Electrons in Solids Simplest Model Free Electron Gas
- Slides: 20
Electrons in Solids Simplest Model: Free Electron Gas Quantum Numbers E, k Fermi “Surfaces” Beyond Free Electrons: Bloch’s Wave Function E(k) Band Dispersion Angle-Resolved Photoemission
Free Electron Gas Quantum numbers of electrons in a solid: E, kx , ky , (kz) Two cuts: Fermi “surface”: I(kx, ky) Band dispersion: I( E , kx) E(k) = ħ 2 k 2/2 m = Paraboloid E Fermi circle Band dispersion kx ky
Fabricating a Two-Dimensional Electron Gas Lattice planes V(z) Inversion Layer (z) Bulk Hamiltonian + boundary conditions V(z) Doped Surface State Surface Hamiltonian (z)
Measuring E, kx, ky in a Two-Dimensional Electron Gas Fermi Surface I(ky, kx) 0. 012 0. 015 0. 022 ky e-/atom: 0. 0015 kx Band Dispersion I(E, kx) 0. 086
Superlattices of Metals on Si(111) 1 monolayer Ag is semiconducting: 3 x 3 Surface doping: 2 1014 cm-2 Equivalent bulk doping: 3 1021 cm-3 Add extra Ag, Au as dopants: 21 x 21
Fermi Surface of a Superlattice Fermi circles are diffracted by the superlattice. Corresponds to momentum transfer from the lattice. ky Angle-resolved photoemission data ky kx Model using Diffraction kx
Fermi “surfaces” of two- and one-dimensional electrons ky 2 D 2 D + superlattice 1 D kx
One-Dimensional Electrons at Semiconductor Surfaces
Beyond the Free Electron Gas
E(k) Band Dispersion
Band Dispersion in a Semiconductor E (e. V) Band gap Empty lattice bands [111] Density of states [100] [110] Wave vector
Two-dimensional bands of graphene E [e. V] Empty EFermi Occupied Empty * K M Occupied K =0 M kx, y K
“Dirac cones” in graphene A special feature of the graphene -bands is their linear E(k) dispersion near the six corners K of the Brillouin zone (instead of the parabolic relation for free electron bands). In a plot of E versus kx, ky one obtains cone-shaped energy bands at the Fermi level.
Topological Insulators A spin-polarized version of a “Dirac cone” occurs in “topological insulators”. These are insulators in the bulk and metals at the surface, because two surface bands bridge the bulk band gap. It is impossible topologically to remove the surface bands from the gap, because they are tied to valence band on one side and to the conduction band on the other. The metallic surface state bands have been measured by angle- and spin-resolved photoemission (left). the
Photoemission (PES, UPS, ARPES) EFermi • Measures an “occupied state” by creating a hole • Determines the complete set of quantum numbers • Probes several atomic layers (surface + bulk)
Measuring the quantum numbers E, k of electrons in a solid The quantum numbers E and k can be measured by angle-resolved photoemission. This is an elaborate use of the photoelectric effect, which was explained as quantum phenomenon by Einstein in 1905. Photon in Electron outside (final state) Electron inside (initial state) Energy and momentum of the emitted photoelectron are measured. Energy conservation: Momentum conservation: Efinal = Einitial + h k||final = k||initial + G|| h = Ephoton Only k|| is conserved (surface!) kphoton 0
Photoemission setup: D(E) Einitial Efinal h Photoemission process: Efinal Photoemission spectrum: e counts EF+h Core Valence Secondary electrons W = width Efinal
Spectrometer with E, kx - multidetection 50 x 50 = 2500 Spectra in One Scan
E, k Multidetection: Energy Bands on a TV Screen E Calculated E(k) Ni (e. V) Measured E(k) EF = 0. 7 0. 9 1. 1 (Å-1) Electrons within ± 2 k. BT of the Fermi level EF are not locked in by the Pauli principle. This is the width of the Fermi-Dirac cutoff at EF. k Spin-split Bands in a Ferromagnet These electrons determine magnetism, superconductivity, specific heat in metals, …
http: //uw. physics. wisc. edu/~himpsel/551/Lectures /Lecture Energy Bands. pdf
- Free electron gas in three dimensions
- Fermi-dirac distribution function at different temperatures
- Drawbacks of free electron theory
- Kronig-penney model energy bands
- Drude free electron theory
- Unpaired electrons in electron configuration
- Valence electrons in rubidium
- Electrons in atoms section 3 electron configuration
- Electron configuration for oxygen
- Difference between dielectric and conductor
- Which substance contains free-floating valence electrons?
- Electron cloud model vs bohr model
- What is the noble gas notation
- Degenerate electron gas
- Electron configuration vs noble gas configuration
- Where do electrons live
- Noble gases configuration
- Gas electron multiplier
- Quasi free electron
- Success of classical free electron theory
- Phonon gas model