ece 875: electronic devices prof. virginia ayres electrical & computer engineering michigan...
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ECE 875:Electronic Devices
Prof. Virginia AyresElectrical & Computer EngineeringMichigan State [email protected]
VM Ayres, ECE875, S14
Chp. 01
Crystals:
Direct space: primitive cells
Reciprocal space: Brillouin zones
Lecture 03, 13 Jan 14
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 zb = a/2 x + a/2 y + 0 zc = 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
Ref. Dissertation Enzo Ungersbock, “Advanced modeling of strained CMOS technology”
= b
c == a
Only shows one of the four inside atoms
VM Ayres, ECE875, S14
Rock salt can be also considered as two inter-penetrating fcc lattices.
VM Ayres, ECE875, S14
Rock salt can be also considered as two inter-penetrating fcc lattices.
Ref: http://sunlight.caltech.edu/chem140a/Ch1aCrystals1.pdf
VM Ayres, ECE875, S14
Rock salt can be also considered as two inter-penetrating fcc lattices.
Ref: http://sunlight.caltech.edu/chem140a/Ch1aCrystals1.pdf
VM Ayres, ECE875, S14
Rock salt can be also considered as two inter-penetrating fcc lattices.
Ref: http://sunlight.caltech.edu/chem140a/Ch1aCrystals1.pdf
VM Ayres, ECE875, S14
Rock salt can be also considered as two inter-penetrating fcc lattices.
Ref: http://sunlight.caltech.edu/chem140a/Ch1aCrystals1.pdf
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
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.
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 zb = a/2 x + a/2 y + 0 zc = 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
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
The bcc and fcc lattices are reciprocals of each other – Pr. 06.
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
VM Ayres, ECE875, S14
Refs:Jacobs, Ayres, et al, NanoLett, 07: 05 (2007)
Jacobs, Ayres, et al, Nanotech. 19: 405706 (2008)
Gallium Nitride Plan view
VM Ayres, ECE875, S14
Refs:Jacobs, Ayres, et al, NanoLett, 07: 05 (2007)
Jacobs, Ayres, et al, Nanotech. 19: 405706 (2008)
Gallium Nitride Cross section view
VM Ayres, ECE875, S14
Reciprocal space (Reciprocal lattice):
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 + lzIn a non-orthogonal system: direction = ha* + kb* + lc*
C-C^
VM Ayres, ECE875, S14
Example: Streetman and Banerjee:
Pr. 1.3: Label the planes illustrated in fig. P1-3:
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Answer:Cubic system: Orthogonal: standard plane and direction in Reciprocal space:
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Answer:Cubic system: Orthogonal: non-standard plane and direction in Reciprocal space:
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^
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 + lzIn a non-orthogonal system: direction = ha* + kb* + lc*
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:
VM Ayres, ECE875, S14
For 1.5(a):
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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
Direct space (lattice) Direct space (lattice) Reciprocal space (lattice)
VM Ayres, ECE875, S14
For 1.5(b):Find the volume of k-space corresponding to the reciprocal space vectors a*, b* and c*
VM Ayres, ECE875, S14
VM Ayres, ECE875, S14
Note:
pick up factors of: (2)3
1a. b x c
1primitive cell volume = Sze Vc = Vcrystal
=
VM Ayres, ECE875, S14
HW01:
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Given: direct space basis vectors a, b, and c for bcc.Find reciprocal space basis vectors a*, b*, and c* for bccCompare the result to direct space a, b, and c for fcc
VM Ayres, ECE875, S14