1. 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)

2. 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

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

4. 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)

5. 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 )

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

7. 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

8. Surface electrons on Si(111)-Ag
Doping by extra Ag atoms
Fermi “surface”
e-/atom:
Band dispersion

9. Fermi “surfaces” of two- and one-dimensional electronskx ky 2D
2D + super-lattice
 1D

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