Modern Methods in Heterogeneous Catalysis Research

Transkript

1 Modern Methods in Heterogeneous Catalysis Research Band Structures Literature: Bergmann Schäfer: Lehrbuch der Experimentalphysik Band 6 Festkörperphsik Charles Kittel: Einführung in die Festkörperphysik

2 Change of Energy Level from atom to solid

3 Band structure The following properties of solids can be explained on a microscopic level by the band structure: Electrical conductivity Thermal properties (heat capacity, heat conductivity) Optical properties Deducing of parameters like effective mass and density of states

4 Metal, Semiconductor, Isolator Bandgap E g > 3eV Bandgap 1eV < E g <3eV No bandgap

5 Electrical conductivity: metal - semiconductor Low temperature: defect scattering Higher temperature: phonon scattering

6

7 Fermi-Dirac-Verteilung Fermi-Verteilung am absoluten Temperaturnullpunkt Fermi-Verteilung bei endlichen Temperaturen Auflösungsvermögen: 1K entspricht 0.34 mev Auflösung ist mit 100 mev begrenzt bei RT

8 Density of states Metal Semiconductor

9 Doping of Semiconductors

10 pn transition

11 LED: Light Emitting Diode

12 Which materials are used? Material GaAlAs GaAsP, AlInGaP GaP SiC GaN, InGaN Colour Red and infrared, <1000 nm Red, orange and yellow green Blue, low efficiency UV, blue and green

13 How to create white light by LEDs?

14 Bandgaps Material Art Energie in ev 0 K 300 K Elemente Diamant indirekt 5,4 5,46 6,4 Si indirekt 1,17 1,12 Ge indirekt 0,75 0,67 Se direkt 1,74 IV-IV Verbindungen SiC 3C indirekt 2,36 SiC 4H indirekt 3,28 SiC 6H indirekt 3,03 II-VI Verbindungen TiO2 3,03 3,2 ZnO direkt 3,43 3,37 ZnS 3,6 CdS 2.42 CdSe 1.74 CdTe 1.45 Material III-V Verbindungen InP InAs InSb InN InxGa1- xn GaN GaP 3C GaSb GaAs AlxGa1- xas AlAs AlSb AlN Art direkt direkt direkt direkt direkt direkt indirekt direkt direkt x<0,4 direkt, x>0,4 indirekt indirekt indirekt Energie in ev 0 K 300 K 1,42 1,27 0,43 0,355 0,23 0,17 0,7 0,7 3,37 3,37 2,26 0,81 0,69 1,52 1,43 1,42 2,16 2,16 1,65 1,58 6,2

15 Schrödinger equation in one electron approximation: 2 h Δ + 2 m How to calculate band structures? V ψ = () r () r Eψ () r The potential is periodic with the elemental cell: V ( r + R) = V ( r) Bloch-Theorem: ψ n ( k, r + R) = ψ ( k, r) exp( ik R) n Bloch-Funktions: ψ n ( k, r) = u ( k, r) exp( ikr) ; u ( k, r + R) = u ( k, r) This equation has to be solved: n n n 2 h 2m Δ ik 2 h m + 2 h k 2m 2 + V () r u ( k, r ) E () k u ( k, r) n = n n

16 Brillouin-Cell of cubic crystals

17 Calculated Band Structures

18 Band structure calculation methods: Tight binding model : each electron is usually associated with only one atom at a time, and treats the other atoms in the solid as perturbations. Kroning-Penney-Model: atoms are barriers to electron motion, while the electrons are otherwise free and independent. While simple, it predicts many of important phenomena, but is not quantitatively accurate. Density functional theory: Traditional methods in electronic structure theory, in particular Hartree-Fock theory and its descendants, are based on the complicated many-electron wavefunction. The main objective of density functional theory is to replace the many-body electronic wavefunction with the electronic density as the basic quantity. Whereas the many-body wavefunction is dependent on 3N variables, three spatial variables for each of the N electrons, the density is only a function of three variables and is a simpler quantity to deal with both conceptually and practically

19 Photoelectron Spectroscopy

20 ARUPS (angle resolved UV photoelectron spectroscopy)

21 ARUPS: K װ is constant

22 ARUPS of Cu(111) k = (2m/ħ 2 )E kin sinθ

23 Inverse Photoemission Spectroscopy (IPES)

24 Photoemission and Inverse Photoemission of ZrO 2 /Si Theory resolution ΔE c = 1.15 ev ΔE g = 5.7 ev ΔE v = 3.40 ev ZrO 2 SiO 2 Si VBM, CBM Determination: Comparison with Theory (where possible) First Principles Theory VBM = E F ev First Principles Theory CBM = E F ev

25 IPES: Si(111)

26 Sample current spectroscopy

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28 Direct Indirect Semiconductor Die Absorption eines Photons ist nur bei einer direkten Bandlücke effektiv möglich, bei einer indirekten Bandlücke muss ein zusätzlicher Quasiimpuls (k) beteiligt werden, wobei ein passendes Phonon erzeugt oder vernichtet wird. Dieser Prozess mit einem Photon allein ist aufgrund des niedrigen Impulses des Lichts wesentlich unwahrscheinlicher, das Material zeigt dort eine schwächere Absorption.

29 Exciton An exciton is a bound state of an electron and an imaginary particle called an electron hole in an insulator or semiconductor, and such is a Coulomb correlated electron-hole pair. It is an elementary excitation, or a quasiparticle of a solid.

30 Exciton

31 Effective Mass In solid state physics, a particle's effective mass is the mass it seems to carry in the semiclassical model of transport in a crystal. It can be shown that, under most conditions, electrons and holes in a crystal respond to electric and magnetic fields almost as if they were free particles in a vacuum, but with a different mass. This mass is usually stated in units of the ordinary mass of an electron m e ( kg). Newton: F = ma a: acceleration h = h 2π reduced Planck s constant k: wave number or momentum since k = Ε(k) energy as a function of k or dispersion relation Effective mass: m * = h 2 d 2 dk Quantum mechanics: 2 1 d ε qe 2 h dk a = 2 p h ε 2 1

32 Cyclotron Resonance Cyclotron: Lorentz-Force: Cyclotron Frequence: ω = e c * m B

33 Landau-Level E 1 = h ω z n h 2 k 2m 2 z *

34 Material Si(4.2K) Ge GaAs InSb ZnO ZnSe Electron effective mass 1.08 me 0.55 me me me 0.19 me 0.17me Group IV III-V II-VI

35 The following methods to measure the band structure were introduced: Angle resolved UV Photoelectron Spectroscopy Inverse Photoelectron spectroscopy Sample current spectroscoy Optical spectroscopy Zyclotron Resonance