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## 3-Dimensional Crystal Structure

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**3-D Crystal StructureBW, Ch. 1; YC, Ch. 2; S, Ch. 2**• General: A crystal structure is DEFINED by primitive lattice vectorsa1, a2, a3. • a1, a2, a3depend on geometry. Once specified, the primitive lattice structureis specified. • The lattice is generated by translating through a DIRECT LATTICE VECTOR: r = n1a1+n2a2+n3a3. (n1,n2,n3) are integers. rgenerates 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 = n1a1+n2a2+n3a3to 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 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**• 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**View of tetrahedral coordination & 2 atom basis. Wurtzite hexagonal close packed (hcp) lattice, 2 atom basis**Diamond & Zincblende crystals**• The primitive lattice is fcc. The fcc primitive lattice is generated byr = 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:Theai’s are NOTmutually 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! Theseare NOTmutually 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 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, b3are definedin terms of the direct primitive lattice vectors a1, a2, a3as 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 conventions The high symmetry pointson the BZ surfaceRoman letters The high symmetrydirections inside the BZ Greek letters The BZ CenterΓ(0,0,0) The symmetry directions: [100] ΓΔX , [111] ΓΛL , [110] ΓΣK 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] ΓΣK [100] ΓΔX [111] ΓΛL 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 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 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 rotationRotationby (2π/n) radians C2 = π (180°), C3 = (⅔)π (120°), C4 = (½)π (90°), C6= (⅓)π (60°) σReflectionsymmetry through a plane i Inversion symmetry Sn Cnrotation, followed by a reflection through a plane to therotation 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.