1

Material Electromagnetic Property

Material partition under electric fieldMaterial partition under magnetic field

Lorentzian modelArtificial material

2

Material Partition under Electric Field

• Dielectrics

• Conductors

3

Different Dielectrics

• (Microscopic) non-polarized dielectrics – well described by the Lorentzian model

• (Microscopic) polarized dielectrics – still not polarized macroscopically due to the random orientation of polarization in microscope, usually described by Clausius-Mossotti’s equation, i.e., Curie’s law modified by the dielectric empty cavity

20

03c

Np

kT

2 20 0

0 0

1/ (1 )

1 / 3 3 3 3c

c

Np Np

kT kT

• Ferroelectric dielectrics – polarized in macroscope (e.g., BaTiO3)

4

Different Dielectrics

rD E � x xx xy xz x

y yx yy yz y

z zx zy zz z

D E

D E

D E

The permittivity tensor can be diagonalized under propercoordinates transform:

Anisotropic dielectrics:

' ' ' '

' ' ' '

' ' ' '

0 0

0 0

0 0

x x x x

y y y y

z z z z

D E

D E

D E

5

Why anisotropic?

3 52 2 2 2 2 2 2

1 1 1 1 2 1( )

( ) 1 2 /

axO

r r r rx y z x a y z r ax r

52 2 2 2 2 2 2 2 2

1 1 1 3 1( )

( ) ( )O

r rx y z x a y z x a y z

Highly symmetric dipole distribution in dielectrics generates highly symmetric Coulomb’s potential, which makes the dielectrics more isotropic, and vice versa

Equal permittivity along all three axes – isotropic crystal

Two of the three permittivities are equal, the 3rd is different – uniaxial crystal (normally refractive index < abnormally refractive index, positive uniaxial, otherwise, negative uniaxial)

All three permittivities are different – biaxial crystal

6

Crystal Classification

• Isotropic – with cubic lattice, e.g., diamond

• Positive (negative) uniaxial – with trigonal, tetragonal, or hexagonal lattice, e.g., quartz, zircon, rutile, ice (beryl, calcite, tourmaline, sodium nitride)

• Biaxial – with triclinic, monoclinic, or orthorhombic lattice, e.g., feldspar, mica, topaz, gypsum

7

Crystal Classification

Zircon Quartz Rutile

Beryl Calcite Tourmaline

Feldspar

Mica

Topaz

8

Material Partition under Magnetic Field• If the electron’s orbit has the full symmetry (s-orbit), the microscopic

current ring has no specific orientation, the unit inherent magnetic moment (torque) is zero; under the external magnetic field, the inherent symmetry of the unit is broken and an induced magnetic moment appears, with its orientation against the external field; materials made of such units form the diamagnetics

• If the electron’s orbit is asymmetric, the microscopic current has a specific orientation, the unit has an inherent magnetic moment (the macroscopic magnetic moment is still zero due to the random orientation from unit to unit); under the external magnetic field, the unit magnetic moment will be aligned in the direction of the external field; materials made of such units are the paramagnetics

• Some materials have aligned magnetic moment units in a small domain, with random alignment only from domain to domain, under even weak external magnetic field, all magnetic moment in different domains can get aligned with (if the unit magnetic moment in a single domain are all aligned), or against (if the unit magnetic moments are contra-aligned in pairs in a single domain) the external field; materials made of such domains are the ferromagnetics (antiferromagnetics)

9

Diamagnetics

Origin of diamagnetics: the unit (atom) has no inherent magnetic moment,once is placed in an external magnetic field, extra current is induced in theunit; the induced current must take the opposite direction against the externalfield, hence the unit induced magnetic moment is in the opposite direction of the external field

222 ( )

2 2

d r dB dJ er dBE r B r E eEr

dt dt dt dt

e

rB

2 2 2 2

( )2 4 6

e e r N e rM J N B N B

m m m

2 2

0 0

1 1

1 16

r M e rN

B m

Finally:

10

Paramagnetics

Origin of paramagnetics: the unit (atom) has a built-in magnetic moment, once is place in an external magnetic field, the unit takes the processionalmotion following the external field

B

a

2

3aM N BkT

Curie’s law from classical

statistical physics

2 22 2 2

2

( 1) ( 1)

3 12B

j j j j eM Ng B Ng B

kT kTm

The correct form (g - Lande’s factor, j - spin quantaof the unit (atom):

Finally: 2 22

0 0 2

1 1

( 1)1 112

r M j j eNg

B kTm

11

Lorentzian Model

• The material is viewed as a group of spring bonded flexible electrons on fixed ion centers

2

2

0)(dt

xdm

dt

dxkxteE The motion of a single electron:

0002

2 )(

m

teEx

m

k

dt

dx

mdt

xd

000

2 )(~

~~~m

Eex

m

kxj

mx

)(

)(~

)(

)(~

~22

00

00

20

jm

Ee

m

kj

mm

Eex

0

20 m

k

0m

j

E

jVm

ENexe

V

NP p

220

20

2200

20

0

)(~

)(

)(~

~)(~The (dipole) polarization:

)(~

)1()(~

)(~

)(~

220

2

00

E

jPED p

The displacement:

Vm

Nep

0

22

12

Lorentzian Model

• Permittivity for insulators and semiconductors

])()(

)(1[

)1(

222220

2

222220

220

2

0

220

2

0

pp

p

j

j

0

1

Normal Normal Abnormal

For frequency far away from 0the real part decays more slowlythan the imaginary part – that’swhy we often take a real dielectricconstant with the lossy part ignored.

13

Lorentzian Model

• Permittivity for metals – the Drude model

])(

1[

))(

1(

22

2

22

2

0

2

0

pp

p

j

j

00

1

Normal

Abnormal

00 If the loss is negligible, 0

we find 0~ 02

2

p

The refractive index becomes imaginary. Therefore, inside metals,there is no EM wave can possibly betraveling – only exponentially decayed(i.e., evanescent) wave is allowed.

14

General Electromagnetic Property

0

Dielectrics:Normal EM wave

propagation

Conductors or plasms:No EM wave direct

propagation, EM wave and electron resonance can happen,

which support the resonance propagation

Ferromagnetics or antiferromagnetics:

No EM wave direct propagation, in weak diamagnetics such as

gyromagnetics, EM can propagate with attenuation, no

reciprocity (chirality)

Undiscovered in nature, can be artificially

synthesized – Meta materials:

EM wave can propagate, with energy and phase moving along opposite

direction

15

Meta-material for Microwave

16

Home Work 3

1. Consider the radiation attenuation due to the electron acceleration in Lorentzian model, find the necessary modification to the general formula for material permittivity.

2. *A glass of water contains uniformly distributed small particles made of the same material in scattered size ranging in the neighborhood of microns. In the visible light range, the relative permittivity of the water is around 1.77 from 2000nm to 100nm, whereas the relative permittivity of the particle material is 1.689 at 2000nm and 1.789 at 100nm. What will happen if we shine the glass of water with a beam of the natural light? Think of an application by employing this effect.

3. *Calculate the relative permittivity and permeability of a 3D microwave meta-material built with normal PCB boards (with the structure shown in the previous slide) for Q-band (with wavelength range from 6mm to 9mm).

Select one of problem 2 or problem 3 to work with, work out both problems for 5 bonus points.