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Low energy loss; electronic structure and dielectric properties pt - - PowerPoint PPT Presentation

Jul 05, 2023 362 likes β€’598 views

Low energy loss; electronic structure and dielectric properties pt 2 FYS5310/FYS9320 Lecture 8 09.03.2017 Recap from last time 2 If the initial states are sharply peaked in energy, then all transitions originate at this energy

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Low energy loss; electronic structure and dielectric properties pt 2

FYS5310/FYS9320 Lecture 8 09.03.2017

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SLIDE 2

Recap from last time

2

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  • If the initial states are sharply peaked in

energy, then all transitions originate at this energy

  • One particular Ei and one particular E

takes you to a single point in the conduction band Ef

  • In effect we are convoluting the

conduction band DOS with a delta function

  • The spectrum reflects a scaled conduction

band DOS

  • But what if the initial states are in the

valence band?

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πΈπ‘“π‘šπ‘’π‘ π‘”π‘£π‘œπ‘‘π‘’π‘—π‘π‘œ βŠ— 𝑑𝐸𝑃𝑇 = 𝑑𝐸𝑃𝑇 𝑀𝐸𝑃𝑇 βŠ— 𝑑𝐸𝑃𝑇 =?

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SLIDE 4

The EELS spectrum as a Joint Density

  • f States

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𝐽(𝐹) ∝ Ψ

𝑔 𝑓𝑗𝒓⋅𝒔 Ψ𝑗 2πœπ‘€π‘ 𝐹𝑗 πœπ‘‘π‘ 𝐹𝑗 + 𝐹 𝑒𝐹𝑗 𝜁𝐺 πœπΊβˆ’πΉ

This is good for core losses: But for single electron transitions in the low loss region we need to consider the convolution of valence DOS with conduction DOS (also called Joint Density of States, JDOS):

No dipole approximation?

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SLIDE 5

The dielectric polarization of the material

𝑸 πœ• = 𝜁0 𝜁 πœ• βˆ’ 1 𝑭(πœ•)

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𝑬 πœ• = 𝜁0𝑭 πœ• + 𝑸 πœ• = 𝜁0𝑭 πœ• + 𝜁0 𝜁 πœ• βˆ’ 1 𝑭 πœ• = 𝜁 πœ• 𝜁0𝑭(πœ•) So what happens if 𝜁 πœ• =0?

The polarization of a material subjected to a time warying electric field is: The displacement (total field) in the material is then:

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SLIDE 6

The dielectric function in the Drude model

  • For free electrons in a uniform

background potential, the dielectric fuction is

  • Where πœ•π‘ž is a harmonic
  • scilator resonance frequency

given by

  • 𝜐 is the scattering time/damping

factor

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𝜁 πœ• = 1 βˆ’ πœ•π‘ž

2

πœ• + π‘—πœ•/𝜐 πœ•π‘ž = π‘œπ‘“2 𝑛0𝜁0

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Thickness measurements and the mean free path

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t e- The electron can lose energy to plasmon excitations many times

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SLIDE 11

Thickness measurements and the mean free path

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𝑄

π‘œ = 1

π‘œ! 𝑒 πœ‡

π‘œ

𝑓

βˆ’π‘’ πœ‡ = π½π‘œ

𝐽𝑒 π‘„π‘œ=0 = 𝑓

βˆ’π‘’ πœ‡ = 𝐽0

𝐽𝑒 𝑒 πœ‡ = ln 𝐽𝑒 𝐽0

Absolute thickness determination is also possible, but need model or experimental detemination for mean free path

F&H

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SLIDE 12

Dielectric function, refractive index, speed of light

  • The real part of the dielectric fuction gives the

refractive index n=οƒ–ο₯

  • The refractive index gives the phase velocity of light

in the material c=c0/n.

  • This is lower than the speed of light in vacuum

nSi(600 nm, Eο‚»2 eV) ο‚» 4

𝑑𝑇𝑗 =

𝑑0 π‘œπ‘‡π‘— β‰ˆ 0,25 𝑑0

𝑀𝑓(200 π‘™π‘Š) β‰ˆ 0,7 𝑑0

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SLIDE 13

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Erni & Browning, Ultramic (2008)

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The KrΓΆger equation

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𝐽 𝐹 = 2𝐽0𝑒 πœŒπ‘0𝑛0𝑀2 𝐽𝑛 βˆ’ 1 𝜁 𝐹 ln 1 + 𝛾 Θ𝐹

2

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SLIDE 15

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E q Only non- relativistic bulk effects

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E q Bulk plus relativistic effects

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E q Bulk plus relativistic plus surface effects

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SLIDE 18

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  • H. R. Daniels, Phd thesis (2003)
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SLIDE 20

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Midgley, Ultramic. (1999)

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SLIDE 21

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Problems for next time

1) Make a plot of the phase velocity of light as a function of refractive indexes n between 1 and 10. The critical acceleration voltage is defined as the voltage giving an electron velocity equal to the phase velocity of a material with refractive index n 2) Make a plot of the critical acceleration voltage as a function of n.

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