FYS Vår 2017 (Kondenserte fasers fysikk)

Transcription

1 FYS Vår 2017 (Kondenserte fasers fysikk) Pensum: Introduction to Solid State Physics by Charles Kittel (Chapters 1-9, 11, 17, 18, 20) Andrej Kuznetsov delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO Tel: , e-post: andrej.kuznetsov@fys.uio.no visiting address: MiNaLab, Gaustadaleen 23a

2 2017 FYS3410 Lectures and Exam (based on C.Kittel s Introduction to SSP, Chapters 1-9, 11, 17,18,20) Module I Periodic Structures and Defects (Chapters 1-3, 20) T 17/ Introduction. Crystal bonding. Periodicity and lattices. Lattice planes and Miller indices. Reciprocal space. W 18/ Bragg diffraction and Laue condition T 24/ Ewald construction, interpretation of a diffraction experiment, Bragg planes and Brillouin zones W 25/ Surfaces and interfaces. Elastic strain in crystals T 31/ Point defects and atomic diffusion in crystals W 01/ Summary of Module I Module II Phonons (Chapters 4, 5, and 18 pp ) T 07/ Vibrations in monoatomic and diatomic chains of atoms; examples of dispersion relations in 3D W 08/ Periodic boundary conditions (Born von Karman); phonons and its density of states (DOS) T 14/ Effect of temperature - Planck distribution; lattice heat capacity: Dulong-Petit, Einstein, and Debye models W 15/ Comparison of different lattice heat capacity models T 21/ Thermal conductivity and thermal expansion W 22/ Vibrational and thermal properties of nanostructures T 28/ Summary of Module II Module III Electrons (Chapters 6, 7, 11 - pp , 18 - pp , and Appendix D) W 01/ Free electron gas (FEG) versus free electron Fermi gas (FEFG) T 07/ DOS of FEFG in 3D; Effect of temperature Fermi-Dirac distribution; heat capacity of FEFG in 3D W 08/ Transport properties of electrons electrons examples for thermal, electric and magnetic fields T 14/ DOS of FEFG in 2D - quantum wells W 15/ DOS in 1D quantum wires, and in 0D quantum dots T 21/ Origin of the band gap; Nearly free electron model W 22/ Kronig-Penney model; Empty lattice approximation; Number of orbitals in a band T 28/ no lecture W 29/ no lecture T 4/ Effective mass method W5/ Summary of Module III Easter break Module IV Semiconductors and Metals (Chapters 8, 9 pp , and 17) T 18/ Approaches for energy band calculations W 19/ Fermi surfaces and metals T 25/ Intrinsic carrier generation in semiconductors elctrons and holes W 26/ Localized levels for hydrogen-like impurities in semiconductors donors and acceptors. Doping. T 02/ Carrier statistics in semiconductors; p-n junctions and metal-semiconductor contacts W 03/ Optical properties of semiconductors and optoelectronic device operation demos with Randi Haakenaasen T 09/ Summary of Module IV Summary and repetition T 16/ Repetition - the course in a nutshell Exam Week 22, June 1-2, your presence is required for 1 h please book your time in advance 3h 1h

3 Fermi Surfaces and Metals Construction of Fermi Surfaces Electron Orbits, Hole Orbits, and Open Orbits Calculation of Energy Bands Experimental Methods in Fermi Surface Studies

4 Fermi Surfaces and Metals Construction of Fermi Surfaces Electron Orbits, Hole Orbits, and Open Orbits Calculation of Energy Bands Experimental Methods in Fermi Surface Studies

5 Reduced Zone Scheme: k 1 st BZ. Reduced Zone Scheme k is outside 1 st BZ. k = k + G is inside.

6 Periodic Zone Scheme ε k single-valued ε k multi-valued ε nk single-valued ε nk = ε nk+g periodic E.g., s.c. lattice, TBA

7 Zone boundary: Construction of Fermi Surfaces

8 3 rd zone: periodic zone scheme

9 Harrison construction of free electron Fermi surfaces Points lying within at least n spheres are in the n th zone.

10 Nearly free electrons: Energy gaps near zone boundaries Fermi surface edges rounded. Fermi surfaces & zone boundaries are always orthogonal.

11 Fermi Surfaces and Metals Construction of Fermi Surfaces Electron Orbits, Hole Orbits, and Open Orbits Calculation of Energy Bands Experimental Methods in Fermi Surface Studies

12 Electron Orbits, Hole Orbits, and Open Orbits Electrons in static B field move on intersect of plane B & Fermi surface.

13 Nearly filled corners: P.Z.S. Simple cubic TBM P.Z.S.

14 Fermi Surfaces and Metals Construction of Fermi Surfaces Electron Orbits, Hole Orbits, and Open Orbits Calculation of Energy Bands Experimental Methods in Fermi Surface Studies

15 Tight Binding Method for Energy Bands 2 neutral H atoms Ground state of H 2 Excited state of H 2 1s band of 20 H atoms ring.

16 Wigner-Seitz Method Wigner-Seitz B.C.: d /d r = 0 at cell boundaries. Wigner-Seitz result for 3s electrons in Na. Table 3.9, p.70 ionic r = 1.91A r 0 of primitive cell = 2.08A n.n. r = 1.86A is constant over 7/8 vol of cell.

17 Cohesive Energy linear chain Na Table 6.1, p.139: F ~ 3.1 ev. K.E. ~ 0.6 F ~ 1.9 ev ev for free atom. 0 ~ 8.2 ev for u ev for k at zone boundary. ~ ~ 6.3 ev Cohesive energy ~ ~ 1.1 ev exp: 1.13 ev

18 Pseudopotential Methods Conduction electron ψ plane wave like except near core region. Reason: ψ must be orthogonal to core electron atomic-like wave functions. Pseudopotential: replace core with effective potential that gives true ψ outside core. Empty core model for Na (see Chap 10) R c = 1.66 a 0. U ~ 50.4 ~ 200 U ps at r = 0.15 With Thomas-Fermi screening.

19 Typical reciprocal space U ps Empirical Pseudopotential Method

20 Fermi Surfaces and Metals Construction of Fermi Surfaces Electron Orbits, Hole Orbits, and Open Orbits Calculation of Energy Bands Experimental Methods in Fermi Surface Studies

21 Experimental Methods in Fermi Surface Studies Experimental methods for determining Fermi surfaces: Magnetoresistance Anomalous skin effect Cyclotron resonance Magneto-acoustic geometric effects Shubnikov-de Haas effect de Haas-van Alphen effect Experimental methods for determining momentum distributions: Positron annihilation Compton scattering Kohn effect

22 Experimental methods for determining Fermi surfaces: Magnetoresistance Anomalous skin effect Cyclotron resonance Magneto-acoustic geometric effects Shubnikov-de Haas effect de Haas-van Alphen effect Experimental methods for determining momentum distributions: Positron annihilation Compton scattering Kohn effect Metal in uniform B field 1/B periodicity

23 De Haas-van Alphen Effect dhva effect: M of a pure metal at low T in strong B is a periodic function of 1/B. 2-D e-gas: PW in (B) dir. # of states in each Landau level (spin neglected) See Landau & Lifshitz, QM: Non-Rel Theory, 112. Allowed levels B = 0 B 0

24 Number of e = 48 D = 16 D = 19 D = 24 For the sake of clarity, n of the occupied states in the circle diagrams is 1 less than that in the level diagrams.

25 Critical field (No partially filled level at T = 0): s = highest completely filled level Black lines are plots of n = s ρ B, n = N = 50 at B = B s. Red lines are plots of n = s N / ( N / ρ B ), n = N = 50 at N / ρ B = s.

26 Fermi Surface of Copper Cu / Au Monovalent fcc metal: n = 4 / a 3 Shortest distance across BZ = distance between hexagonal faces Band gap at zone boundaries band energy there lowered necks Distance between square faces 12.57/a : necking not expected