Proof for Pav for Orientational (Dipolar) Polarization

8/17/2019 Proof for Pav for Orientational (Dipolar) Polarization

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Topics for discussion

• Temperature dependence of orientational

(dipolar) polarizability, αd(T)=po2

/3kT• How does frequency affect the Ionic

Polarization, αi(ω) ? (Lorentz Model)

• Frequency dependence of Electronic

Polarization, αe(ω) ?

• Comparison with ‘f’ dependence of αd(ω), i.e.Debye’s relation

• Dielectric Mixtures & Heterogeneous Media

From Principles of Electronic Materials and Devices, Third Edition , S.O. Kasap (© McGraw-Hill, 2005)


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The frequency dependence of the real and imaginary parts of the dielectric constant in the

presence of interfacial, orientational, ionic, and, electronic polarization mechanisms.



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Piezoelectric transducers

|SOURCE: Photo by SOK


4/21From Principles of Electronic Materials and Devices, Third Edition , S.O. Kasap (© McGraw-Hill, 2005)

A pyroelectric detector based

on LiTaO3

|Courtesy of Molectron Detector Inc.

A 70 MHz pyroelectric detector

|Courtesy of Molectron Detector Inc.


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Fig 7.54


Dipolar Polarization: The Effect of Temperature

In the presence of an applied field a dipole tries to rotate to align with the field

against thermal agitation.

Q: Mean induced dipole moment while accounting for the

thermal energies and the random collisions among the dipoles?


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Torque

θ θ τ sin)sin( o Epa F ==

)/exp( kT E −

E p E pd E p E ooo +−== ∫ θ θ θ θ

cossin0

Prob. that the molecule has an energy E :

Torque experienced by dipole (po=aQ),

Pot. Energy E at an angle θ is ∫ θ τ d

Fraction ‘f’ of molecules oriented at θ α exp(-E/kT) ~exp(po Ecosθ /kT)

Fig 7.54


Considering the 3 D nature of dipole orientation, one

must use solid angles defined by dΩ.

The whole sphere around the dipole corresponds to a

solid sphere of 4π

Furthermore, we need to find the average dipolemoment along E (as it is net induced due to E), pocosθ


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Fig 7.55


The dipole is pointing within a solid angle dΩ

∫

∫Ω

Ω=

π

π

θ

4

0

4

0)cos(

fd

fd p p

o

av

From def n of average,


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Average dipole moment

Dipole moment along E

∫

∫Ω−

Ω−

=π

π

θ

θ θ

4

0

4

0

)/cosexp(

)/cosexp()cos(

d kT E p

d kT E p p

po

oo

av

Boltzmann factor

Integration gives a Langevin function L( x )

pav = po L( x)

x = po E/kT From Principles of Electronic Materials and Devices, Third Edition , S.O. Kasap (© McGraw-Hill, 2005)


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At large fields, L(x)→1,

⇒ pav= po

At low fields, L(x) ≈ x/3,

kT p

kT

E p x p p

od

ooav

2

2

31

33

=⇒

==

α

(Dipolar or orientational

polarizability)

The Langevin function.

Fig 7.56



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Ionic Polarization & Dielectric Resonance

)/( −+−+ += M M M M M r Consider a pair of oppositely charged ions. In the presence of an applied field E along x, the

Na+ and Cl- ions are displaced from each other by a distance x. The net average (or induced)

dipole moment is pi.

Polarizing force=QE Restoring force=Fr ; Frictional force=γdx/dt;

Fig 7.57


β=spring constant associated

with ionic bond, & γ = dependon mech. of energy loss from E2

2

dt

xd M dt

dxγ xQE F F F F r lossr total =+−=++= β


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)exp(22

2

t j E M

Q x

dt

dxγ

dt

xd o

r

I I ω ω =++

Loss coefficient per unit reduced mass, γI =γ/Mw

Resonant or natural vibrational frequency of the IONIC bond, ωI = ( / M r )1/2

Reduced mass Applied field

Soln, x=x oexp(j ωt)

•Forced Oscillator Eqn. (damped motion of a ball attached to a spring in a viscous

medium) and is oscillated by field. – Lorentz Oscillator Model

•xo, pi=(Qxi), αi=pi/E - all are complex, i.e., phase shifted w.r.t. E

2

2

)0( I r

i M

Q

ω α =

)( 22

2

ω γ ω ω α

I I r

ii

j M

Q

E

Qx

E

p

+−=== ⇒

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ ⎟⎟ ⎠

⎞⎜⎜⎝

⎛ +⎟⎟

⎠

⎞⎜⎜⎝

⎛ −

=

I I

I

I

ii

γ j

ω

ω

ω ω

ω

α ω α 2

1

)0()( (ω/ωI)≡ Normalized freq.



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Fig 7.58


• αi” peaks at ω~ωI

(i.e.,ionic bond resonant freq.)

• Sharpness & peak-valuedepend on loss factor γ

• αi’ ~ constant at ω«ωI

•Through ω, αi’ shows rapid

change from + to -, & then it

→0 for ω>>ωI

•At ω=ωI, polarization lags

behind E by 90°.

•dP/dt & dE/dt in same phase,leading to max E transfer

•For ω»ωI ,negligible coupling

•Ionic poln relaxn peak, ω=ωI(b’cause of max. coupling)

Resonant frequency (ωI) for ionic

polarization relaxation are typically in

IR range, i.e. 1012

Hz.


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(a) An ac field is applied to a dipolar medium. The polarization P ( P = Np) is out of phase with

the ac field.

(b) The relative permittivity is a complex number with real (ε r ') and imaginary (ε r '')

parts that exhibit frequency dependence.Fig 7.13



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Debye Equations

2

)(1

]1)0([1

ωτ

ε ε

+

−+=′ r r 2

)(1

]1)0([

ωτ

ωτ ε ε

+

−=′′ r r

ε r = dielectric constant (complex)

ε ′r = real part of the complex dielectric constant

ε ″r = imaginary part of the complex dielectric constant

ω = angular frequency of the applied field

τ = relaxation time



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Cole-Cole plots

Cole-Cole plot is a plot of ε ″r vs. ε ′r as a function of frequency, ω . As thefrequency is changed from low to high frequencies, the plot traces out a

circle if Debye equations are obeyed.

Fig 7.17



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(a) Real and imaginary part is of the dielectric constant, ε r ' and ε r '' versus frequency for (a) a

polymer, PET, at 115 °C and (b) an ionic crystal, KCl, at room temperature.

both exhibit relaxation peaks but for different reasons.

SOURCE: Data for (a) from author’s own experiments using a dielectric analyzer (DEA),(b) from C. Smart, G.R. Wilkinson, A. M. Karo, and J.R. Hardy, International Conference on

lattice Dynamics, Copenhagen, 1963, as quoted by D. G. Martin, “The Study of the Vibration

of Crystal Lattices by Far Infra-Red Spectroscopy,” Advances in Physics, 14, no. 53-56, 1965,

pp. 39-100.Fig 7.16



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The frequency dependence of the real and imaginary parts of the dielectric constant in the presence of interfacial, orientational, ionic, and, electronic polarization mechanisms.

Fig 7.15



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Relative permittivity, εr

Fig 7.58


• From C-M Eqn. Also, we needto consider the electronic

polarizability αe of the two

types of ions.

• εr also complex• Since αi=0 for ω>>ωI, εr → εop

][32)(

1)(−+ ++=

+−

eei

o

i

r

r N α α α ε ω ε

ω ε

)(3321

2)(1)(

22

2

ω γ ω ω ε ε α

ε ε

ω ε ω ε

I I r o

i

o

ii

rop

rop

r

r

j M Q N N

+−==

+−−

+−

)( 22

2

ω γ ω ω α I I r

i

i j M

Q

E

Qx

E

p

+−===

Dielectric Dispersion relation

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ ⎟⎟ ⎠

⎞⎜⎜⎝

⎛ +⎟⎟

⎠

⎞⎜⎜⎝

⎛ −

=

ee

e

e

e

e γ j

ω

ω

ω ω

ω

α ω α

2

1

)0()(

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ ⎟⎟ ⎠

⎞⎜⎜⎝

⎛ +⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −

=

I I

I

I

i

i γ j

ω

ω

ω ω

ω

α ω α

2

1

)0()(

•Lorentz Oscillator Model also applicable to

Elec. Polarizab., by considering resonant freqand loss factor involved in elect. poln:

Dielectric Mixtures and Heterogeneous Media


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Heterogeneous dielectric media examples

(a) Dispersed dielectric spheres in a dielectric matrix.

(b) A heterogeneous medium with two distinct phases I and II.(c) Series mixture rule.

(d) Parallel mixture rule.

Dielectric Mixtures and Heterogeneous Media

Fig 7.59



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•Effective dielectric constant, εreff of the mixture

• ⇒ C= εoεreff A/d

•In Solid Solution, we simply add the polarizabilities (CM Eqn) of

each species of ions weighted by their concentration, e.g. CsCl

•Consider a heterogeneous dielectric that has two mixed phases I

and II with diel. Constt. εr1 and εr2, and volume fraction v1 and v2,

such that v1 + v2 = 1 (fig. b), useful mixture rule isn

r

n

r

n

reff vv 2211 ε ε ε +=

n=depends on type of mixture,

For series connected stack (fig.(c)), n = -1,

For parallel connected stack (fig.(d)), n = 1.



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Maxwell-Garnett formula

• For dispersed dielectric spheres (fig.(a)) havingdielectric constant εr1 such as air pores, in a

continuous dielectric matrix (with εr2)

21

211

2

2

22 r r

r r

r reff

r reff v

ε ε

ε ε

ε ε

ε ε

+

−=

+

−

• Validity – well upto about 20% of volume fraction.

• Useful for predicting εreff of many different types of dielectric

that have dispersed pores.e.g., 41% porosity in SiO2 changes εr from 3.9 to 2.5 (a trick in making

low-K dielectric for interlayer dielectric (ILD) between multilayers of

metal interconnect lines in Microlectronic industry, as they offer highspeed operation of chip due to lower RC time constant)


Proof for Pav for Orientational (Dipolar) Polarization

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