Bandstructures, Part IV: Diamond

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3-Dimensional Crystal Structure
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3-Dimensional Crystal Structure
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3-D Crystal Structure BW, Ch. 1 YC, Ch. 2 S,
Ch. 2 (see lectures from Physics 4309-5304 for
MANY more details! http//www.phys.ttu.edu/cmyles
/Phys4309-5304/lectures1.html)

• 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 n1a1n2a2n3a3.
• (n1,n2,n3) are integers. r generates the lattice
  points. Each lattice point corresponds to a set
  of (n1,n2,n3).

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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 n1a1n2a2n3a3 to get
• the crystal structure of the
• DIRECT LATTICE

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Diamond Zincblende Structures

• 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 2 atoms are the same. Zincblende 2
  atoms are different. The Cubic Unit Cell looks
  like

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

• Weve 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 n1a1n2a2n3a3.
• 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 ais 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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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! These are NOT mutually
• orthogonal!
• Wurtzite Crystals
• 2 atoms per hcp point

Primitive hcp lattice hexagonal unit cell
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Reciprocal LatticeReview? BW, Ch. 2 YC, Ch. 2
S, Ch. 2

• Motivations (More discussion later).
• The Schrödinger 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
  periodicity
• It can be shown that, for this V(r),
  wavefunctions have the form
• ?k(r) eik?r uk(r), where uk(r) uk(rR).
• ?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 b1, b2, b3.
• b1, b2, b3 are defined in terms of the direct
  primitive lattice vectors a1, a2, a3 as
• bi ? 2p(aj ? ak)/O
• i,j,k, 1,2,3 in cyclic permutations, O
  direct lattice primitive cell volume O ? 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

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• 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 2p(-1,1,1)/a b2 2p(1,-1,1)/a
• b3 2p(1,1,1)/a

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• The 1st BZ for the fcc lattice (the primitive
  cell for the bcc k space lattice) looks like
• b1 2p(-1,1,1)/a
• b2 2p(1,-1,1)/a
• b3 2p(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 ? G ? (0,0,0)
• The symmetry directions
• 100 ? G?X ?, 111 ? G?L ?, 110 ? GSK?
• We need to know something about these to
  understand how to interpret energy bandstructure
  diagrams Ek vs k

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Detailed View of BZ for Zincblende Lattice
?????? 110 ? GSK
100 ? G?X ?????
?????? 111 ? G?L
To understand interpret bandstructures, you
need to be familiar with the high symmetry
directions in this BZ!
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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 ks
• ? Using symmetry can save computational effort.

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• Consequences of BZ symmetries for bandstructures
• Wavefunctions ?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 lt?k(r)O?k(r)gt, 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 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 (2p/n) radians
• C2 p (180), C3 (?)p (120), C4 (½)p
  (90), C6 (?)p (60)
• s ? Reflection symmetry through a plane
• i ? Inversion symmetry
• Sn ? Cn rotation, followed by a reflection
• through a plane ? to the rotation axis
• s, I, Sn ? Improper rotations
• Also All of these have inverses.

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Crystal Symmetry Operations

• For Rotations Cn, we need to specify the
  rotation axis.
• For Reflections s, 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 CH4 MoleculeThe Td
Point Group. The same as for diamond zincblende
crystals
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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

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Group Theory

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