3-Dimensional Crystal Structure

3-Dimensional Crystal Structure

• 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 = n1a1+n2a2+n3a3.

(n1,n2,n3) are integers. r generates the lattice points. Each lattice point corresponds to a set of (n1,n2,n3).

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

• 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 = n1a1+n2a2+n3a3 to

get the crystal structure of the

DIRECT LATTICE

Diamond & Zincblende Structures• We’ve 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 LatticeThe Cubic Unit Cell

Zincblende LatticeThe Cubic Unit Cell

Other views of the cubic unit cell

Diamond LatticeThe Cubic Unit Cell

Diamond Lattice

Zincblende (ZnS) Lattice

Zincblende LatticeThe Cubic Unit Cell.

• View of tetrahedral coordination & 2 atom basis:

Zincblende/Diamond face centered cubic (fcc) lattice with a 2 atom basis

Wurtzite Structure• We’ve 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 = n1a1+n2a2+n3a3. • 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 ai’s are NOT mutually orthogonal!

Diamond: 2 identical atoms per fcc point

Zincblende:

2 different atoms per fcc point

Primitive fcc lattice cubic unit cell

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 Crystals2 atoms per hcp point

Primitive hcp lattice hexagonal unit cell

primitive lattice points

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

• Motivations: (More discussion later).

• The Schrödinger Equation & wavefunctions ψk(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:

ψk(r) = eikr uk(r), where uk(r) = uk(r+R).

ψk(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 2π(aj ak)/Ω

i,j,k, = 1,2,3 in cyclic permutations, Ω = direct lattice primitive cell volume Ω 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 bi’s

• 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 bi’s for this are

b1 = 2π(-1,1,1)/a b2 = 2π(1,-1,1)/a b3 = 2π(1,1,1)/a

• The 1st BZ for the fcc lattice (the primitive cell for the bcc k space lattice) looks like:

b1 = 2π(-1,1,1)/a

b2 = 2π(1,-1,1)/a

b3 = 2π(1,1,1)/a

For the energy bands: Now discuss the labeling conventions for the high symmetry BZ points

Labeling conventionsThe high symmetry points on the

BZ surface Roman letters

The high symmetry directions

inside the BZ Greek letters

The BZ Center Γ (0,0,0)

The symmetry directions:

[100] ΓΔX , [111] ΓΛL , [110] ΓΣKWe need to know something about these to understand how to interpret

energy bandstructure diagrams: Ek vs k

Detailed View of BZ for Zincblende Lattice

To understand & interpret bandstructures, you need to be familiar with the high symmetry directions in this BZ!

[100] ΓΔX

[111] ΓΛL

[110] ΓΣK

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 k’s

Using symmetry can save computational effort.

• Consequences of BZ symmetries for bandstructures:

Wavefunctions ψk(r) can be expressed such that they have definite transformation properties under crystal symmetry operations.

QM Matrix elements of some operators O: such as <ψk(r)|O|ψk(r)>, 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 THEORYThis 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 (2π/n) radiansC2 = π (180°), C3 = (⅔)π (120°), C4 = (½)π (90°), C6 = (⅓)π (60°)

σ Reflection symmetry through a plane

i Inversion symmetrySn Cn rotation, followed by a reflection

through a plane to the rotation axis

σ, 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: σ, 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 [kn]:

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.