January 2010

Air ForceOffice of ScientificResearch

MultidisciplinaryUniversityResearchInitiative

High‐Frequency Magnetism in Metamaterials and the 

Landau‐Lifshitz Permeability Argument

Roberto MerlinUniversity of Michigan

1

1

0

0

c t

c t

∂∇× = −

∂∂

∇× =∂

∇ ⋅ =

∇ ⋅ =

BE

DH

D

B

= ε= μ

D EB H

MAXWELL’S EQUATIONS

WHAT IS µ?

εS

COMPOSITE

OUTLINE• HIGH-FREQUENCY MAGNETISM

• LANDAU-LIFSHITZ PERMEABILITY ARGUMENT

• HOMOGENIZATION OF METAMATERIALS: ELECTRODYNAMICS OF CONTINUOUS MEDIA

• ATOMS vs. SPLIT RINGS AND SPHERES

• PLASMON RESONANCES

S MM/ 1<< λ ε μ ≈if then

Sε|← →|

S SIm / 1ε = κ >> λ >> S SRe ~ / 1nε = λ >>

PERFECT DIAMAGNETµS ª 0

PERFECT PARAMAGNETµS ª •

NEGATIVE REFRACTION

PERFECTABSORBER

µMM ≡ εMM

µMM < 0εMM < 0

NEGATIVE REFRACTION

© D. Schurig

ε<0μ<0

V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968)

© D. Schurig

n = 1.5 n = 0.7 n = -1

q

ω

BEYOND THE DIFFRACTION LIMIT: NEGATIVE REFRACTION

ε ≡-1μ ≡-1

SINGULARITY

Wu et al. APL 90, 063107 (2007)

λ= 5 µm

Shelby, Smith and Schultz, Science 292, 77 (2001)

λ= 3 cm

Yen et al.Science 303, 1494 (2004)

λ= 0.025 cm

Shalaev et al.Optics Lett. 30, 3356 (2005)

λ= 1.5µm

Zhang et al.PRL 95, 137404 (2005)

λ= 2µm

Dolling et al.Optics Lett. 32, 53 (2007)

λ= 0.78µm

109              1010 1011           1012              1013  1014            1015FREQUENCY

(Hz)

n<0

λ= 0.5 µm

Grigorenko et al. Nature 438, 335 (2005)

PROBLEMS2D vs. 3D

(impedance sheet vs. bulk)

graphene vs. graphite

vs.kaL<<1?

C. L. Holloway, Metamaterials 3, 100 (2009)MM(at high frequencies)

22M M L

E E

( / )~ ~ ~ 1/

c c H L at E L L

× χ χλ ⎛ ⎞⎛ ⎞ <<⎜ ⎟ ⎜ ⎟∂ ∂ ωχ χ⎝ ⎠ ⎝ ⎠

MP∇

LANDAU-LIFSHITZ PERMEABILITY ARGUMENT

/c t= × + ∂ ∂j M P∇

d2

i Vcω⎛ ⎞≡ − ×⎜ ⎟

⎝ ⎠∫ M r Ptotal magnetic moment

uniqueness and significance of M

cM ~ v2/c2 ~ aL2/λ2 nS ~ λ/ or κS >> λ/

La L<< << λ

when does M represent the magnetic-dipole density?

change in totaltotal magnetic = electricmoment moment

‐iωΔr/2cΔr

METAMATERIALS and MOLECULAR SOLIDS

C/V≡M m

L( / )a>> λm pambiguity is removed if

/ 2i c→ − ωΔ ×m m r pcoordinate change

origin varies from cell to cell

C C| |~ | |~V Vp mE H

high-εS substances

( )2C L C| |~ | |~ /V a Vλp mE H

non-resonant induced moments (molecules)

( ) ( )

( ) ( )

ˆˆ ˆ ˆ0 0 0 0

ˆ ˆ ˆ ˆ0 0 0 0

i j i jkij ijk

s ss s

i j i jij ij

s ss s

p s s p p s sA

m s s m p s s mG

Θα = =

ω −ω ω −ω

γ = =ω −ω ω −ω

∑ ∑

∑ ∑

0

1

3(0) / (0) ...

(0) (0) ...

(0) ...

i ij j ijk j k ij jjk

i ij j ij jj

ij ijk kk

p A x G

m G

q A

=

∗

∗

= α + ∂ ∂ + +

= γ + +

= +

∑

∑

∑

rE E B

B E

E

First Homogenization Step: Induced Multipoles and Single Particle Scattering

E

B

2mol mol mol

mol mol mol mol mol 2mol 0 0 0

~ ~ ~/

GV V V

A⎧ ⎫

α >> >> γ⎨ ⎬λ λ λ⎩ ⎭

Small Spheres (λ0>>RS) vs. Molecules

( )S S

3 3SS S 2

S

1 1 2(sin cos )2 2 ( 1)sin cos

kRF FR R FF F

ε⎛ ⎞ε − − θ − θ θ⎛ ⎞= = = θ =⎜ ⎟ ⎜ ⎟ε + + θ − θ+ θ θ⎝ ⎠⎝ ⎠

p mE H

( ) ( )

2

molED MDmol mol mol mol

ED 0 MD

~ ~V V⎛ ⎞ω ω

α γ ⎜ ⎟ω −ω λ ω −ω⎝ ⎠

2

S SSPH SPH SP SPH S

S 0

3( 1) /104 ( 2)

RV V⎛ ⎞ε −

α = γ ≈ πε⎜ ⎟π ε + λ⎝ ⎠

S S 1kR ε <<

SPH SPH SP SPH3 3

4 8V Vα = γ ≈ −

π π

S S 1kR κ >>

SCATTERING BY A SPLIT-RING2 4

1 0S 2

spr

(1 ) i rc Z

− π ωγ = −ε

21

Sspr

(1 )Zig−

∗α = − −εω

( )1/2

20 0 Sspr 1/2 2 1/2

S 1 S

4 ( ) / 1/( )

r J kaZ i i L c Cka c J ka

−

− −

⎛ ⎞π ω ε= − ω − ω⎜ ⎟ε ε⎝ ⎠

S 0 / 1aκ >> λ >> S 0~ / 1n aλ >> 1/ 20 S/a << λ ε

LC RESONANCE CAVITY-LIKE RESONANCES

nSka ª 5p/4, 9p/4, 13p/4 

PLASMON RESONANCE

. 0d =∫ E lw2=c2/LC

RM, PNAS 106, 1693 (2009)

r0 >> a >>g

LC & SKIN-DEPTH RESONANCES

104 109 1014

0.0

0.5

1.0

Perm

eabi

lity

Frequency (Hz)

μ1

μ2

108 1011 1014

10-8

10-7

10-6

Frequency (Hz)

Ski

n D

epth

(m)

0 1 2 3-1

0

1

2

μ

ω/ω0

0 1 2 3

-3

-2

-1

0

1

2

3

γ/V C

ω/ω0

LC-resonanceLARGE |εs|

3spr 0 5 6

-3 2mol mol mol

~ 10 10r c

e

−γ λ× ≈ −

γ

PLASMON RESONANCES: OPTICAL FREQUENCIES (|ε| < 100)

0 1 2 30.90

0.95

1.00

1.05

1.10

μ

ω/ω0

( )20 0/ /Γ ω ≥ λ

0 1eVω >

LORENTZ-LORENZ, CLAUSIUS-MOSOTTI & LEWIN FORMULAS

CC

CC

/

/

VV

VV

≡ = α

≡ = γ

pP

mM

E

B

4 / 34 / 3

+ π+ π

E PH M

E=

B=

C1 2 H

C

C1 2 H

C

3 83 4

3 83 4

ViV

ViV

⎛ ⎞+ παε = ε + ε = ε ⎜ ⎟− πα⎝ ⎠

⎛ ⎞+ πγμ = μ + μ = μ ⎜ ⎟− πγ⎝ ⎠

MAXWELL-GARNETT BRUGGERMAN

(MIE SCATTERING)

µ

Frequency (GHz)

CONCLUDING REMARKSHIGH-FREQUENCY MAGNETISM

S SIm / 1ε = κ >> λ >> S SRe ~ / 1nε = λ >>

P

P/cλ >> >> λδ ≈ ω

METALSλ  > 2.5 µm

FERROELECTRICSLOW-FREQUENCY TO PHONONS

FAR-INFRARED