ece 875: electronic devices prof. virginia ayres electrical & computer engineering michigan...

ECE 875:Electronic Devices

Prof. Virginia AyresElectrical & Computer EngineeringMichigan State Universityayresv@msu.edu

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:

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 bccCompare the result to direct space a, b, and c for fcc

VM Ayres, ECE875, S14