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![Allow non-equal a-priori probabilities to get less possible moves that are not accepted
Wo
= f DH
ij [ ij ]
ji
Wo
ji
= f[DH ]
In Metropolis this is symmetric
Detailed balance
PWo
P = P W o
P
i ij ij j ji ji
Pij
ji
=
f DHij
[ ]
ji
P f[DH ] ij
exp(-bDH )
Non-Metropolis Monte Carlo
4](https://image.slidesharecdn.com/lecture-12-241119164047-5f478fb4/85/Lecture-12-atomistic-simulation-of-materials-4-320.jpg)
![Free electrons (those giving rise to conductivity) find
an equal potential which is constant throughout the material.
Fermi-Dirac Statistics:
f(E) = 1
exp[(E-Ef)/kT] + 1
1.0
f(E)
0
0.5
Ef
1. At T=0 K: f(E)=1 for E<Ef
f(E)=0 for E>Ef
2. At kT<<Ef (at room temperature kT=0.025 eV)
f(E)=0.5 for E=Ef
T=0 K
kT<<Ef
Fermi Level Referencing
36](https://image.slidesharecdn.com/lecture-12-241119164047-5f478fb4/85/Lecture-12-atomistic-simulation-of-materials-36-320.jpg)
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Lecture-12 atomistic simulation of materials
- 1. Model Hamiltonian and ExperimentalMeasurements MEE 471 Lecture 12 Pakpoom Buabthong pakpoom.b@nrru.ac.th
- 2. Recap • Non-Boltzmann sampling •Non-Metropolis 2
- 3. Non-Boltzmann sampling andUmbrella sampling Simple Sampling Importance Sampling Sample with Boltzmann weight Sample randomly M < A> = å An n=1 exp(-bHn ) M åexp(-bHn ) n=1 An M < A> = å n=1 Non Boltzmann Sampling DH = H - Ho < A> = u Sample with some Hamiltonian Ho M åexp(-b(Hn - Ho )) An u n =1 n=1 M åexp(-b(Hn - Ho )) 3
- 4. Allow non-equal a-prioriprobabilities to get less possible moves that are not accepted Wo = f DH ij [ ij ] ji Wo ji = f[DH ] In Metropolis this is symmetric Detailed balance PWo P = P W o P i ij ij j ji ji Pij ji = f DHij [ ] ji P f[DH ] ij exp(-bDH ) Non-Metropolis Monte Carlo 4
- 5. Outline • Large-scale nanostructures,semiconductor alloy • Mapping electronic structure-calculations into model Hamiltonians • Experimental measurements • UV-Vis • XPS • XAS 5
- 6. Why semiconductor alloys? • Technological interest: tunability of materials properties. Strained layer epitaxy • Scientific interest: spontaneous multilayer ordering 6
- 7. Configurational Statistical Mechanics •Energy of a configuration ? Linear-response theory • Thermodynamic properties ? Monte Carlo Response to small perturbation 7
- 8. Disorder as aperturbation 8
- 9. Disorder as aperturbation Average potential of Ga, In across the lattice Perturbation 9
- 10. Calculate the effectof the perturbation on the energy by evaluating the response of the charge density 10
- 11.
- 12.
- 13.
- 14. Disorder as aperturbation • Calculate material’s energy and stability • Electronic and structural properties • Predict effects on macroscopic thermodynamic properties • Free energy • Entropy • Heat capacity 14
- 15. Formation Energies Elastic StepConfigurational Step 15
- 16. Formation Energies Lattice strain,EOS Atom/lattice arrangement 16
- 17. Full DFT vs.linear response LRT ~ cost-effective LRT ~ response to small perturbation 17
- 18.
- 19. Core-level spectroscopy 19 Photoelectron spectroscopy (XPS) X-rayAbsorption Spectroscopy (XAS)
- 20. XPS measures theenergy of occupied orbitals 20 Photoelectron spectroscopy (XPS) In the experiment, you use a fixed photon energy and measure the photoelectron energy: ℎν = 𝐵. 𝐸. +𝐸!"#$#%&%'$(#) 1) Set this 2) Measure this 3) Calculate the Binding Energy as your answer
- 21. X-ray Absorption Spectroscopy(XAS) measures core-to-valence transitions X-ray Absorption Spectroscopy (XAS) 21
- 22. X-ray Absorption Spectroscopymeasures core-to- valence transitions !"" !""" !"""" "#""! "#"! "#! ! !" !"#A%"&'() $%&'E)*+&,- So far I’m being confusing: I’m showing the arrows go to the continuum, not to valence orbitals Mn 3p Mn 3s O 1s, Mn 2p, Mn 2s Mn 1s 22
- 23. Also known as NearEdge X- ray Absorption Fine Structure (NEXAFS) 23
- 24. 24 M 3d M 1s S1s ~-2400 eV M 4p Dipole-allowed transitions: ΔL=1 (sàp) Quadrupole-allowed transitions: ΔL=2 (sàd) Local symmetry Oxidation states
- 25. An extreme exampleof a strong pre-edge transition 25
- 26. EXAFS spectrum givesgeometry 26 Source: Wikipedia
- 27. The workflow 27 Subtract baseline, normalize 𝑘= 2𝑚 ℎ (𝐸 − 𝐸!) Fourier Transform Adv. Sci. 2019, 6, 1802126
- 28. Core-level spectroscopy 28 Photoelectron spectroscopy (XPS) X-rayAbsorption Spectroscopy (XAS)
- 29. Surface Analysis The Studyof the Outer-Most Layers of Materials (<100 A). • Electron Spectroscopies XPS: X-ray Photoelectron Spectroscopy AES: Auger Electron Spectroscopy EELS: Electron Energy Loss Spectroscopy • Ion Spectroscopies SIMS: Secondary Ion Mass Spectrometry SNMS: Sputtered Neutral Mass Spectrometry ISS: Ion Scattering Spectroscopy 29
- 30. What is XPS? X-rayPhotoelectron Spectroscopy (XPS), also known as Electron Spectroscopy for Chemical Analysis (ESCA) is a widely used technique to investigate the chemical composition of surfaces. 30
- 31. X-ray Photoelectron Spectroscopy SmallArea Detection X-ray Beam X-ray penetration depth ~1µm. Electrons can be excited in this entire volume. X-ray excitation area ~1x1 cm2. Electrons are emitted from this entire area Electrons are extracted only from a narrow solid angle. 1 mm2 10 nm 31
- 32. XPS spectrallines are identified by the shell from which the electron was ejected (1s, 2s, 2p, etc.). The ejected photoelectron has kinetic energy: KE=hv-BE-F Following this process, the atom will release energy by the emission of an Auger Electron. Conduction Band Valence Band L2,L3 L1 K Fermi Level Free Electron Level Incident X-ray Ejected Photoelectron 1s 2s 2p The Photoelectric Process 32
- 33. L electronfalls to fill core level vacancy (step 1). KLL Auger electron emitted to conserve energy released in step 1. The kinetic energy of the emitted Auger electron is: KE=E(K)-E(L2)-E(L3). Conduction Band Valence Band L2,L3 L1 K Fermi Level Free Electron Level Emitted Auger Electron 1s 2s 2p Auger Relation of Core Hole 33
- 34. KE = hv- BE - Fspec Where: BE= Electron Binding Energy KE= Electron Kinetic Energy Fspec= Spectrometer Work Function Photoelectron line energies: Dependent on photon energy. Auger electron line energies: Not Dependent on photon energy. If XPS spectra were presented on a kinetic energy scale, one would need to know the X-ray source energy used to collect the data in order to compare the chemical states in the sample with data collected using another source. XPS Energy Scale- Kinetic energy 34
- 35. XPS Energy Scale-Binding energy BE = hv - KE - Fspec Where: BE= Electron Binding Energy KE= Electron Kinetic Energy Fspec= Spectrometer Work Function Photoelectron line energies: Not Dependent on photon energy. Auger electron line energies: Dependent on photon energy. The binding energy scale was derived to make uniform comparisons of chemical states straight forward. 35
- 36. Free electrons (thosegiving rise to conductivity) find an equal potential which is constant throughout the material. Fermi-Dirac Statistics: f(E) = 1 exp[(E-Ef)/kT] + 1 1.0 f(E) 0 0.5 Ef 1. At T=0 K: f(E)=1 for E<Ef f(E)=0 for E>Ef 2. At kT<<Ef (at room temperature kT=0.025 eV) f(E)=0.5 for E=Ef T=0 K kT<<Ef Fermi Level Referencing 36
- 37. Fermi Level Referencing !"##"$ #"B #"& #"' #"# E#"' E#"& E#"B E#"$ E!"# F *+ ,-./0F*N2-F3+F 4056F.33/FT-/8-.TV.- :0;N0;2F-;-.2<F=->? 5=*?@* F F 37
- 38. hv Because the Fermilevels of the sample and spectrometer are aligned, we only need to know the spectrometer work function, Fspec, to calculate BE(1s). E1s Sample Spectrometer e- Free Electron Energy Fermi Level, Ef Vacuum Level, Ev Fsample KE(1s) KE(1s) Fspec BE(1s) Sample/Spectrometer Energy Level Diagram- Conducting Sample 38
- 39. hv A relative build-upof electrons at the spectrometer raises the Fermi level of the spectrometer relative to the sample. A potential Ech will develop. E1s Sample Spectrometer e- Free Electron Energy BE(1s) Fermi Level, Ef Vacuum Level, Ev KE(1s) Fspec Ech Sample/Spectrometer Energy Level Diagram- Insulating Sample 39
- 40. Binding Energy Referencing BE= hv - KE - Fspec- Ech Where: BE= Electron Binding Energy KE= Electron Kinetic Energy Fspec= Spectrometer Work Function Ech= Surface Charge Energy Ech can be determined by electrically calibrating the instrument to a spectral feature. C1s at 285.0 eV Au4f7/2 at 84.0 eV 40
- 41. Where do BindingEnergy Shifts Come From? -or How Can We Identify Elements and Compounds? Electron-electron repulsion Electron-nucleus attraction Electron Nucleus Binding Energy Pure Element Electron- Nucleus Separation Fermi Level Look for changes here by observing electron binding energies 41
- 42. Elemental Shifts !"#$"#B&'#E)B*&+E,- '.E/E#0 1234132 ! !" #$# %C '%( F* ##+ '$ #,+ -. +%C '# #+' F/ 0CC #% +%+ N2 ,$33 +0 0CC !"#$%&'E)E*$"#*+,-%%&-$%.'E,/#"0+,*+,.1#E%.23,%/# #"#4#E%+ 42
- 43.
- 44. Binding Energy Determination Thephotoelectron’s binding energy will be based on the element’s final-state configuration. Conduction Band Valence Band Fermi Level Free Electon Level Conduction Band Valence Band 1s 2s 2p Initial State Final State 44
- 45.
- 46.