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, k x,k y, (k z ) Fermi “surface”: I(k x, k y ) Band dispersion: I( E, k x ) E(k) = ħ 2 k 2 /2m = Paraboloid kxkx kyky E Fermi circle Band dispersion Two cuts:

Fabricating a Two-Dimensional Electron Gas Lattice planes Inversion Layer Bulk Hamiltonian + boundary conditions Doped Surface State Surface Hamiltonian V(z)  (z)

Measuring E, k x,k y in a Two-Dimensional Electron Gas Fermi Surface I(k y,k x ) Band Dispersion I(E,k x ) e - /atom: kxkx kyky

Superlattices of Metals on Si(111) 1 monolayer Ag is semiconducting:  3x  3 Add extra Ag, Au as dopants:  21x  21 Surface doping: 2  cm -2 Equivalent bulk doping: 3  cm -3

Fermi Surface of a Superlattice Fermi circles are diffracted by the superlattice. Corresponds to momentum transfer from the lattice. kxkx Angle-resolved photoemission data kyky Model using Diffraction kxkx kyky

Fermi “surfaces” of two- and one-dimensional electrons kxkx kyky 2D 2D + super- lattice 1D

One-Dimensional Electrons at Semiconductor Surfaces

Beyond the Free Electron Gas

E(k) Band Dispersion

Band gap E (eV) Density of statesWave vector Empty lattice bands Band Dispersion in a Semiconductor [111] [100] [110]

Two-dimensional bands of graphene Occupied Empty E Fermi E [eV] K  =0 M K Empty  * Occupied  k x,y M 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 k x,k y 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 the 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).

Photoemission (PES, UPS, ARPES) Measures an “occupied state” by creating a hole Determines the complete set of quantum numbers Probes several atomic layers (surface + bulk)  E Fermi

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 Energy and momentum of the emitted photoelectron are measured. Energy conservation: Momentum conservation: E final = E initial + h k || final = k || initial + G || k photon  0 h = E photon Only k || is conserved (surface!) Photon in Electron outside (final state) Electron inside (initial state) Measuring the quantum numbers E,k of electrons in a solid

Photoemission setup: Photoemission process: Photoemission spectrum: h e  counts E final Core Valence D(E) E initial E final E F +h E final W = width Secondary electrons

Spectrometer with E,k x - multidetection 50 x 50 = 2500 Spectra in One Scan

E,k Multidetection: Energy Bands on a TV Screen Electrons within ± 2 k B T of the Fermi level E F are not locked in by the Pauli principle. This is the width of the Fermi-Dirac cutoff at E F. These electrons determine magnetism, super- conductivity, specific heat in metals, … Ni  Å   E k EF=EF= (eV) Spin-split Bands in a Ferromagnet Calculated E(k) Measured E(k) EF=EF=

/Lecture Energy Bands.pdf