Quantum Confinement in Nanostructures Confined in: 1 Direction: Quantum well (thin film) Two-dimensional electrons 2 Directions: Quantum wire One-dimensional electrons 3 Directions: Quantum dot Zero-dimensional electrons Each confinement direction converts a continuous k in a discrete quantum number n. kxkx nznz nyny nyny nznz nxnx kxkx kyky nznz

N atomic layers with the spacing a = d/n N quantized states with k n ≈ n   /d ( n = 1,…,N ) Quantization in a Thin Crystal An energy band with continuous k is quantized into N discrete points k n in a thin film with N atomic layers. n = 2d / n k n = 2  / n = n   /d d E 0  /a  /d E Fermi E Vacuum Photoemission Inverse Photoemission Electron Scattering kk = zone boundary

N atomic layers with spacing a = d/n :  N quantized states with k n ≈ N   /d Quantization in Thin Graphite Films E 0  /a  /d E Fermi E Vacuum Photoemission Lect. 7b, Slide 11 kk 1 layer = graphene 2 layers 3 layers 4 layers  layers = graphite

Quantum Well States in Thin Films discrete for small N becoming continuous for N   Paggel et al. Science 283, 1709 (1999)

Periodic Fermi level crossing of quantum well states with increasing thickness Counting Quantum Well States Number of monolayers N n n

Kawakami et al. Nature 398, 132 (1999) Himpsel Science 283, 1655 (1999) Quantum Well Oscillations in Electron Interferometers Fabry-Perot interferometer model: Interfaces act like mirrors for electrons. Since electrons have so short wavelengths, the interfaces need to be atomically precise. n

The Important Electrons in a Metal Energy  E Fermi Energy Spread  3.5 k B T Transport (conductivity, magnetoresistance, screening length,...) Width of the Fermi function: FWHM  3.5 k B T Phase transitions (superconductivity, magnetism,...) Superconducting gap: E g  3.5 k B T c (T c = critical temperature)

Energy Bands of Ferromagnets States near the Fermi level cause the energy splitting between majority and minority spin bands in a ferromagnet (red and green). Ni Energy Relative to E F [eV] k || along [011] [Å -1 ] Calculation Photoemission data

(Qiu, et al. PR B ‘92) Quantum Well States and Magnetic Coupling The magnetic coupling between layers plays a key role in giant magnetoresistance (GMR), the Nobel prize winning technology used for reading heads of hard disks. This coupling oscillates in sync with the density of states at the Fermi level.

Minority spins discrete, Majority spins continuous Magnetic interfaces reflect the two spins differently, causing a spin polarization. Spin-Polarized Quantum Well States

Filtering mechanisms Interface: Spin-dependent Reflectivity  Quantum Well States Bulk: Spin-dependent Mean Free Path  Magnetic “Doping” Parallel Spin Filters  Resistance Low Opposing Spin Filters  Resistance High Giant Magnetoresistance and Spin - Dependent Scattering

Giant Magnetoresistance (GMR): (Metal spacer, here Cu) Tunnel Magnetoresistance (TMR): (Insulating spacer, MgO) Magnetoelectronics Spin currents instead of charge currents Magnetoresistance = Change of the resistance in a magnetic field