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Phys 446: Solid State Physics / Optical Properties

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Phys 446: Solid State Physics / Optical Properties Phys 446: Solid State Physics / Optical Properties Fall 2015 Lecture 3 Andrei Sirenko, NJIT 1 Solid State Physics Lecture 3 (Ch. 2) Last week: • Crystals, Crystal Lattice, Reciprocal Lattice, Diffraction from crystals • Today: • Scattering factors and selection rules for diffraction • HW2 discussion Lecture 3 Andrei Sirenko, NJIT 2 1 The Bragg Law Conditions for a sharp peak in the intensity of the scattered radiation: 1) the x-rays should be specularly reflected by the atoms in one plane 2) the reflected rays from the successive planes interfere constructively The path difference between the two x-rays: 2d·sinθ the Bragg formula: 2d·sinθ = mλ The model used to get the Bragg law are greatly oversimplified (but it works!). – It says nothing about intensity and width of x-ray diffraction peaks – neglects differences in scattering from different atoms – assumes single atom in every lattice point – neglects distribution of charge around atoms Lecture 3 Andrei Sirenko, NJIT 3 Diffraction condition and reciprocal lattice Von Laue approach: – crystal is composed of identical atoms placed at the lattice sites T – each atom can reradiate the incident radiation in all directions. – Sharp peaks are observed only in the directions for which the x-rays scattered from all lattice points interfere constructively. Consider two scatterers separated by a lattice vector T. Incident x-rays: wavelength λ, wavevector k; |k| = k = 2/; k'kT 2m Assume elastic scattering: scattered x-rays have same energy (same λ) wavevector k' has the same magnitude |k'| = k = 2/ k k' k k' k k' Condition of constructive interference: k ' k T m or Define k = k' - k - scattering wave vector Then k = G , where G is defined as such a vector for which G·T = 2m We got k = k' – k = G |k'|2 = |k|2 + |G|2 +2k·G G2 +2k·G = 0 2k·G = G2 – another expression for diffraction condition Lecture 3 Andrei Sirenko, NJIT 4 2 Ewald Construction for Diffraction Condition and reciprocal space Lecture 3 Andrei Sirenko, NJIT 5 Geometric interpretation of Laue condition: 2k·G = G2 – Diffraction is the strongest (constructive interference) at the perpendicular bisecting plane (Bragg plane) between two reciprocal lattice points. – true for any type of waves inside a crystal, including electrons. – Note that in the original real lattice, these perpendicular bisecting planes are the planes we use to construct Wigner-Seitz cell Lecture 3 Andrei Sirenko, NJIT 6 3 Geometric interpretation of Laue condition: 2k·G = G2 Lecture 3 Andrei Sirenko, NJIT 7 Summary Various statements of the Bragg condition: 2d·sinθ = mλ ; k = G ;2k·G = G2 Reciprocal lattice is defined by primitive vectors: A reciprocal lattice vector has the form G = hb1 + kb2 + lb3 It is normal to (hkl) planes of direct lattice Only waves whose wave vector drawn from the origin terminates on a surface of the Brillouin zone can be diffracted by the crystal First BZ of bcc lattice First BZ of fcc lattice Lecture 3 Andrei Sirenko, NJIT 8 4 Solid State Physics Lecture 3 (continued) (Ch. 2) Atomic and structure factors Experimental techniques: Neutron and electron diffraction Lecture 3 Andrei Sirenko, NJIT 9 Diffraction process: 1) Scattering by individual atoms 2) Mutual interference between scattered rays Scattering from atom i(krt) 2 Consider single electron. Plane wave u Ae k k A i(kRt) Scattered field: u' fe e fe – scattering length of electron R R – radial distance A Two electrons: u' f eikR 1 eikr e R A or, more generally u' f eikR eikr1 eikr2 e R A ikR ikrl similar to single electron with u' fe e e many electrons: ikr R l f f e l Lecture 3 Andrei Sirenko, NJITe 10 l 5 2 intensity: 2 ikrl I ~ f fe e l 2 this is for coherent scatterers. If random then I ~ Nfe 12 Scattering length of electron: f 1cos22 /2 r ee 1 e2 classical electron radius 15 re 2 2.810 m 40 mc f eikrl f n(r)eikrl d 3r In atom, e e l n(r) – electron density f n(r)eikrl d 3r a - atomic scattering factor (form factor) Lecture 3 Andrei Sirenko, NJIT 11 Atomic scattering factor (dimensionless) is determined by electronic distribution. If n(r) is spherically symmetric, then r0 sinΔk r f 4r 2n(r) dr a 0 Δk r in forward scattering k = 0 so f 4 r 2n(r)dr Z a Z - total number of electrons Atomic factor for forward scattering is equal to the atomic Z number (all rays are in phase, hence interfere constructively) Lecture 3 Andrei Sirenko, NJIT 12 6 Scattering from crystal crystal scattering factor: ikrl ikRl fcr e fale l l th Rl - position of l atom, fal - corresponding atomic factor rewrite fcr FS iks j - structure factor of the basis, where F f e aj summation over the atoms in unit cell j ikRc - lattice factor, summation over all and S e l unit cells in the crystal l c Where Rl Rl s j Lecture 3 Andrei Sirenko, NJIT 13 Since k = G, c iGRl i2m the lattice factor becomes S e e N l l Then scattering intensity I ~ |f |2 where iGs j cr fFNNfecr aj j G = Ghkl = hb1 + kb2 + lb3 if sj = uja1 + vja2 + wja3 i(u ja1 v ja2 w ja3 )(hb1 kb2 lb3 ) 2i(hu j kv j lw j ) Then F faje faje j j structure factor structure factor Lecture 3 Andrei Sirenko, NJIT 14 7 F FF FF Fhkl( , , ) f (exp 0) 1 fa Lecture 3 Andrei Sirenko, NJIT 15 Example: structure factor of bcc lattice (identical atoms) 2(ihu kv lw ) structure factor j jj Ffe aj j Two atoms per unit cell: s1 = (0,0,0); s2 = a(1/2,1/2,1/2) i(hkl) F fa 1 e F=2fa if h+k+l is even, and F=0 if h+k+l is odd Diffraction is absent for planes with odd sum of Miller indices For allowed reflections in fcc lattice h,k,and l are all even or all odd 4 atoms in the basis. What about simple cubic lattice ? Lecture 3 Andrei Sirenko, NJIT 16 8 Lecture 3 Andrei Sirenko, NJIT 17 hkl Lecture 3 Andrei Sirenko, NJIT 18 9 F(,,)hkl F(hkl , , ) f [1 exp( i ( h k ) exp( i ( h l ) exp( i ( k l )] F(,,)hkl F F Lecture 3 Andrei Sirenko, NJIT 19 Lecture 3 Andrei Sirenko, NJIT 20 10 Lecture 3 Andrei Sirenko, NJIT 21 Lecture 3 Andrei Sirenko, NJIT 22 11 Low Energy Electron Diffraction (LEED) = h/p = h/(2mE)1/2 E = 20 eV 2.7Å; 200 eV 0.87 Å Small penetration depth (few tens of Å) – surface analysis Lecture 3 Andrei Sirenko, NJIT 23 Reflection high Energy Electron Diffraction (RHEED) • Glancing incidence: despite the high energy of the electrons (5 – 100 keV), the component of the electron momentum perpendicular to the surface is small • Also small penetration into the sample – surface sensitive technique • No advantages over LEED in terms of the quality of the diffraction pattern • However, the geometry of the experiment allows much better access to the sample during observation of the diffraction pattern. (important if want to make observations of the surface structure during growth or simultaneously with other measurements • Possible to monitor the atomic layer-by-atomic layer growth of epitaxial films by monitoring oscillations in the intensity of the diffracted beams in the RHEED pattern. Lecture 3 Andrei Sirenko, NJIT 24 12 MBE and Reflection high Energy Electron Diffraction (RHEED) Lecture 3 Andrei Sirenko, NJIT 25 Real time growth control by Reflection High Energy Electron Diffraction (RHEED) Growth start (BaTiO ) (BaTiO ) (BaTiO ) 3 8 3 8 3 8 (SrTiO ) (SrTiO3)4 3 4 (SrTiO3)4 Ti shutter open RHEED Intensityun.) (arb. Growth end Ba shutter open Sr shutter open 0 200 400 600 800 1000 1200 1400 Lecture 3 Andrei Sirenko, NJITTime (sec.) 26 110 azimuth 13 Neutron Diffraction • = h/p = h/(2mE)1/2 mass much larger than electron 1Å 80 meV Thermal energy kT at room T: 25 meV called "cold" or "thermal' neutrons • Don't interact with electrons. Scattered by nuclei • Better to resolve light atoms with small number of electrons, e.g. Hydrogen • Distinguish between isotopes (x-rays don't) • Good to study lattice vibrations Disadvantages: • Need to use nuclear reactors as sources; much weaker intensity compared to x-rays – need to use large crystals • Harder to detect Lecture 3 Andrei Sirenko, NJIT 27 Summary Diffraction amplitude is determined by a product of several factors: atomic form factor, structural factor Atomic scattering factor (form factor): f n(r)eikrl d 3r reflects distribution of electronic cloud. a In case of spherical distribution r0 sinΔk r f 4r 2n(r) dr a 0 Δk r Atomic factor decreases with increasing scattering angle Structure factor 2i(hu j kv j lw j ) F faje j where the summation is over all atoms in unit cell Neutron diffraction – "cold neutrons" - interaction with atomic nuclei, not electrons Electron diffraction – surface characterization technique Lecture 3 Andrei Sirenko, NJIT 28 14.
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