ECE 875: Electronic Devices

1. ECE 875:Electronic Devices Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu.edu

2. Lecture 03, 13 Jan 14 • Chp. 01 • Crystals: • Direct space: primitive cells • Reciprocal space: Brillouin zones VM Ayres, ECE875, S14

3. Ref. Dissertation Enzo Ungersbock, “Advanced modeling of strained CMOS technology” Only shows one of the four inside atoms c = = a = b Diamond can be considered as two inter-penetrating fcc lattices. Same basis vectors as fcc: a = a/2 x + 0 y + a/2 z b = a/2 x + a/2 y + 0 z c = 0 x + a/2 y + a/2 z Same primitive cell volume: a3/4 Make it diamond by putting a two-atom basis at each vertex of the fcc primitive cell. Pair a 2nd atom at (¼ , ¼ , ¼) x a with every fcc atom in the primitive cell VM Ayres, ECE875, S14

4. Rock salt can be also considered as two inter-penetrating fcc lattices. VM Ayres, ECE875, S14

5. Rock salt can be also considered as two inter-penetrating fcc lattices. Ref: http://sunlight.caltech.edu/chem140a/Ch1aCrystals1.pdf VM Ayres, ECE875, S14

6. Rock salt can be also considered as two inter-penetrating fcc lattices. Ref: http://sunlight.caltech.edu/chem140a/Ch1aCrystals1.pdf VM Ayres, ECE875, S14

7. Rock salt can be also considered as two inter-penetrating fcc lattices. Ref: http://sunlight.caltech.edu/chem140a/Ch1aCrystals1.pdf VM Ayres, ECE875, S14

8. Rock salt can be also considered as two inter-penetrating fcc lattices. Ref: http://sunlight.caltech.edu/chem140a/Ch1aCrystals1.pdf VM Ayres, ECE875, S14

9. Rock salt can be also considered as two inter-penetrating fcc lattices. Ref: http://sunlight.caltech.edu/chem140a/Ch1aCrystals1.pdf The two interpenetrating fcc lattices are displaced (½, ½ , ½) x aNote: also have pairs of atomsdisplaced (½, ½, ½) x a VM Ayres, ECE875, S14

10. Rock salt can be also considered as two inter-penetrating fcc lattices. Ref: http://www.theochem.unito.it/crystal_tuto/mssc2008_cd/tutorials/surfaces/surfaces_tut.html MgO crystallizes in the Rock salt structure VM Ayres, ECE875, S14

11. MgO crystallizes in the Rock salt structure Rock salt can be also considered as two inter-penetrating fcc lattices. Same basis vectors as fcc: a = a/2 x + 0 y + a/2 z b = a/2 x + a/2 y + 0 z c = 0 x + a/2 y + a/2 z Same primitive cell volume: a3/4 Make it Rock salt by putting a two-atom basis at each vertex of the fcc primitive cell. Pair a 2nd atom at (½ , ½, ½) x a with every fcc atom in the primitive cell VM Ayres, ECE875, S14

12. 6 conventional cubic Unit cells 4/6 have same fcc primitive cell and basis vectors fcc: single atom basis Diamond/zb: two atom basis, fcc atoms paired with atoms at (¼, ¼ , ¼ ) x a Rock salt: two atom basis, fcc atoms paired with atoms at (½, ½ , ½) x a Wurtzite = two interpenetrating hcp lattices Same tetrahedral bonding as diamond/zincblende VM Ayres, ECE875, S14

13. The bcc and fcc lattices are reciprocals of each other – Pr. 06. VM Ayres, ECE875, S14

14. Easier modelling Also: crystal similarities can enable heterostructures and biphasic homostructures Wurtzite = two interpenetrating hcp lattices Same tetrahedral bonding as diamond/zincblende VM Ayres, ECE875, S14

15. Gallium Nitride Plan view Refs: Jacobs, Ayres, et al, NanoLett, 07: 05 (2007) Jacobs, Ayres, et al, Nanotech. 19: 405706 (2008) VM Ayres, ECE875, S14

16. Gallium Nitride Cross section view Refs: Jacobs, Ayres, et al, NanoLett, 07: 05 (2007) Jacobs, Ayres, et al, Nanotech. 19: 405706 (2008) VM Ayres, ECE875, S14

17. Reciprocal space (Reciprocallattice): VM Ayres, ECE875, S14

18. C-C ^ HW01: Find Miller indices in a possibly non-standard direction Miller indices: describe a general direction k. Miller indices describe a plane (hkl). The normal to that plane describes the direction. In an orthogonal system: direction = hx + ky + lz In a non-orthogonal system: direction = ha* + kb* + lc* VM Ayres, ECE875, S14

19. Example: Streetman and Banerjee: Pr. 1.3: Label the planes illustrated in fig. P1-3: VM Ayres, ECE875, S14

20. Answer: Cubic system: Orthogonal: standard plane and direction in Reciprocal space: VM Ayres, ECE875, S14

21. Answer: Cubic system: Orthogonal: non-standard plane and direction in Reciprocal space: VM Ayres, ECE875, S14

22. C-C ^ HW01: Si: cubic: orthogonal Find Miller indices in a possibly non-standard direction Hint: check intercept values versus the value of the lattice constant a for Si (Sze Appendix G) VM Ayres, ECE875, S14

23. HW01: Find Miller indices in a possibly non-standard direction Miller indices: describe a general direction k. Miller indices describe a plane (hkl). The normal to that plane describes the direction. In an orthogonal system: direction = hx + ky + lz In a non-orthogonal system: direction = ha* + kb* + lc* VM Ayres, ECE875, S14

24. Non-orthogonal, non-standard directions in Reciprocal space: P. 10: for a given set of direct [primitive cell] basis vectors, a set of reciprocal [k-space] lattice vectors a*, b*, c* are defined: P. 11: the general reciprocal lattice vector is defined: G =ha* + kb* + lc* VM Ayres, ECE875, S14

25. For 1.5(a): VM Ayres, ECE875, S14

26. Direct space (lattice) Direct space (lattice) Reciprocal space (lattice) Conventional cubic Unit cell Primitive cell for: fcc, diamond, zinc-blende, and rock salt Reciprocal space = first Brillouin zone for: fcc, diamond, zinc-blende, and rock salt VM Ayres, ECE875, S14

27. For 1.5(b): Find the volume of k-space corresponding to the reciprocal space vectors a*, b* and c* VM Ayres, ECE875, S14

28. VM Ayres, ECE875, S14

29. Note: pick up factors of: (2p)3 1 a. b x c 1 primitive cell volume = Sze Vc = Vcrystal = VM Ayres, ECE875, S14

30. HW01: VM Ayres, ECE875, S14

31. Given: direct space basis vectors a, b, and c for bcc. Find reciprocal space basis vectors a*, b*, and c* for bcc Compare the result to direct space a, b, and c for fcc VM Ayres, ECE875, S14

32. VM Ayres, ECE875, S14