1 / 32

ECE 875: Electronic Devices

This lecture delves into the fundamental concepts of crystal structures, including primitive cells, direct space arrangements, and reciprocal space with a focus on Brillouin zones. Professor Virginia Ayres from Michigan State University explores various crystal lattices such as diamond and rock salt, highlighting their unique properties and inter-relationships. The discussion leans on advanced modeling techniques relevant to strained CMOS technology. Key examples illustrate the construction of unit cells, mathematics of Miller indices, and the implications for electronic device engineering.

tan
Download Presentation

ECE 875: Electronic Devices

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related