FYS3410 - Vår 2017 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/FYS3410/v16/index.html

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: +47-22857762,

e-post: andrej.kuznetsov@fys.uio.no

visiting address: MiNaLab, Gaustadaleen 23a

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 17/1 12-15 Introduction. Crystal bonding. Periodicity and lattices. Lattice planes and Miller indices. Reciprocal space. 3h

W 18/1 09-10 Bragg diffraction and Laue condition 1h

T 24/1 12-14 Ewald construction, interpretation of a diffraction experiment, Bragg planes and Brillouin zones 2h

W 25/1 08-10 Surfaces and interfaces. Elastic strain in crystals 2h

T 31/1 12-14 Point defects and atomic diffusion in crystals 2h

W 01/2 08-10 Summary of Module I 2h

Module II – Phonons (Chapters 4, 5, and 18 pp.557-561)

T 07/2 12-14 Vibrations in monoatomic and diatomic chains of atoms; examples of dispersion relations in 3D 2h

W 08/2 08-10 Periodic boundary conditions (Born – von Karman); phonons and its density of states (DOS) 2h

T 14/2 12-14 Effect of temperature - Planck distribution; lattice heat capacity: Dulong-Petit, Einstein, and Debye models 2h

W 15/2 08-10 Comparison of different lattice heat capacity models 2h

T 21/2 12-14 Thermal conductivity and thermal expansion 2h

W 22/2 08-10 Vibrational and thermal properties of nanostructures 2h

T 28/2 12-14 Summary of Module II 2h

Module III – Electrons (Chapters 6, 7, 11 - pp 315-317, 18 - pp.528-530, and Appendix D)

W 01/3 08-10 Free electron gas (FEG) versus free electron Fermi gas (FEFG) 2h

T 07/3 12-14 DOS of FEFG in 3D; Effect of temperature – Fermi-Dirac distribution; heat capacity of FEFG in 3D 2h

W 08/3 08-10 Transport properties of electrons electrons – examples for thermal, electric and magnetic fields 2h

T 14/3 12-14 DOS of FEFG in 2D - quantum wells 2h

W 15/3 08-10 DOS in 1D – quantum wires, and in 0D – quantum dots 2h

T 21/3 12-14 Origin of the band gap; Nearly free electron model 2h

W 22/3 08-10 Kronig-Penney model; Empty lattice approximation; Number of orbitals in a band 2h

T 28/3 12-14 no lecture

W 29/3 08-10 no lecture

T 4/4 12-14 Effective mass method 2h

W5/4 08-10 Summary of Module III 2h

Easter break

Module IV – Semiconductors and Metals (Chapters 8, 9 pp 223-231, and 17)

T 18/4 12-14 Approaches for energy band calculations 2h

W 19/4 08-10 Fermi surfaces and metals 2h

T 25/4 12-14 Intrinsic carrier generation in semiconductors – elctrons and holes 2h

W 26/4 08-10 Localized levels for hydrogen-like impurities in semiconductors – donors and acceptors. Doping. 2h

T 02/5 12-14 Carrier statistics in semiconductors; p-n junctions and metal-semiconductor contacts 2h

W 03/5 08-10 Optical properties of semiconductors and optoelectronic device operation demos with Randi Haakenaasen 2h

T 09/5 12-14 Summary of Module IV 2h

Summary and repetition

T 16/5 12-14 Repetition - the course in a nutshell 2h

Exam

Week 22 , June 1-2, your presence is required for 1 h – please book your time in advance

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

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

Reduced Zone Scheme

Reduced Zone Scheme: k 1st BZ.

k is outside 1st BZ.

k = k + G is inside.

Periodic Zone Scheme

εk single-valued

εk multi-valued

εnk single-valued

εnk = εnk+G periodic

E.g., s.c. lattice, TBA

Construction of Fermi Surfaces

Zone boundary:

3rd zone: periodic zone scheme

Harrison construction of free electron Fermi surfaces

Points lying within at least n spheres are in the nth zone.

Nearly free electrons:

Energy gaps near zone boundaries → Fermi surface edges “rounded”.

Fermi surfaces & zone boundaries are always orthogonal.

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

Electron Orbits, Hole Orbits, and Open Orbits

Electrons in static B field move on intersect of plane B & Fermi surface.

Nearly filled corners:

P.Z.S.

P.Z.S.

Simple cubic

TBM

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

Tight Binding Method for Energy Bands

2 neutral H atoms

Ground state of H2 Excited state of H2

1s band of 20 H atoms ring.

Wigner-Seitz result for

3s electrons in Na.

Wigner-Seitz B.C.:

d /d r = 0 at cell boundaries.

Table 3.9, p.70 ionic r = 1.91A

r0 of primitive cell = 2.08A n.n. r = 1.86A

is constant over 7/8 vol of cell.

Wigner-Seitz Method

Cohesive Energy

linear chain

Na

5.15 eV for free atom.

0 ~ 8.2 eV for u0 .

+2.7 eV for k at zone boundary.

Table 6.1, p.139: F ~ 3.1 eV.

K.E. ~ 0.6 F ~ 1.9 eV.

~ 8.2+1.9 ~ 6.3 eV

Cohesive energy ~ 5.15 +6.3 ~ 1.1 eV

exp: 1.13 eV

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)

Rc = 1.66 a0 .

U ~ –50.4 ~ 200 Ups at r = 0.15

With Thomas-Fermi screening.

Typical reciprocal space Ups

Empirical Pseudopotential Method

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

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

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

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)

B = 0

Allowed

levels

See Landau & Lifshitz, “QM: Non-Rel Theory”, §112.

B 0

For the sake of clarity, n of the occupied states in the circle diagrams is 1 less than that in the level diagrams.

Number

of e = 48

D = 16 D = 19 D = 24

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 = Bs . Red lines are plots of n = s N / ( N / ρ B ),

n = N = 50 at N / ρ B = s .

Fermi Surface of Copper

Cu / Au

Monovalent fcc metal: n = 4 / a3

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