Crstal Smmetr
Crstal Structure Crstalline vs. amorphous Diamond graphite soot Binding Covalent/metallic bonds metals Ionic bonds insulators Crstal structure determines properties Binding atomic densit scattering Smmetr controls properties of solids
Periodic Structures lattice: a periodic arra of points in space. -- The environment surrounding each lattice point is identical. unit cell volume: set of atoms that is repeated in lattice not unique basis vectors: Group of atoms attached to each lattice point in order generate the crstal structure. not unique translational smmetr: base vectors or lattice vectors Usuall these vectors are chosen either: -- to be the shortest possible vectors or -- to correspond to a high smmetr unit cell
Eample: -D cells Movement of one translation vector = point of same smmetr not unique b a b Conventional crstallographic unit cell: a larger than primitive cell; chosen to displa high smmetr unit cell Primitive unit cell: has minimum volume and contains onl one lattice point Unit cell defined b translation vectors Man possible unit cells can eplain smmetr
Translation vectors and smmetr A lattice translation vector connects two points in the lattice that have identical smmetr: r n1a n b n3c n 1 n n3 integers In our -D lattice: b a b a b a
Three dimensional cubic cells Simple Cubic strucuture Bod centered cubic structure Face centered cubic structure Heagonal etc.
Miller Indices for Crstal Directions & Planes Because crstals are usuall anisotropic their properties differ along different directions it is useful to regard a crstalline solid as a collection of parallel planes of atoms. Crstallographers and CM phsicists use a shorthand notation Miller indices to refer to such planes. 1. Determine intercepts of the plane with the coordinate aes = 3 = = 1
Miller Indices Notation Epress the intercepts as multiples of the base vectors of the lattice 1. eample let s assume that the lattice is given b: a iˆ 1 b 1ˆj c 3kˆ. The intercept ratios become: 1 3 1 1 a 1 b 1 c 3 a 1 b 1 c 1 3. Form reciprocals: 1 1 1 1 4. Multipl through b the factor that allows ou to epress these indices as the lowest triplet of integers: 1 1 1 1 We call this the 1 plane.
Another eample Find the Miller indices of the shaded plane in this simple cubic lattice: a a a aiˆ b aj ˆ c akˆ a Intercepts: a non-intersecting intercept at Intercept ratios: a a 1 a a a a Reciprocals: 0 1 0 We call this the 010 plane. Note: hkl = a single plane; {hkl} = a famil of smmetr-equivalent planes
Crstal Planes and Directions Crstal directions are specified [hkl] as the coordinates of the lattice point closest to the origin along the desired direction: [001] [010] Note: [hkl] = a specific direction; <hkl> = a famil of smmetrequivalent directions [100] [001] Note that for cubic lattices the direction [hkl] is perpendicular to the hkl plane
Primitive Vectors Primitive vectors define translation smmetr of cube There are man other periodic vectors in 3-D crstal structure: Set of primative vector translations will get to an atom BCC Bad choice FCC primitive vectors primitive vectors Smmetr of electrons must match smmetr of crstal
Primitive Unit cell D: Wigner Seit Cell Lattice sites Lines of equal distance between sites space = G
Wigner Seit Cells Cubic Wigner Seit Cell BCC Wigner Seit Cell FCC Wigner Seit Cell?
Reciprocal Space -ra scattering electron waves Lattice has periodicit in R such that Reciprocal lattice will satisf condition: K and R are 3-D vectors Translate unit cell vectors to reciprocal space k-space Lattice of allowed k-vectors for Bloch waves
Cubic lattice in Reciprocal space BCC in real space FCC in reciprocal space FCC in real space BCC in reciprocal space
Bloch Waves Electron wave function Schrodinger equation Periodic Lattice boundar conditions r r m h L L L
Energ levels Solution in periodic lattice L n k L n k L n k e V r r ik k 1 m k k
Band Diagram of Si Conduction Band electrons Effective Mass ~ slope 1 m * 1 k k k Heav holes Light holes Valence Band holes
Can t make it smaller: push it Effective mass ~ 1 m * 1 k k k ~ a lattice constant Intel 65nm PMOS transistor Compressive Stress NMOS: Tensile SIN cap laer larger Si-Si PMOS: Compressive SiGe laer smaller Si-Si Ge = 11% larger than Si SiGe Increased mobilit Change Si lattice constant