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Kramers - Kronig relations,sum rules

Math basics

Fourier transform

sign function and Dirac delta

Fourier transform ofthe sign function:

Convolution I. II.

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Response function

Response function:

Translational invariance:

Fourier:

Long wavelength limit:

Use convolution II

What do we know about ()?1. Real function.2. t: current time. t must be before t( > 0) for physical effects.

for ( < 0)

Susceptibility

For simplicity let us use tfor .1. (t) is a real function. Im (t) = 0From definition of Fourier transform:

where ()=1()+i2()

2. Causality. (t) = 0 for t< 0.Break up to even and odd functions:

Fourier transform of even is pure Re, odd is Im

Fourier transform pairs

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Kramers Kronig relationsFrom: D.W. Johnson, J. Phys. A, Math. Gen. 8, 490, (1975)

To satisfy causality the even and odd parts must be related:

will ensure that is indeed zero for t

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Kramers Kronig relations done.In practical calculations the divergence at = causes trouble. Eliminate!

Add:

FINALLY:

Works well in practical calculations; no principal part is needed

Notes: Arbitrary constant can be added. KK transform of a constant is 0.

Kramers Kronig relations - examplesDirac delta leadsto 1/ divergence

KK transform of Lorentzian peak

is

Step function: Two divergencies

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Kramers Kronig relations , Applicable to any response function

No absorption -> = const. = 1

Kramers Kronig relations - reflectivity

Loss function(will be discussed)

Reflectivity transmissionAmplitude ration: r, tPower ratio: R, T

Power is measured, phase informationis lost, BUT KK to the rescue!

Phase angle restored

Very important in evaluation of data

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Kramers Kronig relations surface imp.Surface impedance

Kramers Kronig relations - consequences

Finite dc meansRe is not 1.

Due to dc, Im is divergent at low

Each peak in () contributes to the static dielectric constant:

Practical: There is no way to get large dc without having large lossesat some finite .

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Classical models:metal and insulator

Drude model

Charged particles, density nEquation of motion: using

Polarization, relaxation time:

Solve for =P/E

Dielectric function

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Drude conductivity

Simpler form, introducing the

plasma frequency

High frequency limit:

Turn it into conductivity, using general relationship between ,

alternativeway, sameresult

with

Conductivity

Real part: Lorentzian.

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Loss function

Peak at p

Plasma frequency

Real part of crosses zero. Longitudinal waves are possible. At thesame time, major change in reflectivity (transverse waves)

Add dielectric background:Fast electronsp = E (one atom) ---Slow electrons(we are looking at these) Clausius-Mosotti

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Drude model - reflectivity

Effective dielectric function

Zero crossing happens atfast slow

In real metals: phonons, too

Phonons discussed later.Notice difference below andabove plasma frequency.

Transparent above p

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Drude model - experiments

Indium antimonide AluminumW.G. Spitzer, Phys. Rev, 106, 882 (1957) (H. Raether, Springer Tracts in Mod.

Phys. Vol 38 (1965)

Tuning of conduction electrondensity

Drude model - Three regimes

Hagen-Rubens

Skin depth

Square root dep.

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Drude model - tranparency

Relaxation regime: between 1/ and p

Absorptivity independent of frequency

Transparent regime

How to measure p?

p is more important than pl. pl is contaminated by infty.p contains effective mass.What if the effective mass is frequency dependent (interactions!)

Use Tinkhams formula:

Drude model conductivity:

We get

intercept curvature

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Drude model - transmission of thin film

L. Forro et al. Phys. Rev. Letters 65, 1941 (1990)

Insulators: classical

Oscillator model

Same calculation as Drude

Kramers-Heisenberg dielectric function

Bound chargecontributes to dielectric constant

S: oscillator strength

Lyddane-Sachs-Teller relation

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Diel. function and reflectance

Three regimes:Below resonance: usual dielectric constantRight above resonance: similar to DrudeAbove p: transparent againTwo resonances: Transverse (T=0) and Longitudinal (L)

Experiment

To be supplied