Road map (Where are we headed?) oal: Fairly high level understanding of carrier transport and optical transitions in semiconductors Necessary Ingredients Crystal Structure Lattice Vibrations Free Electron Theory Band Structure Doping And Occupation Statistics Scattering Theory Recombination Transport Theory - Tunneling Modern Topics Quantum Effects Strain Reduced Dimensionality Heterostructures, Quantum Wells, Superlattices Computational Methods - -
Crystal Structure (read Kittel, Chapters,2) Solids can be crystalline, polycrystalline, or amorphous. Crystalline: perfect or near perfect regularity Polycrystalline: small crystal grains randomly stuck together (crystallites) Amorphous: fully uncorrelated atomic positions We will primarily deal with crystalline semiconductors and only briefly study the others. Some definitions: Lattice: a periodic array of points in space Basis: atom or set of atoms attached to lattice points A crystal structure is composed of a lattice & a basis. Example: Solid water ice - one H 2 O water molecule (basis) at each lattice site. lattice unit vectors â i, span the space lattice such that all lattice points can be reached by general lattice vector The primitive lattice cell defined by primitive axes has only one lattice point per cell (corner points shared!) - but may (typically) have more than one atom. aas: fcc lattice, one aas per lattice point. Si : also fcc with 2 silicons per lattice point The conventional unit cell is often not a primitive cell. For cubic lattices, the lattice constant a is the edge of conventional cell. fcc lattice: vol. conventional cell a 3 vol. primitive cell a 3 /4 The direction indices, and Miller indices are also based on the conventional cell Direction [0] : vector in direct lattice Miller indices (0) specify a crystal plane: the coordinates of shortest reciprocal lattice vector to that lattice plane. In cubics [h k l] is perpendicular to (h k l), but this is not true in general because lattice unit vectors are not necessarily mutually orthogonal. a - 2 -
Reciprocal lattice - lattice in Fourier Space (K-space) associated with a crystal. reciprocal lattice vectors are given by: bˆ 2π â 2 â 3 â -------------------------- â 2 â bˆ 2 2π-------------------------- â3 â 3 â â 2 â bˆ 3 2π â = = = -------------------------- â 2 3 â â 2 â 3 if â, â 2, â 3 are primitive vectors of the real-space lattice, bˆ, bˆ 2, bˆ 3 are primitive lattice unit vectors of the reciprocal lattice Clearly which is periodic in the space lattice. Now make a Fourier analy- Consider a function sis of nr ( ): nr ( ) nr ( ) = n e i r ( ) nr ( + T) = n e i r ( ) e i ( T) T = ( h bˆ + h 2 bˆ 2 + h 3 bˆ 3 ) ( n â + n 2 â 2 + n 3 â 3 ) This analysis shows that any function which is periodic in the 3 dimensional space lattice can be expressed as a superposition of plane wave functions. The plane-wave functions used in this superposition have propagation vectors which are given by the reciprocal lattice vectors. This is a 3 dimensional analog of a Fourier Series representation of a periodic time function. - 3 -
X-ray diffraction (following Kittel, Ch. 2) Electrons scatter light, scattering amplitude is proportional to electron concentration. nr ( ), electron concentration is periodic in the crystal lattice (Bloch theorem - we will prove this later) phase shift k r } phase shift r k' r } incident x- e ik r 0 scattered x-ray e ik' r The difference in scattering phase between points separated by Δk is the scattering vector, or momentum transfer vector r is e ik ( k' ) r e i( Δk r) The scattering amplitude is: F = crystal dvn( r)e iδk r = crystal dvn e i ( Δk) r for infinite crystal, dve i ( Δk) r = δ( Δk) F = n δ( Δk) finite crystal, volume V F = n V at Δk =, so therefore:, negligible otherwise - 4 -
This says that the allowed scattering vectors must be given by the reciprocal lattice vectors also, we must have k' = k = k since this is elastic scattering Δk = k + = k' The diffraction condition can be expressed as : ( k + ) 2 = k 2 or 2k + 2 = 0 substitute for (both are reciprocal lattice vectors) (equivalent to the familiar Bragg result 2dsinθ = nλ ) Intensity of X-ray Diffraction scattering amplitude for momentum transfer NS structure factor for cell composed of s basis atoms, s nr ( ) = n j ( r r j ) j = (good approx. for atomic cores) then S e i r j = dvn j ( ρ)e i ρ j cell ( ) ρ r r j n j ( ρ): electron concentration of atom j - 5 -
with f j = cell dvn j ( ρ)e i ρ atomic form factor Effect of temperature on x-ray scattering Random thermal motion does not affect the linewidth, only amplitude! For perfect crystal, the linewidth is only set by sample volume. Temperature affects the structure factor - hence intensity. let atomic position fluctuate: assume random, independent ut (), with ut () = 0. Take one term in S. Make a thermal average. f j e i r j e i ( u) e i u u -- ( u) 2 + 2 small u ( u) 2 = 2 u 2 cos 2 θ = -- 2 u 2 (can you prove cos 2 θ = 3 --? ) 3 so e i u = -- 2 u 2 6-6 -
for a classical harmonic oscillator with oscillation freq. ω, mass m :. (You prove this. Here, k is Boltzmann s constant) Since the scattered Intensity is proportional to S 2 Brillouin Zones For a simple 2D square lattice, the reciprocal lattice is also square k 0 -- 2 D D Diffraction condition: 2( k ) = 2 k 2 -- = -- 2 2 D take reciprocal lattice vector. Draw a plane, which is the perpendicular bisector of. By construction, the component of k along D is --. Hence. 2 D So any lying on the plane satisfies the Bragg condition. k The first Brillouin zone is defined as the smallest volume cell enclosed by these perpendicular bisector planes. (Wigner-Seitz primitive cell of the reciprocal lattice). Bragg condition is satisfied for all k vectors lying on the surface of Brillouin zone. Very useful when we come to electron propagation in crystals. ood diagrams of 3-d reciprocal lattices and Brillouin zones in Kittel. D - 7 -
aas conventional unit cell. Silicon has the same structure. Brillouin zone for fcc lattice. Conventional labels for symmetry points and lines are indicated. Two cells in the fcc reciprocal lattice. Two examples of the first Brillouin zone are shown. - 8 -