drude lorentz circuit gonano zich
TRANSCRIPT
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
3 C.A. Gonano, R.E. Zich
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:
4 C.A. Gonano, R.E. Zich
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
5 C.A. Gonano, R.E. Zich
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:
6 C.A. Gonano, R.E. Zich
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:
7 C.A. Gonano, R.E. Zich
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:
8 C.A. Gonano, R.E. Zich
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
C.A. Gonano, R.E. Zich
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
C.A. Gonano, R.E. Zich
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…
C.A. Gonano, R.E. Zich
• 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)
C.A. Gonano, R.E. Zich
• 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”
C.A. Gonano, R.E. Zich
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)
C.A. Gonano, R.E. Zich
• 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)
C.A. Gonano, R.E. Zich
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)
C.A. Gonano, R.E. Zich
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
C.A. Gonano, R.E. Zich
• 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
C.A. Gonano, R.E. Zich
• 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
20 C.A. Gonano, R.E. Zich
• 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
21 C.A. Gonano, R.E. Zich
• 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?
22 C.A. Gonano, R.E. Zich
Extra details • A bit of History…
• About meta-materials
• Circuit models
• Circuit dipoles by Alù & Engheta
23 C.A. Gonano, R.E. Zich
• Circuit meta-materials
• 2-D dielectris and metals
• Circuit metals
• More circuit analog
A bit of History…
24 C.A. Gonano, R.E. Zich
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
25 C.A. Gonano, R.E. Zich
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
26 C.A. Gonano, R.E. Zich
• 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
27 C.A. Gonano, R.E. Zich
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
28 C.A. Gonano, R.E. Zich
• 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
C.A. Gonano, R.E. Zich
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
C.A. Gonano, R.E. Zich
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
C.A. Gonano, R.E. Zich
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
C.A. Gonano, R.E. Zich
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
C.A. Gonano, R.E. Zich
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!
C.A. Gonano, R.E. Zich
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
C.A. Gonano, R.E. Zich
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
C.A. Gonano, R.E. Zich
37
More circuit analogs
Site by Gerard Westendorp, ecleptic engineer
http://westy31.home.xs4all.nl/Electric.html
C.A. Gonano, R.E. Zich
38
This is the last slide
C.A. Gonano, R.E. Zich
THAT’S ALL FOR NOW