Crystal Structure. Insulin crystals. quartz. Gallium crystals. Atoms are arranged in a periodic pattern in a crystal.

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1 Crystal Structure Atoms are arranged in a periodic pattern in a crystal. The atomic arrangement affects the macroscopic properties of a material. Many important materials (silicon, steel) are crystals Gallium crystals quartz Insulin crystals

2 Crystal Structure simple cubic body centered cubic, bcc face centered cubic, fcc + = basis Bravais lattice Crystal

3 Lattice parameters γ c β a α b

4 Crystal planes and directions: Miller indices A plane with the intercepts 1/h, 1/k, 1/l is the (h,k,l) plane. [ ] specific direction < > family of equivalent directions ( ) specific plane { } family of equivalent planes MOSFETs are made on <100> wafers

5 Silicon surfaces Si(100) Si(111) (Source: Sandia Nat.Labs.)

6 Diamond Crystal Structure

7 KOH etching of silicon KOH etches Si {110} > {100} > {111}, producing a characteristic anisotropic V-etch, with sidewalls that form a 54.7 angle with the surface (35.3 from the normal). EtchingAndDecon.pdf

8 simple cubic Po

9 face centered cubic (fcc) Al, Cu, Ni, Ag, Pt, Au, Pb

10 hexangonal close pack (hcp) Ti, Co, Zn, Zr,

11 Close packing HCP FCC HCP = Hexagonal close pack Hexagonal Bravais lattice with two atoms in the basis.

12 body centered cubic bcc W Cr Fe Mo Ta

13 zincblende ZnS GaAs InP GaP InAs AlAs

14 wurtzite ZnO CdS CdSe GaN AlN

15 Structural phase transitions GaAs, Zincblende GaAs, Wurtzite 3C - SiC 4H - SiC 6H - SiC SiC has about 100 polytypes

16 Fluorescent lamp

17 Molecular energy levels

18 Semiconductors valence band conduction band band gap

19 wave vector k A k-vector points in the direction a wave is propagating. wavelength: 2π λ = k momentum: p = k

20 Absorption and emission of photons absorption emission semiconductor hf < E g no absorption

21 What color light does a GaAs LED emit? E = = hf = J hc λ λ = 867 nm infrared

22 Direct and indirect band gaps indirect bandgap: k = 0 phonons are emitted Momentum must be conserved when photons are absorbed or emitted. direct bandgap: k = 0 photons can be emitted

23 Silicon band structure E c = bottom of the conduction band E g = E c - E v E v = top of the valence band k=0 k=0 Electrons with energies in the gap are reflected out of the crystal.

24 Light emitting diodes

25 Metals, semiconductors, insulators E g Metal Semiconductor or insulator E g < 3eV = Semiconductor E g > 3eV = Insulator from: Singh

26 Copper dispersion relation and density of states from Ibach & Lueth

27 Germanium from Ibach & Lueth

28 Band gap Electrons with energies in the gap are reflected out of the crystal.

29 Density of states Silicon Aluminum N(E) filled states empty states filled states empty states

30 Structural phase transition in Sn α Sn transition at 13 C metal β Sn β-sn, white tin, tetragonal α-sn, gray tin, diamond structure

31 Structural phase transitions Si, diamond structure Si II, β-sn, tetragonal silicon makes a diamond to β-sn transition under pressure

32 Fermi function f(e) is the probability that a state at energy E is occupied. f( E) = 1 E E F 1+ exp kt B

33 Silicon density of states T = 300 K T = 300 K electrons in the conduction band

34 Fermi energy The Fermi energy is implicitly defined as the energy that solves the following equation. n = Here n is the electron density. D( E) f ( E) de The density of states, the total number of electrons and the temperature are given. To find the Fermi energy, guess one and evaluate the integral. If n turns out too low, guess a higher E F and if n turns out too high, guess a lower E F.

35 10 47 n = N( E) f ( E) de E [10-19 J] n = m -3

36 free electrons (simple model for a metal) 2 2 ( 2 2 2) p 1 2 E( k) = kx + ky + kz = = 2 mv 3-d density of states 2m 2m E dispersion relation ( m) 3/2 k x k y DE ( ) = 2 0 for E < 0 2π 2 3 E for E > 0

37 Near the bottom of the conduction band, the band structure looks like a parabola. Silicon band structure E c = bottom of the conduction band E g = E c - E v E v = top of the valence band E k x k y [111] k=0 [100]