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

Oct 08, 2023 455 likes β€’700 views

Low energy loss; electronic structure and dielectric properties FYS5310/FYS9320 Lecture 7 02.03.2017 FYS5310 teaching schedule Preliminary schedule only! You should keep the class-times on Wednesdays and Thursdays open unless notified by email

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Low energy loss; electronic structure and dielectric properties

FYS5310/FYS9320 Lecture 7 02.03.2017

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FYS5310 teaching schedule

Preliminary schedule only! You should keep the class-times on Wednesdays and Thursdays open unless notified by email (or in this schedule) that there is no class References to the textbook to Fultz & Howe unless stated otherwise.

Date Time Lecture/lab Topic Chapters Homework

Wednesday 18.01.2017 14:15-16:00 Lecture Introduction to the course. Derivation of the structure factor (01) 4.1, 4.3.1, 6.1 Exercise set 1 (handout) Thursday 19.01.2017 12:15-14:00 Lecture No class (SMN seminar) Wednesday 25.01.2017 13:15-16:00 Lab/Colloquium Going through exercise set 1 + Lecture: The atomic form factor (02) 4.3 Excercise set 2 (handout) Thursday 26.01.2017 12:15-14:00 Lecture No class Wednesday 01.02.2017 14:15-16:00 Lab/colloquium Going though exercise set 2 Thursday 02.02.2017 12:15-14:00 Lecture Uses of EELS and EELS instrumentation (03) 5.1, 5.2; W&C 37 Exercise set 3 (handout) Wednesday 08.02.2017 14:15-16:00 Lab/colloquium Going though exercise set 3 Thursday 09.02.2017 12:15-14:00 Lecture Inelastic form factors (04) 5.4.1-5.4.3 + primer

  • n Dirac notation

Wednesday 15.02.2017 12:15-16:00 Lab/colloquium No class Thursday 16.02.2017 12:15-14:00 Lecture Inelastic form factors, scattering cross sections, dipole selection rules (05) 5.4.4-5.4.7, W&C 39, plus Brehm and Mullin on parity and dipole selectrion rules Wednesday 22.02.2017 12:15-16:00 Lab/colloquium No class Thursday 23.02.2017 12:15-14:00 Lecture Core losses: Quantification and electronic structure (06) 5.4, W&C 39+40 Exercise set 4 (handout) Wednesday 01.03.2017 12:15-16:00 Lab/colloquium Going through excercise set 4 Thursday 02.03.2017 12:15-14:00 Lecture Low energy loss; electronic structure and dielectric properties pt 1 (07) 5.3, W&C 38 Exercise set 5 (handout) Wednesday 08.03.2017 12:15-16:00 Lab/colloquium Computer lab + going through exercise set 5 Thursday 09.03.2017 12:15-14:00 Lecture Low energy loss; electronic structure and dielectric properties pt 2 (08) 5.3, W&C 38 Wednesday 15.03.2017 12:15-16:00 Lab/colloquium No class Thursday 16.03.2017 12:15-14:00 Lecture No class Wednesday 22.03.2017 12:15-16:00 Lab/colloquium Computer lab

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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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Possible transitions contributing to

  • ne point in the energy loss spectrum

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E

Density of states Binding energy

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E

Density of states Binding energy

These transitions are not allowed

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E

Density of states Binding energy

…still no contribution to the EELS spectrum

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SLIDE 7

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E

Density of states Binding energy

What about now?

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E

Density of states Binding energy

Here we see the first transition that contributes to the EELS spectrum

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E

Density of states Binding energy

And this is the final transition that contributes Repeat for the next energy loss E

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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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  • The low loss spectrum

can be used to detect band gaps and so- called critical points in the JDOS.

  • These features are

very important for

  • ptical properties

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Yu & Cardona

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The dielectric fuction

  • Describes the response of the material to an external field
  • Not a constant
  • The real term describes the polarizability
  • The imaginery term describes absorption
  • The Β«single scattering distributionΒ» is given by

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𝜁 πœ• = 𝜁1 πœ• + π‘—πœ2(πœ•) 𝐽 𝐹 = 2𝐽0𝑒 πœŒπ‘0𝑛0𝑀2 𝐽𝑛 βˆ’ 1 𝜁 𝐹 ln 1 + 𝛾 Θ𝐹

2

𝐽𝑛 βˆ’ 1 𝜁 𝐹 = 𝜁2 𝜁1

2 + 𝜁2 2

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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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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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πœ•π‘ž = π‘œπ‘“2 𝑛0𝜁0

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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 23

Problems for next time

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