Effective medium approach to nanostructures · 2018-09-10 · Feynman, Leighton, Sands: The Feynman Lectures on Physics, Vol. 2 (Addison-Wesley, Reading, 1964) §32-5 “The index

Effective medium approach to nanostructures

strengths and weaknesses

Josef Humlíček, CEITEC, Masaryk University

Brno, Czech Republic

EPIOPTICS-15, Erice, July 2018

1

Outline:

• Linear optical response: bulk and nanostructured materials

• Average fields and effective permittivity for small contrast

3D, 2D and 1D systems

• Established mixing rules

• Tests of EMA:

macroscopic scale (glass spheres in liquids)

molecular scale (water solutions of sucrose)

• Differences between mixing rules for binary dielectric mixtures

• EMA and exact solutions for layered structures

• Resonant behavior of EMA mixtures.

2

Linear optical response of bulk

electrons, phonons, ... optical response functions, complex e, s, N in the long-wavelength limit

doped GaAs

0.00 0.02 0.04 0.06 0.08 0.10

-200

-100

0

100

200

2.7x1018

cm-3

Im

Re

n-GaAs

DIE

LE

CT

RIC

FU

NC

TIO

N

PHOTON ENERGY (eV)

5 10 15 20

-10

0

10

20

Im

Re

GaAs

DIE

LE

CT

RIC

FU

NC

TIO

N

PHOTON ENERGY (eV)

l >> a

(nano)structured media

layers, thicknesses << wavelength

e.g., multiple quantum wells

j-th layer: dj, e

j

substrate

Ets

Etp

Ers

Erp

Eis

Eip

s

(meta)

material

ambient

z

x

effe D Ea continuum with

effective permittivity

matrix

substrate

Ets

Etp

Ers

Erp

Eis

Eip

s

(meta)

material

ambient

z

xinclusioninclusions, dimensions << wavelength

e.g., ensembles of quantum dots or wires


Example: multilayers in AlGaN/GaN/AlGaN/Aln/Si HEMTS

metamaterials in buffers between AlGaN/GaN channel

and AlN/Si substrate, studied in IR

j-th layer: dj, e

j

substrate

Ets

Etp

Ers

Erp

Eis

Eip

s

(meta)

material

ambient

z

x


building up the screening charges,

planar interfaces

()

EbE

a

Db

Da

-

-

-

-

-

-

-

-

ebe

a

Db

Ea

eb

Eext

ea D

a

Eb

+

+

+

+

+

+

+

+

(||)

curved interfaces, sphere (ellipsoids)

sparsely distributed in a matrix

Maxwell-Garnett – the nanotechnology of 1904 J.C.M. Garnett, Philos. Trans. R. Soc. Lond. 203, 385 (1904)

(1 ) .eff a bf fe e e e = 1/ (1 ) / / .eff a bf fe e e =

3( )

2 ( )

aeff a b a

b a b a

Df

E f

ee e e e

e e e e

= = .

A general results for small contrast LLL – Landau, Livshitz, Looyenga

The average field quantities,

calculated for a general mixture with a small difference of the permittivities of the components (Landau-Livshitz, Electrodynamics of continuous media):

effe D E ,

( ) ( ) ( ) ( ). e e e E r E + E r , r + r

Neglecting higher-order terms in Taylor expansions leads to

21/3 3 1/3 3

2

1 2 ( )( ) 3 ,

3 6

ee e e e e

e

i.e., the “LL” formula

1/3 3 ,effe e

which was derived in a different way by Looyenga (1965); the “LL(L)” formula.

2( ).

3eff

ee e

e

This can be further approximated by


The (3D) averaging of LL can be fairly easily repeated in reduced dimensions:

where D =3, 2, 1 for the 3D, 2D, and 1D mixtures, respectively.

Interestingly, for D=1 (lamellar structure with fields perpendicular to interfaces):

This is exact for the long-wavelength averaging for general lamellar structures,

not restricted to the low contrast of constituents.

2( ),eff

D

ee e

e

1 1

1D .effe e

LLL (called usually Looyenga) is very popular biochemistry/biology environment:

• very simple and easy to handle,

• the optical contrast in the (complex) mixtures is typically small.


For binary mixtures with the volume fraction f of the components a and b:

The average permittivity:

the deviations from the mean:

/( ), /( ) 1 .a a a b b b a b af V V V f V V V f f

(1 ) ( ),a b a b af f fe e e e e e

( ), (1 )( ),a a a b b b b af fe e e e e e e e e e

the mean of the squared deviation:

2 2 2 2( ) (1 )( ) ( ) (1 )( ) ,a b b af f f fe e e e e

and the LL formula:

2(1 )( ) ( ) .

[ ( )]

b aeff a b a

a b a

f ff

D f

e ee e e e

e e e

Examples of established mixing rules

Some of them implicit:

(e.g., solution of quadratic equation for Bruggeman and CPA)

Example: glass spheres in CCl4

Measurements of Reynolds (Thesis, London 1955)

of glass spheres dispersed in carbon tetrachloride, quoted by Looyenga

long-wavelength limit - low frequencies (≤1 GHz, lvac ≥ 300 cm)

diameter of the glass beads ≤ 1 mm


The slightly bowed composition dependence follows the predictions of various variants of EMA

0.0 0.2 0.4 0.6 0.8 1.0

2.5

3.0

3.5

4.0

4.5

expt.

glass (eb = 4.594)

in CCl4 (e

a = 2.228)

LLCPA

MG

Brugg

Compar1

e

f


0.0 0.1 0.2 0.3 0.4

-0.04

-0.02

0.00

expt.

glass (eb = 4.594)

in CCl4 (e

a = 2.228)

LL, 0.008

CPA,0.006

MG, 0.023

Brugg, 0.011

Compar3

e eff

ee

xp

t

f

A closer look at the deviations from measured values:


0.0 0.1 0.2 0.3 0.4

0.00

0.01

0.02expt.

glass (eb = 4.594)

in CCl4 (e

a = 2.228)

LL

CPA

MG

Brugg

Compar4

f -

f exp

t

fexpt

The typical use of EMA is an estimation of the volume fractions

how do different mixing formulae perform in the case of the known concentrations?

Example: water solution of sucrose

Feynman, Leighton, Sands: The Feynman Lectures on Physics, Vol. 2 (Addison-Wesley, Reading, 1964)

§32-5 “The index of a mixture”

invokes Clausius-Mossotti relation for the effect of local field on induced dipoles


Feynman, Leighton, Sands: The Feynman Lectures on Physics, Vol. 2 (Addison-Wesley, Reading, 1964)

§32-5 “The index of a mixture”

Space filling model of the sucrose (C12H22O11) molecule

Example: water solution of sucrose, no optics first

Extensive data available (importance in food industry) International Critical Tables of Numerical Data, Physics, Chemistry and Technology, First Electronic Edition (Knovel, Norwich, New York 2003). Vol. 2, pp. 334-355.

Measurements of the density (mass per unit volume), masses/molecule and concentrations:

The total volume V is shared by the apparent volumes, V = Vw + Vs;

assuming the apparent volume of the water molecule to be that of the pure water having the density rw (the better the more dilute the solution is):

,w w s sw w s s

N m N mc m c m

Vr

=

.w w s s w ws

w

N m N m N mV

r r

=

Using the mass fraction of sugar (independent of temperature and pressure):

,s sM

w w s s

N mf

N m N m=

we get the apparent volume of one succrose molecule, 1

11.s sM

s

s w M

V mfV

N fr r

=

Example: water solution of sucrose, no optics first

Measured apparent volume of one succrose molecule

International Critical Tables of Numerical Data, Physics, Chemistry and Technology, First Electronic Edition (Knovel, Norwich, New York 2003). Vol. 2, pp. 334-355.

Two other sources (K and B)

0.0 0.2 0.4 0.6 0.8 1.0

0.345

0.350

0.355

0.360

0.365

0.35041+0.00332fM+0.0123f

2

M

B (90oC, 0.35 MPa)

B (25oC, 0.35 MPa)

B (5oC, 0.35 MPa)

20oC

SUCROSE IN WATER

K (25oC)

V1

s (

nm

3)

fM

SucrWrho_AppVolDens

crystal


Polarization of the solution in optical wave, EMA treatment:

the increase of apparent volume Vs with increasing mass fraction, due to the

formation of voids between adjacent sucrose molecules, small enough to prevent filling with water (?) →

three components of the mixture, water, succrose and voids.

trying LLL formula for a small optical contrast;

with volume fractions of water, sugar, and voids

the cube root of effective permittivity of the mixture is

1/3 1/3 1/3 1/3 .eff w w s s vf f fe e e e

, , and , 1,w s v w s vf f f f f f


0.0 0.2 0.4 0.6 0.8 1.0

0.00

0.01

0.02

0.03

0.04

0.0 0.2 0.4 0.6 0.8 1.0-0.1

0.0

0.1

-0.1

0.0

0.1

lD (589.3 nm)

f(r)

v

2.466

2.46

es = 2.47

sucrose in water

t = 20 C

f v

fM

Voids_fvICUMSA-InvRhoF

2.46

es = 2.47

0.08595fM+0.03328f

2

M+0.00715f

3

M

e1

/3-e

1/3

LL

L ,

x10

e1

/3-e

1/3

w

fM

(msd = 1.3x10-5

)

Very precise refractive index data of the solution available for the sodium line 589.3 nm,

at 20C, www.icumsa.org, ICUMSA Method SPS-3 (2000),

up to mass fraction of 85%; note the small level of noise (msd of 1.3E-5 from the cubic polynomial)


Small optical contrast,

permittivity of 1.77686 for pure water,

to 2.26196 at fM=0.85,

→ the sucrose component in EMA below about 2.5,

suggests effective e from LL or LLL formula;

written in terms of the volume fraction of voids:

1/3 1/3 1/3 1/3

1/3 1/3.

1 1

s s wv w

s s

f fe e e e

e e

Note:

es is the permittivity of the hypothetical 100% sugar solution (not a crystal).

The volume fraction of water is

(1 ).w M

w

f fr

r


The value of es :

the zero slope of the fv(fM) dependence for fM→0, which occurs for es =2.466

0.0 0.2 0.4 0.6 0.8 1.0

0.00

0.01

0.02

0.03

0.04

0.0 0.2 0.4 0.6 0.8 1.0-0.1

0.0

0.1

-0.1

0.0

0.1

lD (589.3 nm)

f(r)

v

2.466

2.46

es = 2.47

sucrose in water

t = 20 C

f v

fM


2.46

es = 2.47

0.08595fM+0.03328f

2

M+0.00715f

3

M

e1

/3-e

1/3

LL

L ,

x10

e1

/3-e

1/3

w

fM

(msd = 1.3x10-5

)


Test of the LLL approximation:

comparison of the volume fraction of voids with that from the density

0.0 0.2 0.4 0.6 0.8 1.0

0.00

0.01

0.02

0.03

0.04

0.0 0.2 0.4 0.6 0.8 1.0-0.1

0.0

0.1

-0.1

0.0

0.1

lD (589.3 nm)

f(r)

v

2.466

2.46

es = 2.47

sucrose in water

t = 20 C

f v

fM


2.46

es = 2.47

0.08595fM+0.03328f

2

M+0.00715f

3

M

e1

/3-e

1/3

LL

L ,

x10

e1

/3-e

1/3

w

fM

(msd = 1.3x10-5

)


Fairly good agreement of optical data with EMA for the very small volume of the sugar molecule (0.35 nm3 at RT) !

Confirms the basic lines of Feynman’s approach, except for the use of polarizability of the succrose molecules, obtained from the average of refractive indices of (anisotropic) crystal.

Selected concerns:

• uncertainty in choosing the “EMA rule”,

• oversimplified introduction of the voids

(also in the apparent free volume from the density),

• ...

Differences between mixing rules for binary dielectric mixtures

Normalized differences for real permittivities of a and b

planar structures (parallel and perpendicular) form Wiener bounds

0.0 0.5 1.0

0.0

0.5

1.0

Brugg

LLL

CPA

eeff

f)ea+ fe

b

eeff

f)ea+ fe

b

eb e

a = 1.5

(ee

ff-e

a)/

(eb-e

a)

f

0.0 0.5 1.0

0.0

0.5

1.0

f

CPA

Brugg

LLL

eeff

f)ea+ fe

b

eeff

f)ea+ fe

b

eb e

a = 10

small contrast

(2.466/1.777 ≈ 1.39 for sugar/water)

large contrast

(12/1 for silicon/voids in IR)

Differences between mixing rules for binary dielectric mixtures

comparison of LLL, Bruggeman and CPA for different contrasts

0.0 0.5 1.0

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

10

1

0.1

e ea = 0.01

eb = e

a+e

|(e L

LL-e

CP

A)/

e|

f

0.0 0.5 1.0

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

10

1

0.1

e ea = 0.01

eb = e

a+e

|(e B

rug

g-e

CP

A)/

e|

f

EMA and exact solutions for layered structures

electric field inside the superlattice of absorbing/dielectric materials

0.0 0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

d / lvac

e

0.01 16+2i

0.01 9

Re

Im

E / E

(0)

z / lvac

planar interfaces between homogeneous slabs:

easy solutions of the wave equation (transfer matrices)

j-th layer: dj, e

j

substrate

Ets

Etp

Ers

Erp

Eis

Eip

s

(meta)

material

ambient

z

x

EMA and exact solutions for layered structures

oblique incidence, ellipsometric angles:

a (rather fine) superlattice of d1=d2=lvac/1000

EMA for the field parallel to the interfaces

the total thickness of the film on the e = 6+2i substrate

→ easily observable deviations

0.6 0.8 1.0 1.2 1.4

7

8

9

= 70o

SL

EMA,e||

(

de

g)

d / lvac

0.6 0.8 1.0 1.2 1.4

-15

-10

-5

0

EMA,e||

SL

air - film -substrate

(

de

g)

Resonant behavior of EMA mixtures

Spectacular behavior of the effective optical response possible,

can be traced down to the spectacular behavior of local fields.

Simple case with analytical solution:

the constant field inside an isolated ellipsoid in an infinite host medium,

easily transferrable to diluted mixtures.

The field intensity inside the ellipsoid diverges whenever

11 ( ,0 .=

e

e b

a uL

Lu is the depolarization factor (semiaxes u,v, w of the ellipsoid)

2 2 2 2 20

0,12 ( ) ( )( )( )

u

uvw dtL

t u t u t v t w

=

with the field along u.


In particular, the divergence occurs in

sphere:

u=v=w, Lu=1/3 for εb = −2εa;

cylinder:

field perpendicular to its axis

u=v, w→∞, Lu =1/2 for εb = −εa;

slab:

field perpendicular to the interface

v=w→∞, Lu=1 for εb = 0.


The IR resonance in undoped/doped SL:

500 1000 1500

-50

0

50

e||

e

e

e||

eb

ea

e 1

WAVENUMBER (cm-1)

500 1000 1500

0

50

100

50x(80nm i-AlInAs, 80nm n-InGaAs)

ea

eb

e 2

Conclusion

• EMA can be attractive and helpful.

• EMA can easily fail – if improperly used.

• Using it with caution is recommended.

A selection from the (vast) literature:

L.D. Landau and E.M. Lifshitz, Electrodynamics of continuous media, Second edition (Pergamon Press,

Oxford, 1984), Sec. 9.

R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. II (Addison-Wesley,

1964), § 32-5.

A. Sihvola, Electromagnetic mixing formulas and applications (IEE, Stevenage, 1999).

D.E. Aspnes, Am. J. Phys. 50, 704 (1982).

Ellipsometry at the Nanoscale , M. Losurdo and K. Hingerl, eds., (Springer 2013).

THANK YOU FOR YOUR ATTENTION

32

Documents

Effective medium approach to nanostructures · 2018-09-10 · Feynman, Leighton, Sands: The Feynman Lectures on Physics, Vol. 2 (Addison-Wesley, Reading, 1964) §32-5 “The index