Fys 208, Solid State

Notes about the lectures, Spring Semester, 2001

Updated 21.05.01
          Questions for exercises from 1998  
          quest98.ps (Postscript file)


22.01.01  Introduction.
          Walking through the topics, about molecular physics
          Interesting phenomena
          Quantum Mechanics, Statistical Physics
          Structure

24.01.01  Quantum Mechanics Highlights
          Density - squre modulus of wave function
          Expectation values in classical probability
          Quant.Mech. expectation values
          To every quantity - an operator
          Hamilton operator
          H Psi = E Psi
          Eigenvalues - discrete solutions (sometimes continuous)

          Commutator
          Heisenberg (Heisenberg in acoustics)
          (Born in optics ..... sideremarks)

          Harmonic Oscillator - Newton
          Harmonic Oscillator - Quant. Ladder spectrum (equidistant)
          hbar omega (circular frequency vs. frequency)

          Sideremarks, illustrations:
               Rydberg atoms - see later impurity states in semiconducters

          Crystals as huge molecules
               Born-Oppenheimer approximation
                  Fix the nuclei, do quant. mech. for electrons
                  Electron energies depend on position of nuclei
                  (for 2 atomic nuclei just the distance)

          Splitting of levels, origins of bands
                  This might be seen as origin of the bonding
                  The lower level goes down (the united nuclei)

29.01.01  The linear chain of masses on a spring
          Dispersion relation
          Ashcroft-Mermin chapter 22 pages 430-433
          Mentioned phase velocity, group velocity (will be done later)
          relation between omega and k - the 'wave vector'

          Relation for the density of modes (a note will be prepared) -NOTE
              Densities (for lattice vibrations; explanation to be added)
              modedens.gif
              modedens.m

          String with two types of atoms (two masses, two branches)
          Ashcroft-Mermin chapter 22 pages 433-435,
          see fig. 22.10; two spring constants, not two masses

31.01.01  Vibrations of 2-dim and 3-dim lattices
          Density of modes goes as omega squared
          Einsteins model for heat capacity
          (note, picture of the page, jpg 148kB)
          calculation details, jpg 81kB


05.02.01  Debye's model calculation  note, jpg 127kB
          Table of Debye temperatures table30.jpg, 80kB
          High resolution, same table table50.jpg 150 kB
          Origin of the Boltzmann factor NOTE
          Einstein vs. Debye model: Ashcroft-Mermin page 462
	  .
	  
07.02.01  Elementary transport theory
          Fourier's law - heat conductivity
	  (deriving expression for kappa in Fourier's law)
	  Group velocity

	  Algebraic Method - harmonic oscillator (for phonons)   *NOTE*
	  Eigenmodes (diagonalization of frequencies square...)  *NOTE*

12.02.01  Computer (Matlab) Simulations
	  Chains - showing the addition of wave numbers
	  types of vibrations - waves - demonstration
	  Diagonalization of the coupled Harm.Osc. matrix

	  back to phonons
	  mean free path
	  
	  Umklapp process - see chains above
	  Unharmonic effects: terms of u^3 type
	  Heat conductivity: T-dependence ( T^3 low T, T^(-1) high)

	  Thermal expansion
          .


End of first part

14.02.01  Heat and electrical conductivity of electrons
          CLASSICAL MODELS
          a) HEAT CONDUCTIVITY - see phonons
          b) electrical conductivity
          Drift velocity from acceleration
          Conductivity (relation to resistance)
          Conductivity (sigma) for Cu as an example
          Wiedemann-Franz law derived
          Lorenz number
          CLASSICAL ASSUMPTIONS:
          a) the gas heat capacity (related to Boltzmann)
          b) velocity squared replaced by kT

19.02.01  FERMI GAS MODEL
          Pauli principle
          Independent particles-independent probabilities
          Indistinguishable particles
          -- new form of probability P(r1,r2)=P(r2,r1)
          Slater Determinant
          Available states (density of states in k-space)
          NOTE (scanned) fermi-density.jpg
          Fermi momentum, Fermi energy, Fermi temperature

          Lu: No two leaves are the same (Chinese saying)
          (just opposite of the Indistinguishable particles)

21.02.01  Fermi gas - at nonzero T
          Heat capacity etc  (1 hour only)
          Sommerfeld integrals: Fermi Gas
          In Ashcroft-Mermin: Sommerfeld integrals are pages 45-47 
          FermiSom.ps  (postscript file, overview)
          Scanned:
          sommer-meth-1.jpg (Sommerfeld's method)
          sommer-meth-2.jpg
          sommer-eval.jpg (Evaluation details)

21.02.01  Fermi gas - at nonzero T
	  Heat capacity etc  (1 hour only)
          Wiedemann-Franz law in Fermi-gas version

26.02.01  Bloch theorem

28.02.01  Bloch theorem via Fourier
          NOTE: Bloch.ps (postscript file)

05.03.01  Fourier, bands weak potential limit
          periodicity in the k-space

07.03.01  started semiclassical motion (forces) (1 hour only)
          equation for u-function (see note)

12.03.01  semiclassical motion
          Selfconsistent method (background information - see textbook)

14.03.01  3-dim crystals electronic bands
          the shape of bands
          Notes for the model construction of 3-dim bands
            3dim-1.jpg
            3dim-2.jpg
            3dim-3.jpg

19.03.01  -- no lecture
21.03.01  -- no lecture

26.03.01  Semiconductors
          Energy Gap: diagram of resistivity, photon excitation
          Evaluation of number of electrons in bands (textbook)
          Semiconductor equation (textbook)


28.03.01  Semiconductor equation (textbook)
          Impurity states (textbook)
            a) Donors (P in Si)
               (the 'quasiatoms'; see discussion of Rydberg atoms,
                sideremark 24.01.01)
            b) Acceptors (Al in Si)
               HOLES and ELECTRONS

02.04.01  p-n junction 7 page NOTE
          The placement of Fermi level in acceptor and donor doped
          semiconductors. The mechanisms leading to the common
          position of Ferni level.
          Diffusion current and drift current.
          Fick's law for diffusion. Diffusion constant.
          Mobility.
          Einstein-Nernst relation

04.04.01  p-n junction 7 page NOTE, continued
          Model for charge density distribution
          Model for the depletion zone
          The rectifying function of p-n junction

          Questions for exercises from 1998  
          quest98.ps (Postscript file)

18.04.01  Metals, Fermi surface, thermoelectricity ....
          Explaining the fermi surface in 2 dimensions
          Electron motion in magnetic field
          Cyclotron Frequency ... add formulae here (new *NOTE* ... missing)
          add references to Ashcroft-Mermin

          Metals as poly-micro-crystalline
          Matthiesens rule for conductivity (temperature dependence)
          .
          To be added: typical values of v_F, l, tau (relax. time)
          to be added: Matth. rule relations
          ( rho sigma l) (sigma cross section)
          (sigma conductivity; sigma dep. on tau, tau dep. 1/T)
          (sigma dep. sigma_0 / ( CONST + T )
          .
23.04.01  Hall effect   NOTE Hall effect, 2 pages
          Hall effect in semiconductors, mobility: Not in Ashcroft-Mermin
          .
          Magnetic field/Magnetism basics    NOTE (background information 4 pages)
          (Table of systems of units;  JACKSON's book)
          System of units (physics the same, but even concepts different)
          .
25.04.01  Magnetic properties of materials
          Diamagnetism and Paramagnetism      NOTE  3 pages
          Diamagnetism-atomic origin
          Understanding of negative susceptibility
          Ashcroft-Mermin chapter 31 pages 648-649
          .
30.04.01  Paramagnetic response
          The g-factor
          Thermic effects, evaluation of average spin projection
          Brillouin function for general J and for J=1/2
          Ashcroft-Mermin chapter 31 pages 653-656
          Paramagnetism of metals (picture explains easily)
          Ashcroft-Mermin chapter 31 pages 661-662
          (Ash.Mer. too detailed on this point)
          .
02.05.01  Ferromagnetism          NOTE 3 pages
          The effective spin-spin interaction
          Mean Field theory
          (the explanation of permanent magnetization)
          .
07.05.01  Superconductivity
          .
          Ashcroft-Mermin chapter 34.
09.05.01  Crystal systems, Crystal Structure
          Crystal lattices
          .Ashcroft-Mermin chapter 4, chapter 7; parts only
	  Determination of crystal structures
	  Ashcroft-Mermin chapter 6, parts only
          
          .
          .