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. Energy and momentum of the emitted photoelectron are measured. Energy conservation: Momentum conservation: Efinal = Einitial + h k||final = k||initial + G|| kphoton  0 h = h f = Ephoton Only k|| is conserved (surface!) Photon in Electron outside (final state) Electron inside (initial state)

Photoemission process: Photoemission setup: Photoemission process: Photoemission spectrum:  = h  W h e counts Efinal Core Valence D(E) Einitial EF+h W = width Secondary electrons

Measuring E(k) of Cu by angle-resolved photoemission Efinal  E ,   k|| h  k 2D surface state S1 is independent of k, h .

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)

Inverse Photoemission (BIS, IPES, KRIPES)  EFermi Measures an “empty state” by filling it with an electron Very low cross section (down from photoemission by 2  5·10-5 )

Surface states on Si(111)7x7 Rest Atom Adatom Photoemission Inverse photoemission Back bond + adsorbate

Two-Dimensional Electron Gas (2DEG) Angle-resolved photoemission measures E, kx, ky . Fermi “surface”: I(kx, ky) Band dispersion: I( E , kx) Free electron: E(k) = ħ2 k2/2m = Paraboloid Two cuts: kx ky E Fermi circle Band dispersion

Surface electrons on Si(111)-Ag Doping by extra Ag atoms Fermi “surface” e-/atom: 0.0015 0.012 0.015 0.022 0.086 Band dispersion

Fermi “surfaces” of two- and one-dimensional electrons kx ky 2D 2D + super-lattice  1D

E,k multidetection: Energy bands on a TV screen Calculated E(k) (eV) Measured E(k) E Ni EF EF 0.7 0.9 1.1 (Å-1) E,  multidetection: 50x50 = 2500 channels Electrons within 2 kBT of the Fermi level EF determine magnetism, superconductivity, … k