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Chapter 5 Metals: Energy Bands

Chapter 5 Metals: Energy Bands. -. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. Free Electron Model?. V = 0. g (e). g (e). e. n. e. n F. e F. 3. 2. 1. Free Electron Model. Energy gap. Energy band.

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Chapter 5 Metals: Energy Bands

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  1. Chapter 5 Metals: Energy Bands

  2. - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Free Electron Model? V = 0

  3. g(e) g(e) e n e nF eF 3 2 1 Free Electron Model Energy gap Energy band

  4. Contents 1. Introduction 2. Energy spectra in atoms, molecules, and solids 3. Nearly Free Electron Model 4. Wave Equation of Electron in a Periodic Potential 5. Density of States and Fermi Surface 6. Velocity of Bloch Electron and Effective Mass 7. Electrical Conductivity

  5. + + - - - K L M - - - M 5.2. Energy spectra in atoms, molecules, and solids - Energy spectrum of a Li atom : 1S2 2S1 V(x) V x 2P 2S 1S Atomic orbitals

  6. 2P 2P 2P 2P 2P 2P 2P 2P 2P 2P 2P 2S 2S 2S 2S 2S 2S 2S 2S 2S 2S 2S + + + + + + + + + + + + 1S 1S 1S 1S 1S 1S 1S 1S 1S 1S 1S - - Energy spectrum of a Li molecule (Li2) Discrete doublets

  7. H2 molecule H2+ ion V(x) r x    x The more overlapped, V  But the more overlapped, Er  Li2 molecule s* antibonding or antisymmetric orbital 2s* 2S r s bonding or symmetric orbital 2s   Molecular orbitals

  8. + + 2P As (orbital E) , splitting  2S 1S splitting: small 1S → Li2[(1s)2(1s*)2(2s)2] Li[1S22S1] + Li[1S22S1]

  9. antibonding molecular orbital atomic orbital Bonding molecular orbital - Energy spectrum of Li solid 2P 2S 1Li 2Li 3Li 4Li 5Li NLi N closely spaced sublevels 2P Energy band 2S

  10. x 2Pss* 2Sss* 2P 2S 1ss* + + + 1S Atom Molecule (Li2) Crystal orbitals Solid wavefunctions Solid delocalized orbitals 2P gap 2S energy gap 1S

  11. x Li metal 2P 2P 2P 2S 2S 2S + + + 1S 1S Atom Molecule (Li2) Solid 2P No energy gap 2S 1S

  12. - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 5.3 Nearly Free Electron (NFE) Model Free electron model Free electron wavefunctions, Cf: wave equation for traveling wave → Band electrons Perturbed only weakly by the periodic potential of the ions cores

  13. Free electron model e k What happens at ? Standing waves! 1st BZ boundaries No solution of Schrödinger Eq.

  14. t1 t2 t3 t4 t5 What happened in phonons on the boundary of the 1st BZ? Standing wave! k = p/aor – p/a traveling wave x standing wave x

  15. Electron waves at In phase traveling wave Incident wave t1 x t2 Reflected wavelets Standing wave

  16. Standing waves at Incident wave t1 t2 2 different standing waves formed! x Reflected wavelets

  17. 2 different standing waves formed! Different directions! No sign change for x → -x Sign change for x → -x

  18. g(e) e Energy gap Energy band - - Origin of the energy gap Probability density of an electron: For pure traveling wave, For standing waves, Traveling waves x V(x)

  19. - - : equal portions Traveling waves x V(x) Er(+) < E (traveling electron) Er(+) < Er(-) Er(-) > E (traveling electron) Eg = Er(-) - Er(+)

  20. Free electron model NFE model e e 1st BZ Eg k k 0 -p/a 0 p/a Eg = Er(-) - Er(+)at k = ±p/a Origin of the Eg →2 standing waves: piling up electrons at different regions NFE model →rough approximation for simple metals (Na, K, Al, etc)

  21. 5.4 Wave Equation of Electron in a Periodic Potential 5.4.1 Bloch Functions Eigenvalue Eq.: Eigenvalue Eigenvector or Eigenfunction Periodic zone scheme Reduced zone scheme Extended zone scheme 5.4.2 Wave Equation of Electron in a Periodic Potential

  22. Schrödinger eq. (1.14) - 5.4 Wave Equation of Electron in a Periodic Potential 5.4.1 Bloch Functions V(r) Schrödinger Eq.: periodic potential crystal potential : a lattice vector (Bloch function) Bloch theorem:  One electron wavefunction (translational symmetry)

  23. (Bloch function) V(r) Bloch function: a crystal orbital, delocalized throughout the solid : periodic in the crystal

  24. 5.4.2 Wave Equation of Electron in a Periodic Potential → Energy bands in solids

  25. Li metal 2P 2S Schrödinger Eq.: (Homework) Eigenvector or Eigenfunction Eigenvalue Eq.: Eigenvalue e(k) E3,k Many solutions E gap 3rd band → Energy Eigenvalue E2,k 2nd band E1,k (n: band index) 1st band k 0

  26. w/(4C1/M)1/2 Ex) ka = 1.2p ka = -0.8p k -p/a p/a -0.8p/a 2p/a 1.2p/a 0 First Brillouin Zone How about eigenvalues, En(k)? (Homework) (Reciprocal lattice vector) e Ex) ka = 1.2p ka = -0.8p G(2p/a) k -p/a p/a -0.8p/a 2p/a 1.2p/a 0 First Brillouin Zone

  27. e(k) E3,k E gap 3rd band E2,k 2nd band E1,k 1st band k 0 (for 1-D) Free electron model e(k) 4th Band 3rd Band 2nd Band 1st Band e(k) k 0 1st BZ 2nd BZ 2nd BZ k Periodic zone scheme 0 Reduced zone scheme 1st BZ

  28. NFE model e(k) e(k) E2g E1g k 0 k 0 1st BZ Reduced zone scheme Extended zone scheme : smoothing sharp corner in band intersections potential

  29. 5.5. Density of States and Fermi Surface 5.5.1 Number of Orbitals in a Band 5.5.2 Density of States and Fermi Surface

  30. 5.5.1 Number of Orbitals in a Band Linear crystal → N atoms in L (length) L 1 2 3 5 a 4 6 7 N 8 1st BZ Zone boundary # of states inside the 1st zone: (# of unit cells) Each band has N states inside the 1st BZ. Maximum # of electrons in a single band: 2N = (# of unit cells) (# of crystal orbitals inside the 1st BZ)

  31. Questions? 1. For a single atom of valence one per unit cell, the band is ( ) % filled with electrons. 2. For a single atom of valence two per unit cell, the band is ( ) % filled with electrons. 3. For two atoms of valence one per unit cell, the band is ( ) % filled with electrons.

  32. NFE model Free electron model e e k k 0 -p/a 0 p/a 5.5.2 Density of States and Fermi Surface 1st BZ Eg ky p/a Energy contour kx -p/a p/a -p/a

  33. g(e) N at 0ºK e eF Density of states ky p/a g(e)de: # of states per unit volume between e and e+de de For low k, kx -p/a p/a -p/a For spin degeneracy,

  34. ky p/a g(e) de -p/a p/a e -p/a et Near the top of the band, (Homework)

  35. g(e) g(e) et et e e For overlapped bands, 3d 2p 2s 4s divalent metals transition metals # of electrons actually occupying between e and e +d e, dn(e):

  36. Fermi surface (FS) FS : the surface in k-space inside which all the states are occupied by valence electrons. Why FS? Crystal potential → FS? FS shape? ky p/a For small n, → free-electron behavior → Spherical FS -p/a p/a ky Ex) Li, Na, K,…. As n , FS shape distorted -p/a

  37. g(e) g(e) e e g(e) e Monovalent metals → partially filled Ex) Na one atom per cell → one electron per cell eF → 50% filled 2p 2s Divalent metals →2 bands overlapped →2 electrons per atom eF Insulators → completely filled N at 0ºK (semiconductor) eF

  38. g(e) e eF Q: eF in monovalent metal? → free-electron behavior Q: me dependence of eF?

  39. ky FS Q: FS in polyvalent metals? large n FS → intersecting ZBs For empty lattice model kx ky FS [111] 1st zone ky kx FS 1st zone kx 2nd zone [111] 2nd zone Extended zone scheme Reduced zone scheme

  40. ky FS Energy bands? e(k) ZB[111] ZB[100] kx [111] 1st zone eF ky FS kx k[111] k[100] [111] 2nd zone Reduced zone scheme

  41. 5.6 Velocity of Bloch Electron and Effective Mass Velocity of Bloch electron For Bloch electron, For free electron, (wave packet) ky p/a ky p/a kx -p/a p/a kx -p/a p/a -p/a (m*: effective mass) -p/a

  42. Q: v(any given electron) → constant? Zero v at ZB? Yes! Exceptions: - Collision with phonons e(k) - Electric or magnetic field For low k, For high k, k 0 Standing waves ! k 0

  43. Effective Mass Q: (Bloch electron at low k) vs. (free electron): the same? Difference → mass! (free electron) (Bloch electron) Under electric field, In momentum space Curvature of E-band

  44. e(k) p/a 0 Small mass k e(k) Large mass 0 k 0 k For low k, m*→constant m* As k , m*  For k > kc, m* < 0 k p/a kc 0 Acceleration < 0 →opposite to the applied force Lattice force!

  45. 0 k e(k) Q: How are electrons moved under ? p/a 0 k electron

  46. Holes e(k) → vacant orbitals in a band → positive charge, +e (+ve m*, -e) ke k 0 one hole → near ZB electron removed m*(ke) < 0, e(k) (+ve mh*, +e) +e hole k 0

  47. e h e(k) Under electric field, →All electrons move except the hole →one vacant site movement CB VB a vacant site je ve vh 0 ke jh e(k) +e VB: Valence band hole CB: Conduction band k 0

  48. What happens for electrons under E ? 5.7 Electrical conductivity At t, ky ky Fermi surface At t = 0, Fermi sphere kx kx Net current

  49. At t, g(e) ky kx eF e (energy increment by ) t: collision time (uncompensated electrons)

  50. g(e) e For spherical FS, s g(eF) (for free electron model) Metal : g(eF) , Metal Insulator s Insulator : g(eF) ~ 0, s~ 0 vF(insulator) > vF(metal) eF eF

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