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

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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'kT  2m 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 = 2m 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

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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 

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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

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4 Solid State Physics Lecture 3 (continued) (Ch. 2)

Atomic and structure factors Experimental techniques: Neutron and electron diffraction

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Diffraction process: 1) Scattering by individual atoms 2) Mutual interference between scattered rays Scattering from atom i(krt) 2 Consider single electron. Plane wave u  Ae k  k   A i(kRt) Scattered field: u' fe e fe – scattering length of electron R R – radial distance A Two electrons: u' f eikR 1 eikr  e R A or, more generally u' f eikR eikr1  eikr2  e R

A ikR ikrl similar to single electron with u' fe e e many electrons:  ikr R l f  f e l Lecture 3 Andrei Sirenko, NJITe  10 l

5 2

intensity: 2 ikrl 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  2.810 m 40 mc

f eikrl  f n(r)eikrl d 3r In atom, e  e  l n(r) – electron density

f  n(r)eikrl 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 sinΔk r f  4r 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)

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6 Scattering from crystal

crystal scattering factor: ikrl ikRl fcr  e   fale l l th Rl - position of l atom, fal - corresponding atomic factor

rewrite fcr FS

iks j - structure factor of the basis, where F  f e  aj summation over the atoms in unit cell j

ikRc - lattice factor, summation over all and S  e l  unit cells in the crystal l c Where Rl  Rl  s j

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Since k = G, c iGRl i2m 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 ) 2i(hu j kv j lw j ) Then F   faje   faje j j structure factor structure factor

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7 F FF FF

Fhkl( , , ) f (exp 0) 1 fa

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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(hkl) 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 ?

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hkl 

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9 F(,,)hkl F(hkl , , ) f [1 exp( i ( h k ) exp( i ( h l ) exp( i ( k l )] F(,,)hkl F F

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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

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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.

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12 MBE and Reflection high Energy Electron Diffraction (RHEED)

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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)eikrl d 3r reflects distribution of electronic cloud. a  In case of spherical distribution r0 sinΔk r f  4r 2n(r) dr a  0 Δk r Atomic factor decreases with increasing scattering angle

 Structure factor 2i(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

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