Electronic structure of Metals

1. Electronic structureofMetals

2. Free electron model Traditional model: Explains qualitatively the high electric conductivity. EK=3/2 n k T heat capacity = 3/2 n k Failure:no large heat capacities; no large susceptibility (expected for free charge carriers)

3. Quantized free electron • Particle in a box • Fermi-Dirac statistics • Work function • Photoelectric effect • Explains low magnetic susceptibility

5. Degeneracy As the number of atoms becomes very large (10^20): We need more levels to fill the electrons in. Degeneracy increases (no. of orbitals with the same energy) Better to work with the so-called wave vector k

7. All degenerate combinations of nx, ny, nz have the same energy with a specific value for the k-vector. • these combinations are at the same distance (k) from the origin. k

8. The highest energy level is called the Fermi level: EF • The Fermi energy is determined by the number of electrons N in the system • What is the relation between N and kF? • Only positive values for kx, ky, kz possible! • Count all energy states (all volume elements) in above volume. VOLUME: kF

9. Each volume element is • Number of volume elements (energy states): • What is the relation between N and kF? Spin degeneracy

10. Fermi velocity is the velocity of electrons at Fermi level • Few estimates for Na: • bcc; a=4.2Å • 1 valence electron per atom • Valence electron density= • Fermi enenrgy = 3.5 eV • Fermi velocity ≈108 cm/s

11. Density of state DOSnumber of states (N) with energy Ei.e. have energy between E and E+dE Number of states up to the energy E

12. At non-zero temperature: The Fermi-distribution is included

13. The problem of heat capacity • In the kinetic gas theory: Each monoatomic gas particle has the energy • Classical theory predicts this value for one mole of free electrons (1 mole of group AI metal) • Value measured is only 0.01 of expected • Solution: Pauli exclusion principle, Fermi distribution

14. When we heat a metal at zero Kelvin, not every electron can gain the energy kBT . • Only those electrons within the range EF,EF-kBT can become excited. • The fraction of particles that become excited • Number of excited electrons • The total electronic thermal • energy is • Gives the right heat capacity values

15. Fraction of electrons above EF in Ag at 300 K? • At 300 K, thermal energy kT = 4 x 10-21 J = 0.025 eV • Only electrons with this energy below EF will be affected • Fraction is roughly kT/EF = 0.46% (Exact value is 9kT/16EF) • Only small fraction contributes to conduction

16. Fermi-Dirac Statistics distribution of carriers on the available energy levels satisfies three conditions: • The Pauli exclusion principle: • The equilibrium condition: Minimumfree EnthalpyG • The conservation of particles (or charge) condition; i.e.constant number of carriers regardless of the distribution

18. TF=EF / kT at which a gas of classical particles would have to be heated to have the average energy per particle equal to the fermi energy at 0 K.

19. Still not satisfactory: • Can not answer • why are there metals, semiconductors and non metals? • why is the mean free path for some electron states very large? • Oversimplification • Potential in crystal not constant but periodic

20. Solution of Schroedinger equation yields: k: wave vector;│k│= 2p/l (Describes electron wave) When Bragg’s condition is met (at p/a for 1-dim. crystal, certain energy value are not obtainable (gap)

22. MO-Approach all states between a ± 2b (J ± 2K)

23. Na

24. Why is Be metallic?

27. Insulators and Semiconductors

28. Positive and negative charge carriers

29. n-type semiconductor

30. p-type semiconductor

31. Pure diamond:colorless B added Diamond: blue N added Diamond: yellow

32. Inorganic solids • GaP, GaAs: semiconductors, Isoelectronic with Si • KCl: insulator Isoelectronic with Si • Band model applicable??? NaCl Na+:1s2 2s2 2p6 empty 3s, 3p band Cl-: 1s2 2s2 2p6 3s2 3p6 full 3p band C-band E gap V-band Transition corresponds to e-transfer from Cl- to Na+

34. Correlation between electronegativity difference and gap energies Ehhomopolar band gap Egactual gap Ccharge transfer term C related to Dc

35. Color of some band-gap semiconductors

36. Yellow cadmium sulfide CdS, (E, = 2.6 eV), and black cadmium selenide CdSe (Eg = 1 .6 eV), which have the same structure and form a solid-solution series. Plate IX illustrates the yellow-orange-red-black sequence of these mixed crystals as the band-gap energy decreases. Mixed crystals such as Cd4SSe3 form the painter's pigment cadmium orange and are also used to color glass and plastic.

37. Transition metals t2g band dxy, dxz, dyz overlap s = 103W-1cm-1 pale green (d-d transition) s = 10-14 W-1cm-1

38. Guidelines • Good d-overlap occurs if: 1) Formal charge on cation is small TiO: metallic, TiO2: insulator; Cu2O, MoO2:semi; CuO, MoO3 insulator 2) Cation from early transition metals TiO: metallic, NiO, CuO: poor semiconductors Li-spinels 3) Cation in second or third transition series Cr2O3 poor conductor, Mo and W oxides: good conductors 4) Anion reasonably electropositive NiO: poor conductor, NiS, NiSe, NiTe: good conductors Effects 1, 2 and 3: keep d-orbitals spread out Effect 4: reduction of ionicity and band gap

39. Effect of crystal structure • Fe3O4: inverse spinel [Fe3+]tetr[Fe2+,Fe3+]octO4 almost metallic conductivity Oct. sites close to each other (edge sharing) charge migration (Fe2+ Fe3+) • Mn3O4: normal spinel[Mn2+]tetr[Mn3+2]octO4 tetr. and oct. sites far from each other, no charge transfer • Li-spinels:LiMn2O4[Li+]tetr[Mn3+,Mn4+]octO4semiconductor LiV2O4[Li+]tetr[V3+,V4+]octO4metallic

40. Molecular crystals VdW Forces