Advanced Functional Properties 6PC20

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Advanced Functional Properties 6PC20

• Lecture 2 Kramers-Kronig relations

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Musical analogy to Fourier transform
Music Sound Air pressure P(t) changing
with time
Music Instruments playing together
Orchestration
Time domain description
Frequency domain description
P(t)
J.S. Bach Brandenburg Concertos 1, BWV 1046-51
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Summary
What are we talking about ?
Functional property response to an external
perturbation
In what language can we discuss this response ?

Linear response functions G(t) and c(w)
What is the relation between G(t) and c(w) ?
G(t) and c(w) are related by Fourier transform

and
What do we know in advance of G(t) and c(w) ?
Causality implies G(t)0 for tlt0. Causality
also implies something for c(w)

But WHAT ???
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Dielectric response
Stimulus AC voltage at frequency ?
Dielectric material
Linear response AC current at frequency ?
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Is this boring ?
Dynamic Random Access Memory
Information stored in the form of charging in
capacitor Problems -bleeding refresh every
20 ms - area on Si wafer
Add comments
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Charged capacitor
-

Electrode plates with large area and small
separation Electric field lines are parallel
and have constant density The electric field
inside the capacitor must be uniform
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How strong is the uniform electric field ?
Surface S
Top Area A
Gauss Law
d
Volume Ad
q/V r charge density
Limit d?0
Gauss Law in local 1D form
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How strong is the electric field in the capacitor
?
Between plates No charges, so E constant
In a metal no electric field (E0) Charge on the
electrode must be at the surface
Eout

E0
With s surface charge density
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How strong is the electric field in the capacitor
?
Between plates No charges, so E constant
In a metal no electric field (E0) Charge on
plate at the surface
Eout

E0
With s surface charge density
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How strong is the electric field in the capacitor
?
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Polar molecules between the electrodes
The total surface charge density s Is the sum of
the free and mobile electrical charges on the
inside of the metal surface (sfree) and the
charge bound to the molecule just on the outside
of the surface resultinjg form polarization P of
the dielectric

s free
d-
d-
d-
P
V
d
d
d
s free

D Dielectric displacement P Polarization
We assume that the polarization is a linear
response to E
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The mesurement of the frequency domain
polarization response function of a dielectric
material
Dielectric material
C0 capacitance if there were No dielectric
material between the capacitor plates, but only
vacuum.
Response of the circuit
Response of the material
Stimulus
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The measurement and its outcome
Frequency domain response function of the
material
Low frequency measurement with electrodes
In literature one often finds
High frequency measurement without electrodes
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Dielectric function for a layer of C60
Dresselhaus Annu. Rev. Mater. Sci. 1995.25487-523
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Dielectric response function of water
Segelstein, D., 1981 "The Complex Refractive
Index of Water", M.S. Thesis, University of
Missouri--Kansas City
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Characteristic features when comparing e1(w) and
e2(w)
At frequencies where e1(w) shows a feature
generally also e2(w) shows a feature Features are
not always the same we see different types,
for example
Type 1
Type 2
e1
e1
e2
e2
freq
freq
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Causality ?
Could it be that the correspondence in feature in
e1(w) and e2(w) are the result of the causality
condition for the response function ?
Fourier transform ?
Causality implies G(t)0 for tlt0.
Causality also implies something for
c(w)
But WHAT ???
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Anatomy of the time domain response function G(t)
Properties of G(t) - real -
G(t) 0 for tlt0
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Fourier transform is a linear transform
Fourier
even
f(t)
g(?)
Real even
Real even
odd
Real odd
Imaginary odd
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Fourier transform of a product of functions
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Fourier transform of a product of functions gives
a convolution
Fourier transform of a product of functions gives
the convolution of their individual Fourier
transforms
f(t)
g(?)
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Fourier transform is a linear transform
f(t)
g(?)
Product of functions Gives convolution
ic2(w) is the convolution of c1(w) with the
Fourier transform of the sign function
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feven (t) real, even
cos(-a) cos(a) sin(a) - sin(a)
Real,even
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feven (t) real, even
cos(-a) cos(a) sin(a) - sin(a)
Imaginary , odd
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Properties of the Fourier transform
f(t)
g(?)
f(t)
g(?)
Real even
Real even
Real odd
Imaginary odd
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Hilbert Transform
Derived using
Realizing that
and repeating the whole procedure we get
These integral transformations are called Hilbert
transforms
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Negative frequencies do not make sense
c1(-w) c1(w) even
In addition, using a similar procedure we find
These integral transformations are called
Kramers-Kronig transforms
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Kramers Kronig relations first written down In
1926 by (independently)

• Hendrik Anthony Kramers (1894-1952)
• Working at Leiden University

Ralph Kronig 1904 - 1995 Working for some time
at Delft University

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Summary linear response theory
Fourier transformation
G(t)
c(w)c1(w)ic2(w)
Causality implies that
G(t) 0 for tlt0.
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Exercise lecture 2

• 1. Dielectric spectrum of water
• The static relative dielectric constant er of
  water is around 80.
• Can you relate this value to the data in the
  graph showing e1(w) and e2(w) for water ? Please
  explain !
• The graph of e2(w) for water shows a minimum
  Calculate the wavelength of the electromagnetic
  wave corresponding to this minimum. What is the
  significance of this minimum for life on earth ?
  
• At very high frequencies the value of e2(w) goes
  to zero while e1(w) goes to 1.
• (see graph of e(w) for water.
• - What is the wavelength of the waves associated
  with these high frequencies ?
• - What type of radiation does it correspond to ?
• What does e2(w) ? 0 and e1(w) ? 1 for high
  frequency electromagnetic oscillations tell you
  about the strength of the interaction of the
  corresponding waves with water ?
• The graph of e2(w) for water also shows a
  maximum. Calculate the wavelength of the
  electromagnetic wave corresponding to this
  maximum ? Explain why the microwaves in you
  microwave oven have approximately the same
  frequency.

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Excercises Lecture 2
2. Fourier transform and convolution Use the
matlab program FourierSIN.m and FourierCOS.m to
calculate the Fourier transform of the functions
a. Draw a schematic sketch of of the real
and imaginary components of the Fourier
transform. b. Determine the symmetry properties
(even / uneven) of the real and imaginary
component of the Fourier transforms. c. What
happens to the Fourier transform when you change
the value of w ? What happens when you vary the
value of w (omega) ? (qualitative answer
suffices) d. According to mathematics, the
Fourier transform of a product of two functions
is a convolution of the two Fourier transforms of
the two separate functions . Use your knowledge
and intuition on the Fourier transforms of
Gaussian curves and monochromatic waves and
discuss qualitatively whether the convolution of
predicted by mathematic can account for your
observations under c.