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

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

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VM Ayres, ECE875, S14 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: a 3 /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 Ref. Dissertation Enzo Ungersbock, “Advanced modeling of strained CMOS technology” = b c = = a Only shows one of the four inside atoms

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

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

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

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

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

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VM Ayres, ECE875, S14 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 a Note: also have pairs of atoms displaced (½, ½, ½) x a

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

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VM Ayres, ECE875, S14 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: a 3 /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

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VM Ayres, ECE875, S14 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

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

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

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

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

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VM Ayres, ECE875, S14 Reciprocal space (Reciprocal lattice):

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VM Ayres, ECE875, S14 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* C-C ^

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

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

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

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VM Ayres, ECE875, S14 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) C-C ^

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VM Ayres, ECE875, S14 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*

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VM Ayres, ECE875, S14 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* Non-orthogonal, non-standard directions in Reciprocal space:

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

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

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

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VM Ayres, ECE875, S14

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Note: pick up factors of: (2 ) 3 1 a. b x c 1 primitive cell volume = Sze V c = V crystal =

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VM Ayres, ECE875, S14 HW01:

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VM Ayres, ECE875, S14 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

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VM Ayres, ECE875, S14

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