐸 (𝑥 )

Solução: (𝑥 )=𝐴𝑒𝑖𝑘𝑥

𝐸=ℏ𝑘2

− ℏ2

2𝑚𝜕2 (𝑥 )𝜕 𝑥2 +∑

𝐺𝑈𝐺𝑒− 𝑖𝐺𝑥 (𝑥 )=𝐸 (𝑥 )

𝐺=h 2𝜋𝑎

Elétron numa rede periódica – aproximação por poço retangular periódico

Mostra a existência de banda proibida, imposta por condições de contorno

Simulation of Band Gap Structures of 1D Photonic Crystal

• Journal of the Korean Physical Society, Vol. 52, February 2008, pp. S71S74

Electronic and Photonic Crystals

atoms in diamond structure

wavevector

dielectric spheres, diamond lattice

wavevector

interacting: hard problem non-interacting: easy problem

Electronic & Photonic Modelling

Electronic Photonic

• strongly interacting —tricky approximations

• non-interacting (or weakly), —simple approximations (finite resolution) —any desired accuracy

• lengthscale dependent (from Planck’s h)

• scale-invariant —e.g. size 10 l 10

Option 1: Numerical “experiments” — discretize time & space … go

Option 2: Map possible states & interactions using symmetries and conservation laws: band diagram

Fun with Math

dielectric function e(x) = n2(x)

First task:get rid of this mess

eigen-operator eigen-value eigen-state

H 0+ constraint

Electronic & Photonic Eigenproblems

Electronic Photonic

simple linear eigenproblem(for linear materials)

nonlinear eigenproblem(V depends on e density ||2)

—many well-known computational techniques

Hermitian = real E & , … Periodicity = Bloch’s theorem…

A 2d Model System

square lattice,period a

dielectric “atom”e=12 (e.g. Si)

Periodic Eigenproblems

if eigen-operator is periodic, then Bloch-Floquet theorem applies:

H (x , t)e

ik x t H k (

x )can choose:

periodic “envelope”planewave

Corollary 1: k is conserved, i.e. no scattering of Bloch wave

Corollary 2: given by finite unit cell,so are discrete n(k)H k

Solving the Maxwell Eigenproblem

H(x,y) ei(kx – t)

ik 1e ik Hn

c 2 Hn

ik H 0

where:

constraint:

Want to solve for n(k),& plot vs. “all” k for “all” n,

Finite cell discrete eigenvalues n

Limit range of k: irreducible Brillouin zone

2 Limit degrees of freedom: expand H in finite basis

3 Efficiently solve eigenproblem: iterative methods

QuickTime™ and aGraphics decompressorare needed to see this picture.00.10.20.30.40.50.60.70.80.91Photonic Band GapTM bands

Solving the Maxwell Eigenproblem: 1

1 Limit range of k: irreducible Brillouin zone

—Bloch’s theorem: solutions are periodic in k

first Brillouin zone= minimum |k| “primitive cell”

irreducible Brillouin zone: reduced by symmetry

Solving the Maxwell Eigenproblem: 2a

2 Limit degrees of freedom: expand H in finite basis (N)

H H(xt ) hmbm (x t )m1

solve: ˆ A H 2 H

Ah 2Bh

Am bmˆ A b Bm bm bf g f * g

finite matrix problem:

Solving the Maxwell Eigenproblem: 2b

( ik)H 0— must satisfy constraint:

Planewave (FFT) basis

H(x t ) HGeiGxt

HG G k 0constraint:

uniform “grid,” periodic boundaries,simple code, O(N log N)

Finite-element basisconstraint, boundary conditions:

Nédélec elements[ Nédélec, Numerische Math.

35, 315 (1980) ]

nonuniform mesh,more arbitrary boundaries,

complex code & mesh, O(N)[ figure: Peyrilloux et al.,

J. Lightwave Tech.21, 536 (2003) ]

Solving the Maxwell Eigenproblem: 3a

Ah 2Bh

Faster way:— start with initial guess eigenvector h0

— iteratively improve— O(Np) storage, ~ O(Np2) time for p eigenvectors

Slow way: compute A & B, ask LAPACK for eigenvalues— requires O(N2) storage, O(N3) time

(p smallest eigenvalues)

Solving the Maxwell Eigenproblem: 3b

Ah 2BhMany iterative methods:

— Arnoldi, Lanczos, Davidson, Jacobi-Davidson, …, Rayleigh-quotient minimization

Solving the Maxwell Eigenproblem: 3c

Ah 2BhMany iterative methods:

— Arnoldi, Lanczos, Davidson, Jacobi-Davidson, …, Rayleigh-quotient minimization

for Hermitian matrices, smallest eigenvalue 0 minimizes:

02 min

h' Ahh' Bh

minimize by preconditioned conjugate-gradient (or…)

“variationaltheorem”

Band Diagram of 2d Model System(radius 0.2a rods, e=12)

MG X M Girreducible Brillouin zone

gap forn > ~1.75:1

The Iteration Scheme is Important(minimizing function of 104–108+ variables!)

Steepest-descent: minimize (h + af) over a … repeat

02 min

h' Ahh'Bh

Conjugate-gradient: minimize (h + ad)— d is f + (stuff): conjugate to previous search dirs

Preconditioned steepest descent: minimize (h + ad) — d = (approximate A-1) f ~ Newton’s method

Preconditioned conjugate-gradient: minimize (h + ad)— d is (approximate A-1) [f + (stuff)]

The Iteration Scheme is Important(minimizing function of ~40,000 variables)QuickTime™ and aGraphics decompressorare needed to see this picture.EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑJJJJJJJJJJJJJJJJJJJJJJJJJJJJ0.0000010.000010.00010.0010.010.111010010001000010000010000001101001000

# iterations

preconditionedconjugate-gradient no conjugate-gradient

no preconditioning

The Boundary Conditions are Tricky

E|| is continuous

E is discontinuous

(D = eE is continuous)

Any single scalar e fails: (mean D) ≠ (any e) (mean E)

Use a tensor e:

The e-averaging is ImportantQuickTime™ and aGraphics decompressorare needed to see this picture.BBBBBBBBBBBBBJJJJJJJJJJJJJHHHHHHHHHHHHH0.010.111010010100

resolution (pixels/period)

backwards averaging

tensor averaging

no averaging

correct averagingchanges order of convergencefrom ∆x to ∆x2

(similar effectsin other E&M

numerics & analyses)

Gap, Schmap?

G X M G

But, what can we do with the gap?

Intentional “defects” are good

3D Photonic Crystal with Defects

microcavities waveguides (“wires”)

Intentional “defects” in 2dQuickTime™ and aGraphics decompressorare needed to see this picture.a

QuickTime™ and aGraphics decompressorare needed to see this picture.QuickTime™ and aGraphics decompressorare needed to see this picture.QuickTime™ and aGraphics decompressorare needed to see this picture.(Same computation, with supercell = many primitive cells)

Microcavity Blues

For cavities (point defects)frequency-domain has its drawbacks:

• Best methods compute lowest- bands, but Nd supercells have Nd modes below the cavity mode — expensive

• Best methods are for Hermitian operators, but losses requires non-Hermitian

Time-Domain Eigensolvers(finite-difference time-domain = FDTD)

Simulate Maxwell’s equations on a discrete grid,+ absorbing boundaries (leakage loss)

• Excite with broad-spectrum dipole ( ) source

Response is manysharp peaks,

one peak per modecomplex n [ Mandelshtam,

J. Chem. Phys. 107, 6756 (1997) ]

signal processing

decay rate in time gives loss

QuickTime™ and aGraphics decompressorare needed to see this picture.EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE05010015020025030035040045000.511.522.533.54Signal Processing is Tricky

complex n

?signal processingQuickTime™ and aGraphics decompressorare needed to see this picture.EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE-1-0.8-0.6-0.4-0.200.20.40.60.8012345678910

Decaying signal (t) Lorentzian peak ()

a common approach: least-squares fit of spectrum

fit to:

A( 0)2 G2

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Portion of decaying signal (t) Unresolved Lorentzian peak ()

actual

signalportion

problem: have to run long enough to completely decay

There is a better way, which gets complex to > 10 digits

Unreliability of Fitting ProcessQuickTime™ and aGraphics decompressorare needed to see this picture.EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE0200400600800100012000.50.60.70.80.911.11.21.31.41.5 = 1+0.033i

= 1.03+0.025i

sum of two peaks

Resolving two overlapping peaks isnear-impossible 6-parameter nonlinear fit

(too many local minima to converge reliably)

Sum of two Lorentzian peaks ()

There is a better way, which gets

complex for both peaksto > 10 digits

Quantum-inspired signal processing (NMR spectroscopy):

Filter-Diagonalization Method (FDM)[ Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ]

Given time series yn, write:

yn y(nt) ake i k nt

…find complex amplitudes ak & frequencies k

by a simple linear-algebra problem!

Idea: pretend y(t) is autocorrelation of a quantum system:

ˆ H i t

yn (0) (nt) (0) ˆ U n (0)

time-∆t evolution-operator:

ˆ U e i ˆ H t /

Filter-Diagonalization Method (FDM)[ Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ]

yn (0) (nt) (0) ˆ U n (0)

ˆ U e i ˆ H t /

We want to diagonalize U: eigenvalues of U are ei∆t

…expand U in basis of |(n∆t)>:

Um,n (mt) ˆ U (nt) (0) ˆ U m ˆ U ˆ U n (0) ymn1

Umn given by yn’s — just diagonalize known matrix!

Filter-Diagonalization Summary[ Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ]

Umn given by yn’s — just diagonalize known matrix!

A few omitted steps: —Generalized eigenvalue problem (basis not orthogonal) —Filter yn’s (Fourier transform):

small bandwidth = smaller matrix (less singular)

• resolves many peaks at once • # peaks not known a priori • resolve overlapping peaks • resolution >> Fourier uncertainty

Do try this at home

Bloch-mode eigensolver: http://ab-initio.mit.edu/mpb/

Filter-diagonalization: http://ab-initio.mit.edu/harminv/

Photonic-crystal tutorials (+ THIS TALK): http://ab-initio.mit.edu/

/photons/tutorial/

Photonic Crystals:Periodic Surprises in Electromagnetism

Steven G. JohnsonMIT

A “Defective” Lecture

cavity waveguide

The Story So Far…a

Waves in periodic media can have:

• propagation with no scattering (conserved k)• photonic band gaps (with proper e function)

Eigenproblem gives simple insight:

n (k )

ˆ k Hermitian –> complete, orthogonal, variational theorem, etc.

k x t ) H k (

x )Bloch form:

band diagram

Properties of Bulk Crystalsby Bloch’s theorem

QuickTime™ and aGraphics decompressorare needed to see this picture. (cartoon)

conserved wavevector k

photonic band gap

band diagram (dispersion relation)

d/dk 0: slow light(e.g. DFB lasers)

backwards slope:negative refraction

strong curvature:super-prisms, …

(+ negative refraction)

synthetic mediumfor propagation

Applications of Bulk Crystals

Zero group-velocity d/dk: distributed feedback (DFB) lasers

negative group-velocity ornegative curvature (“eff. mass”):Negative refraction,Super-lensing

source imageobject image

negat ive refraction medium

super-lens Veselago (1968)

divergent dispersion (i.e. curvature): Superprisms

[Kosaka, PRB 58, R10096 (1998).]

using near-band-edge effects

[ C. Luo et al.,Appl. Phys. Lett. 81, 2352 (2002) ]

Photonic Crystals:Periodic Surprises in Electromagnetism

Steven G. JohnsonMIT

Fabrication of Three-Dimensional Crystals

Those Clever Experimentalists

Image: http://cst-www.nrl.navy.mil/lattice/struk/a4.html

The Mother of (almost) All Bandgaps

The diamond lattice:

fcc (face-centered-cubic)with two “atoms” per unit cell

(primitive)a

fcc = most-spherical Brillouin zone

+ diamond “bonds” = lowest (two) bands can concentrate in lines

Recipe for a complete gap:

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GXXULWKfreq

The First 3d Bandgap StructureK. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).

11% gap

overlapping Si spheres

MPB tutorial, http://ab-initio.mit.edu/mpb

for gap at l = 1.55µm,sphere diameter ~ 330nm

Lembram?

Make that? Are you crazy? …maybe!

[ F. Garcia-Santamaria et al., APL 79, 2309 (2001) ]http://www.icmm.csic.es/cefe/Fab/Robot/robot_strategy.htm

fabrication schematic

carefully stack bccsilica & latex spheresvia micromanipulation

…dissolve latex

& sinter (heat and fuse) silica

make Si inverse(12% gap)

Make that? Are you crazy? …maybe!

dissolvelatex spheres

6-layer [001] silica diamond lattice

4-layer [111] silica diamond lattice

[ F. Garcia-Santamaria et al., Adv. Mater. 14 (16), 1144 (2002). ]

A Layered StructureWe’ve Seen Already

rod layer

hole layer

(diamond-like: rods ~ “bonds”)

[ S. G. Johnson et al.,Appl. Phys. Lett. 77, 3490 (2000) ]

Up to ~ 27% gapfor Si/air

hole layer

Making Rods & Holes Simultaneously

substrate

top view

side view

substrate

A A A A

expose/etchholes

substrate

A A A A

backfill withsilica (SiO2)& polish

substrate

A A A A

deposit anotherSi layer

layer 1

B B B B

substrate

layer 1

A A A A

B B B B

A A A A

dig more holesoffset& overlapping

B B B B

substrate

layer 1

A A A A

B B B B

A A A A

backfill

B B B B

C C C C

substrate

layer 1

layer 2

layer 3

A A A A

B B B B

C C C C

A A A A

etcetera

(dissolvesilicawhendone)

oneperiod

B B B B

C C C C

substrate

layer 1

layer 2

layer 3

A A A A

B B B B

C C C C

A A A A

etcetera oneperiod

hole layers

B B B B

C C C C

substrate

layer 1

layer 2

layer 3

A A A A

B B B B

C C C C

A A A A

etcetera oneperiod

rod layers

Tb existe forma alternativa

A More Realistic Schematic

[ M. Qi, H. Smith, MIT ]

e-beam Fabrication: Top View

e-beam Fabrication: Side Views(cleaving worst sample)

substrate

layer 1A A A A

A A A A

B B B B

C C C C

2layer 3

layer 5

layer 7

Adding “Defect” Microcavities450nm

Easiest defect: don’t etch some B holes— non-periodically distributed: suppresses sub-band structure— low Q = easier to detect from planewave

Outros resultados

Yes, it works: Gap at ~4µm[ K. Aoki et al., Nature Materials 2 (2), 117 (2003) ]

50nm accuracy:

(gap effects are limited by finite lateral size)

20 layers

Lithography is a Beast[ S. Kawata et al., Nature 412, 697 (2001) ]

l = 780nm

resolution = 150nm

(3 hours to make)

For a physicist, this is cooler…[ S. Kawata et al., Nature 412, 697 (2001) ]

(300nm diameter coils, suspended in ethanol, viscosity-damped)

A Two-Photon Woodpile Crystal[ B. H. Cumpston et al., Nature 398, 51 (1999) ]

(much work on materialswith lower power 2-photon process)

Difficult topologies

• Arbitrary lattice• No “mask”• Fast/cheap prototyping

[ fig. courtesy J. W. Perry, U. Arizona ]

Atualmente existem vários métodos e processos para a construção de cristais

fotônicos, fora aqueles citados anteriormente.

É um novo campo com grandes expectativas.

Como Químicos e Físicos se entendem sobre estado sólido

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