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• lectures accompanying the book: Solid State Physics: An Introduction, by Philip Hofmann (2nd edition 2015, ISBN10: 3527412824, ISBN-13: 978-3527412822, Wiley-VCH Berlin. www.philiphofmann.net 1 Finite solids / nano-structures • Small is different: • quantum confinement • high relative surface area • Magnetism in nano-structures • Interesting new physics in two and one dimensions 2 Small is different change of bulk properties increased importance of surfaces Periodic boundary conditions Max Born and Theodore von Karman (1912) • • We want to get rid of the surface restrictions, i.e. we want a solid which is finite in size but has no surfaces (!). If we move by one crystal size L, we have to get the same. The surface was invented by the devil - Wolfgang Pauli 4 Periodic boundary conditions chain with lattice spacing a and macroscopic side length L=Na: 1D cubic crystal with lattice spacing a and macroscopic side length L=na: 3D 5 Finite chain with 10 unit cells and one atom per unit cell 6 • N atoms give N so-called normal modes of vibration. • For long but finite chains, the points are very dense. Quantum-size effects thick film (particle in a potential well) λ1 λ2 d Quantum size effects Thin films: free electrons in the other direction 9 • parabolic dispersion parallel to the film • quantization in film direction Quantum size effects for an infinite well, the wave function has to vanish at the boundaries for a finite well Quantum size effects in thin Bi films small very small small very long (hundreds of Å) Magic film height n layers low g(EF) n+2 layers n+1 layers low g(EF) high g(EF) substrate 12 n+1 layers high g(EF) n layers low g(EF) Magic film height n layers low g(EF) n layers low g(EF) n+2 layers low g(EF) substrate 13 n layers low g(EF) Semiconductor nanoparticles bulk semiconductors: smallest energy to create e-h pair semiconductor nano-crystal quantum confinement 14 Coulomb interaction New genuine surface states if k was complex at the surfaces Electronic structure of Bi Debye temp. density of states el-ph interaction strength Superconductivity in Bi nanoclusters • • • • Bulk Bi is not superconducting. Granular films of Bi clusters are superconducting at several K. T is increasing for decreasing cluster c size. T depends on the environment. c Superconductivity caused by metallic surface? - Electronic structure of Bi(110) Superparamagnetism • Bulk magnet: flipping one spin costs energy ≈ kBθC. • Nano-magnet: all the spins in the single domain can be rotated at the same time. This costs much less energy.