Drude Lorentz circuit Gonano Zich

Author: Carlo Andrea Gonano

Co-author: Prof. Riccardo Enrico Zich

Politecnico di Milano, Italy

EUCAP 2014,

6-11 April, The Hague,The Netherlands

Drude-Lorentz model

in circuit form

Contents 1. Introduction

2. The Drude-Lorentz Model

7. Limits of the models

8. Conclusions

9. Questions and Extras

4. Request for a circuit model

5. Extending KCL

6. Circuit dipoles

7. Multi-resonance model

8. Anisotropic materials

2 C.A. Gonano, R.E. Zich

Introduction ESTIMATING PERMITTIVITY

• The Drude-Lorentz model (DLM) is a method for estimate the permittivity e of a bulk material

• Atomic, simple and linearized model, but quite effective and widespread

• Calculating e is a common problem in the project of metamaterials

• Need to design the structure at a microscopic level and in a simple way


CIRCUITS ARE A POWERFUL TOOL FOR THIS PURPOSE

• Problem: the DLM is not a circuit model

Material micro-macro structure – image by Pendry &Smith

The Drude-Lorentz model (I)

• Atoms (or molecules): positive nucleus surrounded by a uniform “cloud” of electrons

• If the particle is invested by an EM wave, the external Eext field will displace negative from positive charge

• The induced moment dipole p is so:


Model’s basic hypothesis

polarized atom

)()()( eeeeeee xxqxqxqp

xqp e

where and

ex

ex

are the centers of positive and negative charge qe

Rayleigh hypothesis • Usually the atom’s length l is much

smaller of the wavelength l


Rayleigh hypothesis

• So the system experiences almost the same E field almost everywhere

If l << l , then the E field can be approximated as conservative

The E field admits a potential V, thus:

• Magnetic effects can be neglected, since:

Rayleigh hypothesis:

0

t

B

• The system does not irradiate

l

2 0

E0

x

E

VE

circuit feature

The Drude-Lorentz model (II)

• In the DLM the quantistic forces are linearized and reproduced by springs and dampers:

• For each electron will so hold:


xqp e

exte Eexkdt

xd

dt

xdm

2

2

• The nucleus mass is much greater than the electrons’ one: mp >> me

extee Eqepkdt

pd

dt

pdm

2

2

SO WE GET A HARMONIC EQUATION

FOR THE DIPOLE MOMENT P

electro-mechanical model

Permittivity calculus (I) • Supposing the Eext field is harmonic, the dipole moment p will be:

• The previous relation can be compacted in Laplace as:


where:

ti

e

e eEm

qe

ip

0

0

22

0 )2()(

1

emk /0 resonance pulsation

)2/( 0 em damping coeff.

)()()( sEssp ext

• The polarization P is defined as the sum of dipole moments per unit of volume:

)()( spV

NsP

ol

P

HOW CAN WE DRAW THE PERMITTIVITY?

• The global E field is the sum of the external one and of that produced by the dipoles themselves: polext EEE

Permittivity calculus (II) • Gathering all the equation, we have the system:


PN

ED

pol

0

11

e

AFTER SOME ALGEBRA, FINALLY IT’S POSSIBLE TO EXPRESS THE PERMITTIVITY AS:

polext EEE

PEsEssD

0)()()( ee

)()()( sEsV

NsP ext

ol

P

global Electric field in the bulk material

permittivity implicit definition

polarization induced by the external field

E field produced by dipoles in ND Dimensions

)(1

1

)(1

11)(

0 sV

N

N

sV

N

Ns

ol

P

D

ol

P

D

e

e

Request for a circuit model

• An equivalent circuit model would be useful:

Easy method for designing

Fully electrical (no springs and dampers)

Suitable for structures of growing complexity

9

LIMITS:

• The E field should be conservative, in order to define a potential

C.A. Gonano, R.E. Zich

• The classic DLM is an electro-mechanical model

l VE

OK!

Rayleigh hypothesis

• In a normal circuit you cannot have locally a net charge…

10

Extending KCL

• Normally, none of circuit devices can accumulate charge, as stated by Kirchhoff Current Law at every node

HOW TO DESCRIBE A LOCAL NET CHARGE?

• Though, a real capacitor is made of two plates, each with net charge

01

N

n

nE i

dt

Qd

Extended KCL

01

N

n

ni

KCL


i 1

i 2

...

i N

Qe

So, a plate allows to store a net charge on a circuit node

CHARGE CONSERVATION ON A NODE

11

Plates and half-capacitors

• The classic capacitor is made of two plates connected by a insulator

121,2,2

1VVCQQ EE


constitutive equation

LET’S NOTICE THAT:

• In circuit approach, charge QE is defined on a node Eulerian quantity

ytQtxqtp Ee

)()()(

• In the DLM, charge qe is defined on a particle

dipole moment

Lagrangian quantity

01

N

n

nE i

dt

Qdi 1

i 2

...

i N

Qe

12

Circuit dipole (I)

THE ANALOGY IS QUITE EASY…

• The Drude-Lorentz dipole oscillates and an RCL circuit can oscillate too…


• So we construct a circuit dipole of fixed length y , invested by the external E field

• The current flowing through the element can be defined as:

dt

Qdi E

ytQtp E )()( dt

dpiy

yQp E

EQ

EQ

tieE

0

Circuit dipole

current – dipole moment relation

13

Circuit dipole (II)


• The external E field can be modeled with a voltage generator

ytEtV exte )()(

yQp E

EQ

EQ

tieE

0

• The charge Qe can accumulate in the plates facing the vacuum (ground)

NO NEED FOR CLOSED LOOPS!

• The RCL dipole is so ruled by a harmonic equation for the current i:

e

t

VdiC

iRdt

idL 0

)(1

oscillating RCL circuit dipole

14

LET’S COMPARE THE MODELS’ EQUATIONS

Harmonic matching

Drude-Lorentz dipole

e

e

qe

m

y

L

2

eqey

R

2

eqe

k

yC

2

1

inertia (no link to m) (quantistic) damping “elasticity”


extee Eqepkdt

pd

dt

pdm

2

2

dt

dpiy ytEtV exte )()(

Circuit harmonic oscillator

e

t

VdiC

iRdt

idL 0

)(1

Matching the equations, we get:

linking relations:

SO WE HAVE THE CIRCUIT EQUIVALENT FOR THE DLM

15

Multi-resonance model (I)


• The classic DLM can be easily extended to multi-orbital atoms

with:

e

j

jjj

jm

en

sss

2

22 )2(

1)(

)()()( sEssp extjj

• In a linear model, the j-th dipole moment and j-th polarizability are:

• The global dipole and polarizability can be easily calculated:

m

j

j spsp1

)()(

m

j

j ss1

)()(

• Let be nj electrons in the j-th orbital, oscillanting with pulsation j and damping j

16

Multi-resonance model (II)


SO WE HAVE THE CIRCUIT EQUIVALENT FOR THE DLM

• Usually the electrons are modeled as non-interacting

• The circuit equivalent consists of RCL parallel dipoles

22 en

m

y

L

j

ej

22 eny

R

j

jj

22

1

en

k

yC j

j

j

inertia

damping

“elasticity”

• The external E field is uniform, and so the voltage: jVV jee

• For each j-th branch the impedance is: j

jjjsC

RsLZ1

17

Multi-resonance model (III)


SO THE GLOBAL DIPOLE AND POLARIZABILITY ARE:

• The current i j and dipole moment pj for each branch will be:

)()(

2

sZs

ys

m

j jZZ 1

11

jj is

yp 1

• The global impedance Z(s) is:

)()(

)(2

sEsZs

ysp ext

)()()( sEssp ext

j

ej

Z

Vi

j

ej

Z

V

syp

1

polarizability – impedance relation

18

Anisotropic materials


• If the material is anisotropic, the polarizability could be different for each k-th direction

EXAMPLE: DIAGONAL POLARIZABILITY MATRIX

• The unit particle (i.e., atom or molecule) can be modeled with many RCL dipoles mutually orthogonal

extEp

3

2

1

00

00

00

Anisotropic polarizability

Passing from a 1-D dipole to 3-D cross structure

19

Orthogonal dipoles


• More generally, each dipole is associated to a k-th direction and has its own impedance Zk(s)

• Each impedance Zk(s) is virtually connected to the ground

THANKS TO HALF-CAPACITORS, THERE’S NO NEED FOR RING

STRUCTURE

• Voltage generators and impedances can be splitted in two halves

• The global polarizability for the k-th direction will be:

)()(

2

sZs

ys

k

k

polarizability – impedance relation

Limits of the models


• Conservative E field, so no magnetic effects:

As previously stated, our task is not to improve the DLM itself, but to rephrase it in circuit form

Accordingly, those two models have similar limits:

• Linearized quantistic forces and non-interacting electrons

• Formally dipoles cannot radiate (circuit feature)

VE

0

t

B

• Particles polarization is null at rest (not true for molecules like H2O)

HOWEVER, DLM IS CURRENTLY WIDE-USED

FOR ITS EFFECTIVENESS AND EASINESS

Conclusions and future tasks

• We did it with simple RCL circuits , introducing the concept of half-capacitor


• The DLM is electro-mechanic, but it can be rephrased in circuit form

• The circuit model could be a powerful design tool for metamaterial’s unit elements

MAIN FEATURES

• We showed that the circuit concept is suitable also for multi-resonance and anisotropic materials

• Differently from other models, the circuit structure is defined for every frequency

That’s all, in brief…

THANKS FOR THE ATTENTION.

QUESTIONS?

EXTRAS?


Extra details • A bit of History…

• About meta-materials

• Circuit models

• Circuit dipoles by Alù & Engheta


• Circuit meta-materials

• 2-D dielectris and metals

• Circuit metals

• More circuit analog

A bit of History…


Paul Karl Ludwig Drude

• In 1900 paper “Zur Elektronentheorie der Metalle”, Paul Drude applied the kinetic theory to describe the electrical conduction in metals

• Gas of free electrons bouncing on massive positive ions, damping due to collision

• Ohm’s law and conductivity s well explained Electron gas;

image in free domain

Em

enJ

e

2

Permittivity can be calculated too:

)(1

)(2

2

0 e

p

mi

e

e

A bit of History… • In 1905 paper “Le movement des electrons dans les

metaux”, H. A. Lorentz tried to describe the interaction between matter and EM waves


Hendrik Antoon Lorentz

• Lorentz proposed that the electrons are bound to the nucleus by a spring-like force obeying to the Hooke’s law

Both Drude and Lorentz models are formally not quantistic,though quite prophetic and effective

• The Drude model can be reobtained by placing k = 0 (no spring)

Designing metamaterials


• In ElectroMagnetics a bulk material can be characterized by its permittivity e and permeability m

• Macroscopic level, distributed parameters

Examples of metamaterial’s structure by John Pendry and David J. Smith

• Sometimes you need to design the structure at a microscopic level and in a simple way

THIS IS A COMMON PROBLEM IN THE PROJECT

OF METAMATERIALS

S k

Negative refraction

Metamaterials

• composite materials engineered to exhibit peculiar properties, e.g. negative refraction

• Sub-wavelength modular structure, but macroscopically homogeneous


BUT WHAT ARE METAMATERIALS?

Back-propagation

• Popularly, materials with negative e and m

meem 0

0,0 cn

Many paradoxal properties…

• wave seems to propagate backward

• negative phase velocity: 0v

• wavevector k has the opposite sign of Poynting vector S

Metamaterials: applications


• In 1999 an artificial material with m <0, e0 in microwave band was realized at Boeing Labs

• Increasing interest in these metamaterials during the last decade

(hundreds articles per year!)

Relatively new research field,

with many applications ranging from:

Example of metamaterial’s structure by David J. Smith

• Cloaking devices

• Selective frequency antennas

• Super-lens, with high resolution, able to focus light through plane interface

(no need for curve slabs!) lens slab with negative index

Circuit models • Circuit models allow to “lump”

system’s properties in few variables

• Easy method for designing

• Already known and used in many context: Power Systems, RF antennas, Electronics…

• Suitable for structures of growing complexity

29

LIMITS:

Lumped-circuital model for nano-antenna

• The system must be sub-wavelength sized: l 0

E

The E field will admit a potential V, thus no trasversal waves!

• …so a Quasi-Steady Approx is implict


30

Plates and half-capacitors WHY HALF-CAPACITORS?

• The classic capacitor is made of two plates connected by a insulator

121,2,2

1VVCQQ EE


i 1

i 2

...

i N

Qe

• Net charge storing allowed

• No need to “close” a circuit in a loop

constitutive equation

LET’S NOTICE THAT:

• In circuit approach, charge QE is defined on a node Eulerian quantity

ytQtxqtp Ee

)()()(

• In the DLM, charge qe is defined on a particle

dipole moment

Lagrangian quantity

Insulators & conductors

• Insulator: just polarization current

• Low EM inertia, bound electrons

31

Dielectric (not dispersive)

Anyway,which are the circuit elements equivalent to dielectric and metals?

VCiI x

SC

0'ee

• Conductor: current can flow

• Ohmic losses

Metal at = 0

VGI x

SG

s

• Conductor: current can flow

• High EM inertia, free electrons (ideal Drude-metal)

Metal at high

S

xL

P

0

2

1

e VLi

I

1

Material’s properties Extensive eq. Circuit element


Circuit dipoles by Alù & Engheta

32

LET’S CONSIDER NOW A PRACTICAL EXAMPLE! Obviously, each materials can be described with more than one element

• External exciting field E is associated to an impressed current generator

Isolated homogeneous sphere with tieEE

0

lR

illuminated with electric field

Lumped circuit equivalent

• The particle is crossed by a current I pol and the impedance is Z nano = V/ I pol

0

2

0)( ERiIimp

ee

1 RiZnano e

• A parallel “fringe” capacitor accounts for the dipolar fields in vacuum around the particle: 1

0 2

RiZ fringe e


33

Circuital model by Andrea Alù & Nader Engheta

dielectric metal

WHICH IS THE ROLE OF PERMITTIVITY e ?

Circuit dipoles by Alù & Engheta (II)

• e’ > 0, e’’ > 0 and low inertia

Z nano is capacitive and resistive

Dielectric sphere

• e’ < 0, e’’ > 0 and high inertia Z nano is inductive and resistive

Metal sphere

''')( eee i

)Re(e RCnano

• No inductors, no resonance!

)Im(e RGnano

1 RiZnano e

12 )Re(

e RLnano )Im(e RGnano

• // inductor and capacitor , so resonance at

2 fringenanoCL 02))(Re( ee

Local Plasmon Resonance


34

DIFFERENT STRUCTURES CAN BE MODELLED…

Circuit meta-materials

composite

dipoles

dielectric-metal array

Nano-trasmission line Negative-refraction line These are just

examples: 2-D and

3-D circuits are

also available!


2-D dielectrics and metals

35

• Ideal dielectric element, non dispersive and without losses, is described like a pure capacitive lattice

• Insulator: just polarization current, which cannot flow very far

• No resonance

• Ideal metal element, without magnetic induction and losses, is described like an inductive lattice

• Electrons are bounded just at contours, so quantistic forces are reproduced by half-capacitors on the border

• Resonances thanks to external coupling with vacuum

HOW TO MODEL 2-D OR 3-D BULK MATERIALS?

Linking them together, we get a dielectric-metal interface for SPP


36

Circuit metals

• Real metals are quite well described by circuit lattice analogous to Drude-Lorentz model

WHAT ABOUT MORE REAL METALS?

• No magnetic effects: induction caused just by electron mass inertia

• E field always conservative, so no irradiation

LIMITS

• Basic inductor lattice, like ideal metals

• Plates on the border account for contour bounded charge

• Resistors account for internal losses

• Shunt plates accounts for internal capacitive effects

Real-metal circuital model


37

More circuit analogs

Site by Gerard Westendorp, ecleptic engineer

http://westy31.home.xs4all.nl/Electric.html


http://westy31.home.xs4all.nl/Electric.html

38

This is the last slide


THAT’S ALL FOR NOW

Drude Lorentz circuit Gonano Zich

Engineering