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3-Dimensional Crystal Structure

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General: A crystal structure is DEFINED by primitive lattice vectors a 1, a 2, a 3. a 1, a 2, a 3 depend on geometry. Once specified, the primitive lattice structure is specified. The lattice is generated by translating through a DIRECT LATTICE VECTOR: r = n 1 a 1 +n 2 a 2 +n 3 a 3. (n 1,n 2,n 3 ) are integers. r generates the lattice points. Each lattice point corresponds to a set of (n 1,n 2,n 3 ). 3-D Crystal Structure BW, Ch. 1; YC, Ch. 2; S, Ch. 2

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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 = n 1 a 1 +n 2 a 2 +n 3 a 3 to get the crystal structure of the DIRECT LATTICE

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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

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Zincblende/Diamond Lattices Diamond Lattice The Cubic Unit Cell Zincblende Lattice The Cubic Unit Cell Other views of the cubic unit cell

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Diamond Lattice The Cubic Unit Cell Diamond Lattice

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Zincblende (ZnS) Lattice Zincblende Lattice The Cubic Unit Cell.

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

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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

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Wurtzite Lattice Wurtzite hexagonal close packed (hcp) lattice, 2 atom basis View of tetrahedral coordination & 2 atom basis.

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Diamond & Zincblende crystals The primitive lattice is fcc. The fcc primitive lattice is generated by r = n 1 a 1 +n 2 a 2 +n 3 a 3. The fcc primitive lattice vectors are: a 1 = (½)a(0,1,0), a 2 = (½)a(1,0,1), a 3 = (½)a(1,1,0) NOTE: The a i ’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

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Wurtzite Crystals The primitive lattice is hcp. The hcp primitive lattice is generated by r = n 1 a 1 + n 2 a 2 + n 3 a 3. The hcp primitive lattice vectors are: a 1 = c(0,0,1) a 2 = (½)a[(1,0,0) + (3) ½ (0,1,0)] a 3 = (½)a[(-1,0,0) + (3) ½ (0,1,0)] NOTE! These are NOT mutually orthogonal! Wurtzite Crystals 2 atoms per hcp point Primitive hcp lattice hexagonal unit cell primitive lattice points

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Reciprocal Lattice Review? BW, Ch. 2; YC, Ch. 2; S, Ch. 2 Motivations: (More discussion later). The Schrödinger Equation & wavefunctions ψ 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 periodicity It can be shown that, for this V(r), wavefunctions have the form: ψ k (r) = e ik r u k (r), where u k (r) = u k (r+R). ψ 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

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Reciprocal Lattice: A set of lattice points defined in terms of the (reciprocal) primitive lattice vectors b 1, b 2, b 3. b 1, b 2, b 3 are defined in terms of the direct primitive lattice vectors a 1, a 2, a 3 as b i 2π(a j a k )/Ω i,j,k, = 1,2,3 in cyclic permutations, Ω = direct lattice primitive cell volume Ω a 1 (a 2 a 3 ) The reciprocal lattice geometry clearly depends on direct lattice geometry! The reciprocal lattice is generated by forming all possible reciprocal lattice vectors: (ℓ 1, ℓ 2, ℓ 3 = integers) K = ℓ 1 b 1 + ℓ 2 b 2 + ℓ 3 b 3

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The First Brillouin Zone (BZ) The region in k space which is the smallest polyhedron confined by planes bisecting the b i ’s The symmetry of the 1 st 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 b i ’s for this are b 1 = 2π(-1,1,1)/a b 2 = 2π(1,-1,1)/a b 3 = 2π(1,1,1)/a

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The 1 st BZ for the fcc lattice (the primitive cell for the bcc k space lattice) looks like: b 1 = 2π(-1,1,1)/a b 2 = 2π(1,-1,1)/a b 3 = 2π(1,1,1)/a

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For the energy bands: Now discuss the labeling conventions for the high symmetry BZ points Labeling conventions The 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] ΓΣK We need to know something about these to understand how to interpret energy bandstructure diagrams: E k vs k

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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

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The fcc 1 st 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 E k one need only compute it for one of the equivalent k’s Using symmetry can save computational effort.

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Consequences of BZ symmetries for bandstructures: Wavefunctions ψ 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 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

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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.

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Group Theory Notation: Crystal symmetry operations (which transform the crystal into itself) Operations relevant for the diamond & zincblende lattices: E Identity operation C n n-fold rotation Rotation by (2π/n) radians C 2 = π (180 ° ), C 3 = (⅔)π (120 ° ), C 4 = (½)π (90 ° ), C 6 = (⅓)π (60 ° ) σ Reflection symmetry through a plane i Inversion symmetry S n C n rotation, followed by a reflection through a plane to the rotation axis σ, I, S n “Improper rotations” Also: All of these have inverses.

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Crystal Symmetry Operations For Rotations: C n, 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 [k n]: The vector to the plane (k,ℓ,n) or (kℓn) Also: k (bar on top) - k, ℓ (bar on top) -ℓ, etc.

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Rotational Symmetries of the CH 4 Molecule The T d Point Group. The same as for diamond & zincblende crystals

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Diamond & Zincblende Symmetries ~ CH 4 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

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Group Theory Applications: It is used to simplify the computational effort necessary in the highly computational electronic bandstructure calculations.

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