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 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). 3-D Crystal Structure BW, 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 theDIRECT LATTICE

Diamond & Zincblende StructuresWeve 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 LatticesDiamond LatticeThe Cubic Unit CellZincblende LatticeThe Cubic Unit CellOther views of the cubic unit cell

Diamond LatticeThe Cubic Unit CellDiamond Lattice

Zincblende (ZnS) LatticeZincblende LatticeThe Cubic Unit Cell.

View of tetrahedral coordination & 2 atom basis:Zincblende/Diamond face centered cubic (fcc) lattice with a 2 atom basis

Wurtzite StructureWeve also seen: Many semiconductors have theWurtzite 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 LatticeWurtzite hexagonal close packed (hcp) lattice,2 atom basis View of tetrahedral coordination & 2 atom basis.

Diamond & Zincblende crystalsThe primitive lattice is fcc. The fcc primitive lattice is generated by r = n1a1+n2a2+n3a3. The fcc primitive lattice vectors are:a1 = ()a(0,1,0), a2 = ()a(1,0,1), a3 = ()a(1,1,0) NOTE: The ais are NOT mutually orthogonal!

Diamond: 2 identical atoms per fcc point Zincblende: 2 different atoms per fcc pointPrimitive fcc lattice cubic unit cell

Wurtzite 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 = ()a[(1,0,0) + (3)(0,1,0)]a3 = ()a[(-1,0,0) + (3)(0,1,0)] NOTE! These are NOT mutually orthogonal!Wurtzite Crystals2 atoms per hcp pointPrimitive hcp lattice hexagonal unit cell primitive lattice points

Reciprocal LatticeReview? BW, Ch. 2; YC, Ch. 2; S, Ch. 2Motivations: (More discussion later). The Schrdinger 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 periodicityIt can be shown that, for this V(r), wavefunctions have the form: k(r) = eikr uk(r), where uk(r) = uk(r+R). 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 asbi 2(aj ak)/ i,j,k, = 1,2,3 in cyclic permutations, = direct lattice primitive cell volume 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: (1, 2, 3 = integers)K = 1b1+ 2b2 + 3b3

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

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

b1 = 2(-1,1,1)/a

b2 = 2(1,-1,1)/a

b3 = 2(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 directionsinside the BZ Greek lettersThe BZ Center (0,0,0)The symmetry directions: [100] X , [111] L , [110] KWe need to know something about these to understand how to interpret energy bandstructure diagrams: Ek vs k

Detailed View of BZ for Zincblende LatticeTo understand & interpret bandstructures, you need to be familiar with the high symmetry directions in this BZ![100] X [111] L [110] K

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 ks Using symmetry can save computational effort!!

Consequences of BZ symmetries for bandstructuresWavefunctions 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 , used in calculating probabilities for transitions rom one band to another when discussing optical & other properties (later in the course), can be shown by symmetry to vanish: Some transitions are forbidden. This givesOPTICAL & other SELECTION RULES

Math of High SymmetryThe Math tool for all of this is GROUP THEORYAn 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 latticesE Identity operationCn n-fold rotation Rotation by (2/n) radiansC2 = (180), C3 = () (120), C4 = () (90), C6 = () (60) Reflection symmetry through a planei Inversion symmetrySn Cn rotation, followed by a reflection through a plane to the rotation axis , I, Sn Improper rotationsAlso: All of these have inverses!

Crystal Symmetry OperationsFor Rotations: Cn, we need to specify the rotation axisFor Reflections: , we need to specify reflection planeWe usually use Miller indices (from SS physics)k, , n integersFor Planes: (k,,n) or (kn): The plane containingthe origin & is to the vector [k,,n] or [kn]For Vector directions: [k,,n] or [kn]: Thevector to the plane (k,,n) or (kn)Also: k (bar on top) - k, (bar on top) -, 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 TheoryApplications: It is used to simplify the computational effort necessary in the highly computational electronic bandstructure calculations.