Electrons in Solids

1. 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

2. E Fermi circle Band dispersion kx ky Free Electron Gas Quantum numbers of electrons in a solid:E,kx,ky,(kz) Fermi “surface”:I(kx,ky) Band dispersion:I(E, kx) E(k) = ħ2 k2/2m= Paraboloid Two cuts:

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

4. Fermi Surface I(ky,kx) e-/atom: 0.0015 0.012 0.015 0.022 0.086 ky kx Band Dispersion I(E,kx) Measuring E,kx,ky in a Two-Dimensional Electron Gas

5. Superlattices of Metals on Si(111) 1 monolayer Ag is semiconducting: 3x3 Add extra Ag,Au as dopants: 21x21 Surface doping: 21014 cm-2 Equivalent bulk doping: 31021 cm-3

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

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

8. One-Dimensional Electrons at Semiconductor Surfaces

9. Beyond the Free Electron Gas

10. E(k) Band Dispersion

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

12. Two-dimensional bands of graphene E[eV]  Empty EFermi Occupied K =0 M K Empty * kx,y M K Occupied 

13. “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 parabolicrelationfor free electron bands). In a plot of E versus kx,ky one obtains cone-shaped energy bands attheFermi level.

14. 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).

15.  EFermi 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)

16. Measuring the quantum numbers E,k of electrons in a solid The quantum numbers E and k can be measured by angle-resolvedphotoemission. 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= EphotonOnly k|| isconserved (surface!) Photon inElectron outside (final state) Electron inside (initial state)

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

18. Spectrometer with E,kx -multidetection 50x50 = 2500 Spectra in One Scan

19. E,k Multidetection: Energy Bands on a TV Screen Calculated E(k) (eV) Measured E(k) E Ni EF= EF= (Å-1) 0.7 0.9 1.1 Electrons within±2kBTof the Fermi levelEFare not locked in by the Pauli principle. This is the width of the Fermi-Dirac cutoff at EF. These electrons determine magnetism, super-conductivity, specific heat in metals, … k Spin-split Bands in a Ferromagnet

20. http://uw.physics.wisc.edu/~himpsel/551/Lectures/Lecture Energy Bands.pdf