1. PHYS208 28.02.2008 page 1 PHYS208-Solid State Physics 2008 Part 2 February-April 2008 Electrons in Periodic Structures Band Theory for Electrons Transport - Effective Mass Semiconductors

2. PHYS208 28.02.2008 page 2 Last slide from lecture on 27. February 2008

3. PHYS208 28.02.2008 page 3 PHYS208-Solid State Physics 2008 Part I January-February 2008 Introduction Phonons Classical Electrons Fermi Gas Electrons

4. PHYS208 28.02.2008 page 4 Lecture Thursday 28. February 2008 Room 292, 13:15 – 16:00 Topic: Electrons – Bloch states (Periodic poentials) Accompanying texts: Notes from 2007 Bloch – Fourier notes (in web collection) Comment: After the lecture version

5. PHYS208 28.02.2008 page 5 Text

6. PHYS208 28.02.2008 page 6 Bloch Theorem part 1 tilda ~

7. PHYS208 28.02.2008 page 7 THIS IS Bloch's Theorem the function u is periodic with the periodicity of the potential

8. PHYS208 28.02.2008 page 8 What to do with Bloch theorem ?

9. PHYS208 28.02.2008 page 9 Text

10. PHYS208 28.02.2008 page 10 Text

11. PHYS208 28.02.2008 page 11 Text

12. PHYS208 28.02.2008 page 12 Text

13. PHYS208 28.02.2008 page 13 Text

14. PHYS208 28.02.2008 page 14

15. PHYS208 28.02.2008 page 15 Lecture Thursday 6. March 2008 Room 292, 13:15 – 16:00 Topic: Electrons – Bloch states (Periodic potentials) Accompanying texts: Bloch – Fourier notes (in web collection) Comment: we had a crash of the drawing program; End of lecture somewhat disturbed 22 __ a G = 22 __ a G =

16. PHYS208 28.02.2008 page 16 Last Time Dirac Notation Exposes the structure For examole the operator, Or the completeness

17. PHYS208 28.02.2008 page 17 Bloch Theorem part 1 – Last lecture

18. PHYS208 28.02.2008 page 18 THIS IS Bloch's Theorem the function u is periodic with the periodicity of the potential PHYSICS: YOU GET BACK THE PLANE WAVES

19. PHYS208 28.02.2008 page 19 Periodicity in the k-space related to the periodicity in the normal ('configuration' space)

20. PHYS208 28.02.2008 page 20 Brillouin zone -  /a to  /a Expanding the wavefunction – all possible k-values; The distance between k-values – from periodicity on L=N a (many =N atoms with a between neighbours) Large L -> small k Expanding the potential – all possible K-values; The distance between k-values – from periodicity on a (over 1 atom with a to the neighbour) Small a -> large K (G=2  /a steps) 22 __ a G =

21. PHYS208 28.02.2008 page 21 Text

22. PHYS208 28.02.2008 page 22 These pictures can be really understood most easily from the Fourier Analysis (English Notes.... )

23. PHYS208 28.02.2008 page 23 Text Rearranging band matrices A band matrix can be transformed BY REARRANGING THE BASIS i.e. just changing the order of the basis states The diagonalization: the two slosest states are pushed away from each other a=b case – 'degenerate' and a,b, different

24. PHYS208 28.02.2008 page 24 BAND THEORY Where are the BANDS?

25. PHYS208 28.02.2008 page 25 Text

26. PHYS208 28.02.2008 page 26 Text

27. PHYS208 28.02.2008 page 27 Text

28. PHYS208 28.02.2008 page 28 Lecture Wednesday 12. March 2008 Room 316, 14:15 – 16:00 and Lecture Thursday 13. March 2008 Room 292, 13:15 – 16:00 Topics: Band Theory – 2 and 3 dimensions Accompanying texts: 3dim.pdf Webpages: 2007 and 2006 notes Comment: nearly final version TWO LECTURES COMBINED 22 __ a G =

29. PHYS208 28.02.2008 page 29 Physics of Bloch states: we get the plane waves Why is it good? Transport, currents Currents controlled by the phase change

30. PHYS208 28.02.2008 page 30 LCAO – Tight Binding - English Notes Last time – currents - illustration at AMOS/pub/

31. PHYS208 28.02.2008 page 31

32. PHYS208 28.02.2008 page 32 Superposition of Coulomb potentials -1/|r-R|

33. PHYS208 28.02.2008 page 33 n=10 F1=figure; xp=1:3*n; yp=1:3*n; n3=3*n; x=n+0.1:0.1:2*n; y=n+0.1:0.1:2*n; %% size n1=10*(n) % mesh of field_x values and field_y values for iy=1:n1 field_x(:,iy)=x'; end for iy=1:n1 field_y(iy,:)=y; end V=zeros(n1); del=0.01; % delta is a small delta to avoid zero divisions for tsx=1:n3 for tsy=1:n3 V=V-1./sqrt((field_x-xp(tsx)).^2+(field_y-yp(tsy)).^2+ del); end hold on; s=median(median(V)); u=max(max(V)); V=bottom(V,s-u+s); fi2=pcolor(V); colormap(gray);shading interp ; fi1=contour(V,30); axis 'square'; F2=figure; surfc(V) shading interp Matlab program to draw the figure, previous page

34. PHYS208 28.02.2008 page 34 LCAO Linear Combination of Atomic Orbitals

35. PHYS208 28.02.2008 page 35 LCAO Linear Combination of Atomic Orbitals Tight Binding Model (opposite to the weak coupling) LCAO – Tight Binding - English Notes Bloch Theorem – first part – as approximation for LCAO coefficients

36. PHYS208 28.02.2008 page 36 LCAO – Tight Binding our way Compare with English Notes Our handwritten note 3dim.pdf

37. PHYS208 28.02.2008 page 37 These integrals need illustration – see next page

38. PHYS208 28.02.2008 page 38 Done at the lecture wednesday See also next page

39. PHYS208 28.02.2008 page 39 Done after the lecture thursday

40. PHYS208 28.02.2008 page 40

41. PHYS208 28.02.2008 page 41

42. PHYS208 28.02.2008 page 42

43. PHYS208 28.02.2008 page 43 The steps from plane wave coefficients to the displacements and related sin ka and cos ka 3dim.pdf

44. PHYS208 28.02.2008 page 44 One dimension: this we shall use later..... 3dim.pdf

45. PHYS208 28.02.2008 page 45

46. PHYS208 28.02.2008 page 46 % Plotting the 3-dim FYS208 % Brillouin, 2dim crystal, tight binding % can be run by Matlab< fff.html T41=41; T20=20; clear x; clear y; clear f; clear u; for i=1:T41 x(i)=-pi+ pi/T20*(i-1); end for i=1:T41 y(i)=-pi+ pi/T20*(i-1); end for i=1:T41 for j=1:T41 f(i,j)=3.0-cos(x(i))-cos(y(j)); end u=3.0-cos(x); figure(3); % Mesh surface: % surfl(f) % Add shading shading interp colormap(gray); % Plot in grey and add contours.... This is good ! surfc(f) shading interp figure(4); contour(f,12);axis square; % set(gca,'AspectRatio',[1 1]) % Matlab program to draw the figures, previous page clear x; clear y; clear f; clear u; T41=81; for i=1:T41 x(i)=-pi+ pi/T20*(i-1); end for i=1:T41 y(i)=-pi+ pi/T20*(i-1); end for i=1:T41 for j=1:T41 f(i,j)=3.0-cos(x(i))-cos(y(j)); end u=3.0-cos(x); figure(5); shading interp colormap(gray); surfc(f) shading interp figure(6); c= contour(f,12);;axis square; clabel(c); % set(gca,'AspectRatio',[1 1]) for the next page: colormap flag

47. PHYS208 28.02.2008 page 47

48. PHYS208 28.02.2008 page 48 Dynamics of electrons in a lattice ----- next topic

49. PHYS208 28.02.2008 page 49 Lecture Wednesday 26. March 2008 Lecture Thursday 27. March 2008 Topics: Effective mass – transport in periodic structures Bloch state properties Associated texts: EffectiveMass2008_SLIDES.pdf groupvelocity_Mcdonald_1996.pdf on Studentportal Group velocity Expectation value of momentum -> assign velocity as /m Quantum mechanics and forces ? Constant force -> inclined plane potential (linear function) Wave packet; Transport Model Bloch E = a – b cos (ka) -> understanding of effective mass Comment: Edited April 1st,2008

50. PHYS208 28.02.2008 page 50 Group Velocity - history... See groupvelocity_Mcdonald_1996.pdf on Studentportal

51. PHYS208 28.02.2008 page 51 Deriving the Schrödinger-like equation for the periodic function u(r) To derive the expectation value of momentum p, we differentiate the whole Schrödinger-like equation for u(r) with respect to k

52. PHYS208 28.02.2008 page 52

53. PHYS208 28.02.2008 page 53 Wave packets; Special types of superposition solutions; Explain the classical Transport in quantal terms; Correspond to pulses in light and radiowaves Narrow in MOMENTUM broad in MOMENTUM Broad in SPACE narrow in SPACE

54. PHYS208 28.02.2008 page 54 http://web.ift.uib.no/AMOS/MOV/WAVE/

55. PHYS208 28.02.2008 page 55

56. PHYS208 28.02.2008 page 56 New version of the drawing, see also the following

57. PHYS208 28.02.2008 page 57

58. PHYS208 28.02.2008 page 58 The key is the velocity graph as k increases, the position in band «increases», i.e. the energy in band increases But the velocity in that region DECREASES Thus NEGATIVE MASS INFINITE MASS: The velocity does not change (or only very little) close to its maximum or minimum If push does not change velocity – mass is very large, even infinite

59. PHYS208 28.02.2008 page 59 Changes to be done - noticed in the lectures REMEMBERED: DONE Remember polar plot page http://web.ift.uib.no/AMOS/PHYS261/2007/POLAR/ -> Group velocity files – add our to our text – done – 2008 version -> say that momentum is calculated - edited -> McDonald to Studentportal Wavepackets in matlab PAK98 directory i-index effective mass (misprint; corrected) missing square paranthesis in group velocity ____________________________________________ STILL REMEMBER: more on matlab wavepackets; Java wavepacket

60. PHYS208 28.02.2008 page 60 UNFINISHED NOTE FOLLOWS

61. PHYS208 28.02.2008 page 61 Lecture Wednesday 2. April 2008 Topics: SEMICONDUCTORS intro Comment: Preliminary, wednesday 16:30

62. PHYS208 28.02.2008 page 62

63. PHYS208 28.02.2008 page 63

64. PHYS208 28.02.2008 page 64

65. PHYS208 28.02.2008 page 65

66. PHYS208 28.02.2008 page 66

67. PHYS208 28.02.2008 page 67 Temperature dependence of conductivity

68. PHYS208 28.02.2008 page 68 E g from photoconductivity Threshold effect changing h (or h  ) – energy of the radiation