ECE 875:
Electronic Devices
Prof. Virginia Ayres
Electrical & Computer Engineering
Michigan State University
ayresv@msu.edu
Lecture 03, 13 Jan 14
Chp. 01
Crystals:
Direct space: primitive cells
Reciprocal space: Brillouin zones
VM Ayres, ECE875, S14
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
VM Ayres, ECE875, S14
the primitive cell
Rock salt can be
also considered
as two interpenetrating 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
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
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
VM Ayres, ECE875, S14
the primitive cell
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
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
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
Reciprocal space (Reciprocal lattice):
VM Ayres, ECE875, S14
HW01:
C-C
^
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
Example: Streetman and Banerjee:
Pr. 1.3: Label the planes illustrated in fig. P1-3:
VM Ayres, ECE875, S14
Answer:
Cubic system: Orthogonal: standard plane and direction in Reciprocal
space:
VM Ayres, ECE875, S14
Answer:
Cubic system: Orthogonal: non-standard plane and direction in
Reciprocal space:
VM Ayres, ECE875, S14
HW01:
C-C
^
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
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
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
For 1.5(a):
VM Ayres, ECE875, S14
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
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: (2p)3
1
a. b x c
=
1
primitive cell volume = Sze Vc = Vcrystal
VM Ayres, ECE875, S14
HW01:
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
VM Ayres, ECE875, S14
VM Ayres, ECE875, S14