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PPT – Bandstructures, Part IV: Diamond & Zincblende Lattices: Crystal Structures, Reciprocal Lattices, Brillouin Zones & Symmetries. PowerPoint presentation | free to download

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3-Dimensional Crystal Structure

3-Dimensional Crystal Structure

3-D Crystal StructureBW, Ch. 1 YC, Ch. 2 S,

Ch. 2

- General A crystal structure is DEFINED by

primitive lattice vectors a1, a2, a3. - a1, a2, a3 depend on geometry. Once specified,

the primitive lattice structure is specified. - The lattice is generated by translating through a
- DIRECT LATTICE VECTOR
- r n1a1n2a2n3a3.
- (n1,n2,n3) are integers. r generates the lattice

points. Each lattice point corresponds to a set

of (n1,n2,n3).

- Basis (or basis set) ?
- The set of atoms which, when placed at each

lattice point, generates the crystal structure. - Crystal Structure ?
- Primitive lattice structure basis.
- Translate the basis through all possible
- lattice vectors r n1a1n2a2n3a3 to
- get the crystal structure of the
- DIRECT LATTICE

Diamond Zincblende Structures

- Weve seen Many common semiconductors have
- Diamond or Zincblende crystal structures
- Tetrahedral coordination Each atom has 4

nearest-neighbors (nn). Basis set 2 atoms.

Primitive lattice ? face centered cubic (fcc). - Diamond or Zincblende ? 2 atoms per fcc lattice

point. - Diamond The 2 atoms are the same.
- Zincblende The 2 atoms are different.
- The Cubic Unit Cell looks like

Zincblende/Diamond Lattices

Diamond Lattice The Cubic Unit Cell

Zincblende Lattice The Cubic Unit Cell

Other views of the cubic unit cell

Diamond Lattice

Diamond Lattice The Cubic Unit Cell

Zincblende (ZnS) Lattice

Zincblende Lattice The Cubic Unit Cell.

- View of tetrahedral coordination 2 atom basis

Zincblende/Diamond ? face centered cubic (fcc)

lattice with a 2 atom basis

Wurtzite Structure

- Weve also seen Many semiconductors have the
- Wurtzite Structure
- Tetrahedral coordination Each atom has 4

nearest-neighbors (nn). Basis set 2 atoms.

Primitive lattice ? hexagonal close packed (hcp). - 2 atoms per hcp lattice point
- A Unit Cell looks like

Wurtzite Lattice

Wurtzite ? hexagonal close packed (hcp)

lattice, 2 atom basis

- View of tetrahedral coordination 2 atom basis.

- Diamond Zincblende crystals
- The primitive lattice is fcc. The fcc primitive

lattice is generated by r n1a1n2a2n3a3. - The fcc primitive lattice vectors are
- a1 (½)a(0,1,0), a2 (½)a(1,0,1), a3

(½)a(1,1,0) - NOTE The ais are NOT mutually orthogonal!
- Diamond
- 2 identical atoms per fcc point
- Zincblende
- 2 different atoms per fcc point

Primitive fcc lattice cubic unit cell

primitive lattice points

- Wurtzite Crystals
- The primitive lattice is hcp. The hcp primitive

lattice is generated by - r n1a1 n2a2 n3a3.
- The hcp primitive lattice vectors are
- a1 c(0,0,1)
- a2 (½)a(1,0,0) (3)½(0,1,0)
- a3 (½)a(-1,0,0) (3)½(0,1,0)
- NOTE! These are NOT mutually
- orthogonal!
- Wurtzite Crystals
- 2 atoms per hcp point

Primitive hcp lattice hexagonal unit cell

Reciprocal LatticeReview? BW, Ch. 2 YC, Ch. 2

S, Ch. 2

- Motivations (More discussion later).
- The Schrödinger Equation wavefunctions ?k(r).

The solutions for electrons in a periodic

potential. - In a 3d periodic crystal lattice, the electron

potential has the form - V(r) ? V(r R) R is the lattice

periodicity - It can be shown that, for this V(r),

wavefunctions have the form - ?k(r) eik?r uk(r), where uk(r) uk(rR).
- ?k(r) ? Bloch Functions
- It can also be shown that, for r ? points on the

direct lattice, the wavevectors k ? points on a

lattice also - ? Reciprocal Lattice

- Reciprocal Lattice A set of lattice points

defined in terms of the (reciprocal) primitive

lattice vectors b1, b2, b3. - b1, b2, b3 are defined in terms of the direct

primitive lattice vectors a1, a2, a3 as - bi ? 2p(aj ? ak)/O
- i,j,k, 1,2,3 in cyclic permutations, O

direct lattice primitive cell volume O ? a1?(a2 ?

a3) - The reciprocal lattice geometry clearly depends

on direct lattice geometry! - The reciprocal lattice is generated by forming

all possible reciprocal lattice vectors (l1, l2,

l3 integers) - K l1b1 l2b2 l3b3

- The First Brillouin Zone (BZ)
- ? The region in k space which is the smallest

polyhedron confined - by planes bisecting the bis
- The symmetry of the 1st BZ is determined by the

symmetry of direct lattice. It can easily be

shown that - The reciprocal lattice to the fcc direct lattice
- is the body centered cubic (bcc) lattice.
- It can also be easily shown that the bis for

this are - b1 2p(-1,1,1)/a b2 2p(1,-1,1)/a
- b3 2p(1,1,1)/a

- The 1st BZ for the fcc lattice (the primitive

cell for the bcc k space lattice) looks like - b1 2p(-1,1,1)/a
- b2 2p(1,-1,1)/a
- b3 2p(1,1,1)/a

- For the energy bands Now discuss the labeling

conventions for the high symmetry BZ points - Labeling conventions
- The high symmetry points on the
- BZ surface ? Roman letters
- The high symmetry directions
- inside the BZ ? Greek letters
- The BZ Center ? G ? (0,0,0)
- The symmetry directions
- 100 ? G?X ?, 111 ? G?L ?, 110 ? GSK?
- We need to know something about these to

understand how to interpret energy bandstructure

diagrams Ek vs k

Detailed View of BZ for Zincblende Lattice

?????? 110 ? GSK

100 ? G?X ?????

?????? 111 ? G?L

To understand interpret bandstructures, you

need to be familiar with the high symmetry

directions in this BZ!

The fcc 1st BZ Has High Symmetry!A result of

the high symmetry of direct lattice

- The consequences for the bandstructures
- If 2 wavevectors k k? in the BZ can be

transformed into each other by a symmetry

operation - ? They are equivalent!
- e.g. In the BZ figure There are 8 equivalent BZ

faces ? When computing Ek one need only compute

it for one of the equivalent ks - ? Using symmetry can save computational effort.

- Consequences of BZ symmetries for bandstructures
- Wavefunctions ?k(r) can be expressed such that

they have definite transformation properties

under crystal symmetry operations. - QM Matrix elements of some operators O
- such as lt?k(r)O?k(r)gt, used in calculating

probabilities for transitions from one band to

another when discussing optical other

properties (later in the course), can be shown by

symmetry to vanish - So, some transitions are forbidden. This gives
- OPTICAL other SELECTION RULES

Math of High Symmetry

- The Math tool for all of this is
- GROUP THEORY
- This is an extremely powerful, important tool for

understanding - simplifying the properties of crystals of high

symmetry. - 22 pages in YC (Sect. 2.3)!
- Read on your own!
- Most is not needed for this course!
- However, we will now briefly introduce some

simple group theory notation discuss some

simple, relevant symmetries.

Group TheoryNotation Crystal symmetry

operations (which transform the crystal into

itself)

- Operations relevant for the diamond zincblende

lattices - E ? Identity operation
- Cn ? n-fold rotation ? Rotation by (2p/n) radians
- C2 p (180), C3 (?)p (120), C4 (½)p

(90), C6 (?)p (60) - s ? Reflection symmetry through a plane
- i ? Inversion symmetry
- Sn ? Cn rotation, followed by a reflection
- through a plane ? to the rotation axis
- s, I, Sn ? Improper rotations
- Also All of these have inverses.

Crystal Symmetry Operations

- For Rotations Cn, we need to specify the

rotation axis. - For Reflections s, we need to specify reflection

plane - We usually use Miller indices (from SS physics)
- k, l, n ? integers
- For Planes (k,l,n) or (kln) The plane

containing - the origin is ? to the vector k,l,n or kln
- For Vector directions k,l,n or k?n
- The vector ? to the plane (k,l,n) or (kln)
- Also k (bar on top) ? - k, l (bar on top) ? -l,

etc.

Rotational Symmetries of the CH4 MoleculeThe Td

Point Group. The same as for diamond zincblende

crystals

Diamond Zincblende Symmetries CH4

- HOWEVER, diamond has even more symmetry, since

the 2 atom basis is made from 2 identical atoms. - The diamond lattice has more translational

symmetry - than the zincblende lattice

Group Theory

- Applications
- It is used to simplify the computational effort

necessary in the highly computational electronic

bandstructure calculations.

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