33-D Crystal Structure BW, 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 aDIRECT 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).
4Primitive lattice structure + basis. 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 possiblelattice vectors r = n1a1+n2a2+n3a3 toget the crystal structure of theDIRECT LATTICE
5Diamond & Zincblende Structures We’ve seen: Many common semiconductors haveDiamond or Zincblende crystal structuresTetrahedral 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
6Zincblende/Diamond Lattices The Cubic Unit CellZincblende LatticeThe Cubic Unit CellOther views of the cubic unit cell
7Diamond LatticeDiamond LatticeThe Cubic Unit Cell
8Zincblende (ZnS) Lattice Zincblende LatticeThe Cubic Unit Cell.
9Zincblende/Diamond View of tetrahedral coordination & 2 atom basis: face centered cubic (fcc)lattice with a 2 atom basis
102 atoms per hcp lattice point Wurtzite StructureWe’ve also seen: Many semiconductors have theWurtzite StructureTetrahedral coordination: Each atom has 4 nearest-neighbors (nn). Basis set: 2 atoms. Primitive lattice hexagonal close packed (hcp).2 atoms per hcp lattice pointA Unit Cell looks like
11Wurtzite Lattice Wurtzite hexagonal close packed (hcp) lattice, 2 atom basisView of tetrahedral coordination & 2 atom basis.
12Diamond & 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 pointZincblende:2 different atoms per fcc pointPrimitive fcc latticecubic unit cell
13NOTE! These are NOT mutually primitive lattice pointsWurtzite CrystalsThe primitive lattice is hcp. The hcp primitive lattice is generated byr = 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 mutuallyorthogonal!2 atoms per hcp pointPrimitive hcp latticehexagonal unit cell
14Reciprocal Lattice Review? 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 periodicityIt 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 FunctionsIt can also be shown that, for r points on the direct lattice, the wavevectors k points on a lattice also Reciprocal Lattice
15bi 2π(aj ak)/Ω K = ℓ1b1+ ℓ2b2 + ℓ3b3 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 asbi 2π(aj ak)/Ωi,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
16The First Brillouin Zone (BZ) The region in k space which is the smallest polyhedron confinedby planes bisecting the bi’sThe 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 latticeis the body centered cubic (bcc) lattice.It can also be easily shown that the bi’s for this areb1 = 2π(-1,1,1)/a b2 = 2π(1,-1,1)/ab3 = 2π(1,1,1)/a
17The 1st BZ for the fcc lattice (the primitive cell for the bcc k space lattice) looks like: b1 = 2π(-1,1,1)/ab2 = 2π(1,-1,1)/ab3 = 2π(1,1,1)/a
18 ΓΔX ,  ΓΛL ,  ΓΣK For the energy bands: Now discuss the labeling conventions for the high symmetry BZ pointsLabeling conventionsThe high symmetry points on theBZ surface Roman lettersThe high symmetry directionsinside the BZ Greek lettersThe BZ Center Γ (0,0,0)The symmetry directions: ΓΔX ,  ΓΛL ,  ΓΣKWe need to know something about these to understand how to interpret energy bandstructure diagrams: Ek vs k
19Detailed View of BZ for Zincblende Lattice  ΓΣK ΓΔX  ΓΛLTo understand & interpret bandstructures, you need to be familiar with the high symmetry directions in this BZ!
20 Using symmetry can save computational effort. The fcc 1st BZ: Has High Symmetry! A result of the high symmetry of direct latticeThe 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.
21OPTICAL & other SELECTION RULES 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 givesOPTICAL & other SELECTION RULES
22GROUP THEORY Math of High Symmetry The Math tool for all of this is 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.
23Group Theory Notation: Crystal symmetry operations (which transform the crystal into itself) Operations relevant for the diamond & zincblende lattices:E Identity operationCn 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 planei Inversion symmetrySn Cn rotation, followed by a reflectionthrough a plane to the rotation axisσ, I, Sn “Improper rotations”Also: All of these have inverses.
24Crystal Symmetry Operations For Rotations: Cn, we need to specify the rotation axis.For Reflections: σ, we need to specify reflection planeWe usually use Miller indices (from SS physics)k, ℓ, n integersFor Planes: (k,ℓ,n) or (kℓn): The plane containingthe 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.
25Rotational Symmetries of the CH4 Molecule The Td Point Group Rotational Symmetries of the CH4 Molecule The Td Point Group. The same as for diamond & zincblende crystals
26Diamond & 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 symmetrythan the zincblende lattice
27Group Theory Applications: It is used to simplify the computational effort necessary in the highly computational electronic bandstructure calculations.