3- Dimensional Crystal Structure
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.