Solid State Physics Lecture 2 (Ch. 2.1-2.3, 2.6-2.7)

Last week: Phys 446: • Crystals, • Crystal Lattice, Solid State Physics / Optical • Reciprocal Lattice, and Properties • Types of bonds in crystals Today: Fall 2007 • Diffraction from crystals • Importance of the reciprocal lattice concept

Lecture 2 Andrei Sirenko, NJIT 1 Lecture 2 Andrei Sirenko, NJIT 2

Crystal Lattice Reciprocal Lattice

Lecture 2 Andrei Sirenko, NJIT 3 Lecture 2 Andrei Sirenko, NJIT 4 Diffraction of waves by crystal lattice The Bragg Law

• Most methods for determining the atomic structure of crystals are Conditions for a sharp peak in the based on scattering of particles/radiation. intensity of the scattered radiation: • X-rays is one of the types of the radiation which can be used 1) the x-rays should be specularly • Other types include electrons and neutrons reflected by the atoms in one plane • The wavelength of the radiation should be comparable to a typical 2) the reflected rays from the interatomic distance of a few Å (1 Å =10-10 m) successive planes interfere constructively ν hc hc λ(Å) = 12398/E(eV) ⇒ The path difference between the two x-rays: 2d·sinθ ⇒ E = h = λ ⇒ λ = E few keV is suitable energy the Bragg formula: 2d·sinθ = mλ for λ = 1 Å The model used to get the Bragg law are greatly oversimplified • X-rays are scattered mostly by electronic shells of atoms in a solid. (but it works!). Nuclei are too heavy to respond. – It says nothing about intensity and width of x-ray diffraction peaks • Reflectivity of x-rays ~10-3-10-5 ⇒ deep penetration into the solid ⇒ x-rays serve as a bulk probe – neglects differences in scattering from different atoms – assumes single atom in every lattice point – neglects distribution of charge around atoms Lecture 2 Andrei Sirenko, NJIT 5 Lecture 2 Andrei Sirenko, NJIT 6

The Bragg Law and Diffraction grating Meaning of d for 2D Compare Bragg Law 2d·sinθ = mλ 2d·sinθ = mλ

X-ray Diffraction

d Lecture 2 Andrei Sirenko, NJIT 7 Lecture 2 Andrei Sirenko, NJIT 8 Meaning of d for 3D X-rays are EM waves http://www.desy.de/~luebbert/CrystalCalc_Cubic.html 2d·sinθ = mλ d 111

n d = hkl 222 Intercepts: a,a,a hkl Reciprocals: a/a, a/a, a/a ++ = 1, 1, 1 222 Miller index for this plane : (1 1 1) abc na⋅ da111 =≈3.13 A for Si with = 5.431 A Lecture 23 Andrei Sirenko, NJIT 9 Lecture 2 Andrei Sirenko, NJIT 10

X-rays and X-ray tube X-rays and Synchrotrons Bragg Law 2d·sinθ = mλ Bragg Law 2d·sinθ = mλ for m=1 2d > λ Synchrotron radiation X-ray tube Natural Synchrotron Radiation

Stars and Galaxies

Accelerating electron emits light Electronic transitions

Lecture 2 Andrei Sirenko, NJIT 11 Lecture 2 Andrei Sirenko, NJIT 12 Synchrotron Diffraction condition and reciprocal lattice Radiation produced by Von Laue approach: relativistic – crystal is composed of identical electrons in atoms placed at the lattice sites T accelerators – each atom can reradiate the incident (since 1947) radiation in all directions. – Sharp peaks are observed only in the directions for which the x-rays scattered from all lattice points interfere constructively.  k  k' Consider two scatterers separated by a lattice vector T. k = k'= Incident x-rays: wavelength λ, wavevector k; |k| = k = 2π/λ; k k' Assume elastic scattering: scattered x-rays have same energy (same λ) ⇒ wavevector k' has the same magnitude |k'| = k = 2π/λ   •Synchrotron Radiation from a storage ring Condition of constructive interference: ( k ' − k ) ⋅ T = m λ or ()k'−k ⋅T = 2πm is the most bright manmade source of white light •Useful for materials studies from Far Infrared and UV to X-ray Define ∆k = k' - k - scattering wave vector Then ∆k = G , where G is defined as such a vector for which G·T = 2πm Lecture 2 Andrei Sirenko, NJIT 13 Lecture 2 Andrei Sirenko, NJIT 14

We obtained the diffraction (Laue) condition: ∆k = G where G·T = 2πm We got ∆k = k' – k = G ⇒ |k'|2 = |k|2 + |G|2 +2k·G ⇒ G2 +2k·G = 0 Vectors G which satisfy this relation form a reciprocal lattice 2k·G = G2 – another expression for diffraction condition A reciprocal lattice is defined with reference to a particular Bravais lattice,

which is determined by a set of lattice vectors T. Now, show that the reciprocal lattice vector G = hb1 + kb2 + lb3 is orthogonal to the plane represented by Miller indices (hkl) Constricting the reciprocal lattice from the direct lattice:

Let a1, a2, a3 - primitive vectors of the direct lattice; T = n1a1 + n2a2 + n3a3 plane (hkl) intercepts axes at points x, y, and z given in units a , a and a Then reciprocal lattice can be generated using the primitive vectors 1 2 3 By the definition of the Miller indices:

where V = a1·(a2×a3) is the volume of the unit cell

Then vector G = m1b1 + m2b2 + m3b3 We have bi·aj = δij define plane by two non-collinear vectors u and v lying within this plane: Therefore, G·T = (m1b1 + m2b2 + m3b3)·(n1a1 + n2a2 + n3a3) = 2π(m1n1+ m2n2+ m3n3) = 2πm The set of reciprocal lattice vectors determines the possible scattering wave prove that G is orthogonal to u and v: vectors for diffraction analogously show Lecture 2 Andrei Sirenko, NJIT 15 Lecture 2 Andrei Sirenko, NJIT 16 Now, prove that the distance between two adjacent parallel planes of Ewald Construction for Diffraction the direct lattice is d = 2π/G. Condition and reciprocal space The interplanar distance is given by the projection of the one of the vectors xa1, ya2, za3, to the direction normal to the (hkl) plane, which is the direction of the unit vector G/G ⇒

The reciprocal vector G(hkl) is associated with the crystal planes (hkl) and is normal to these planes. The separation between these planes is 2π/G 2k·G = G2 ⇒ 2|k|Gsinθ = G2 ⇒ 2·2πsinθ /λ = 2π/d ⇒ 2dsinθ = λ ∆k θ 2dsinθ = mλ - get Bragg law k k' Lecture 2 Andrei Sirenko, NJIT 17 Lecture 2 Andrei Sirenko, NJIT 18

Reciprocal Space: Accessible Area for Diffraction 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 2 Andrei Sirenko, NJIT 19 Lecture 2 Andrei Sirenko, NJIT 20 Solid State Physics Diffraction process: Lecture 2 (continued) (Ch. 2.4-2.5, 2.9-2.12) 1) Scattering by individual atoms 2) Mutual interference between scattered rays Scattering from atom i(k⋅r−ωt) 2π Consider single electron. Plane wave u = Ae k = k = λ A i(kR−ωt) Atomic and structure factors Scattered field: u'= f e f – scattering length of electron e R e Experimental techniques R – radial distance A Neutron and electron diffraction 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 similar to single electron with A ikR i∆k⋅rl many electrons: u'= f e e i∆k⋅r e ∑ f = f e l Lecture 2 Andrei Sirenko, NJIT 21 Lecture 2R Andrei lSirenko, NJITe ∑ 22 l

2 Atomic scattering factor (dimensionless) is determined by intensity: 2 i∆k⋅rl I ~ f = fe ∑e electronic distribution. l If n(r) is spherically symmetric, then 2 this is for coherent scatterers. If random then I ~ Nfe r 2 1 2 0 Scattering length of electron: 2 sin()∆k ⋅r fe = []()1+ cos 2θ / 2 re f = 4πr n(r) dr a ∫ 0 ∆k ⋅r 2 1 e −15 classical electron radius re = ≈ 2.8×10 m 2 2 4πε 0 mc in forward scattering ∆k = 0 so f = 4π r n(r)dr = Z a ∫ Z - total number of electrons i∆k⋅r i∆k⋅r 3 In atom, f e l → f n(r)e l d r e ∑ e ∫ l n(r) – electron density Atomic factor for forward scattering is equal to the atomic number

f = n(r)ei∆k⋅rl d 3r (all rays are in phase, hence interfere constructively) a ∫ - atomic scattering factor (form factor) Lecture 2 Andrei Sirenko, NJIT 23 Lecture 2 Andrei Sirenko, NJIT 24 Scattering from crystal

i∆k⋅r i∆k⋅R Since ∆k = G, c crystal scattering factor: l l iG⋅Rl i2πm fcr = ∑e = ∑ fale the lattice factor becomes S = ∑e = ∑e = N l l l l th Rl - position of l atom, fal - corresponding atomic factor

2 iGs⋅ j Then scattering intensity I ~ |fcr| where fFNNfe=⋅= rewrite fcr = FS⋅ cr∑ aj j

i∆k⋅s j - structure factor of the basis, where F = f e ∑ aj summation over the atoms in unit cell G = Ghkl = hb1 + kb2 + lb3 if sj = uja1 + vja2 + wja3 j i(u ja1 +v ja2 +w ja3 )(hb1 +kb2 +lb3 ) 2πi(hu j +kv j +lw j ) i∆k⋅Rc - lattice factor, summation over all Then F = f e = f e and S = e l ∑ aj ∑ aj ∑ unit cells in the crystal j j l structure factor c Where Rl = Rl + s j

Lecture 2 Andrei Sirenko, NJIT 25 Lecture 2 Andrei Sirenko, NJIT 26

Example: structure factor of bcc lattice (identical atoms)

2(πihu++ kv lw ) structure factor jjj Ffe= ∑ aj j

Two atoms per unit cell: s1 = (0,0,0); s2 = a(1/2,1/2,1/2)

πi(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 ?

Lecture 2 Andrei Sirenko, NJIT 27 Lecture 2 Andrei Sirenko, NJIT 28 Lecture 2 Andrei Sirenko, NJIT 29 Lecture 2 Andrei Sirenko, NJIT 30

Experimental XRD techniques

Rotating crystal method – for single crystals, epitaxial films θ-2θ, rocking curve, ϕ - scan

Powder diffraction

Laue method – white x-ray beam used most often used for mounting single crystals in a precisely known orientation

Lecture 2 Andrei Sirenko, NJIT 31 Lecture 2 Andrei Sirenko, NJIT 32 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 2 Andrei Sirenko, NJIT 33 Lecture 2 Andrei Sirenko, NJIT 34

Applications of X-ray Diffraction for crystal and Applications of X-ray Diffraction for hetero-structures (one or more crystalline films grown on a substrate) thin-film analysis

Lecture 2 Andrei Sirenko, NJIT 35 Lecture 2 Andrei Sirenko, NJIT 36 High Angular Resolution X-ray Diffraction Setup B11 Tiernan

X-ray Diffraction Setup

Lecture 2 Andrei Sirenko, NJIT 37 Lecture 2 Andrei Sirenko, NJIT 38

Example of High Angular Resolution X-ray Diffraction Low Energy Electron Diffraction (LEED) analysis of a SiGe film on Si substrate λ= h/p = h/(2mE)1/2 E =20 eV →λ≈2.7Å; 200 eV → 0.87 Å Small penetration depth (few tens of Å) – surface analysis

Lecture 2 Andrei Sirenko, NJIT 39 Lecture 2 Andrei Sirenko, NJIT 40 Reflection high Energy Electron Diffraction (RHEED) MBE and 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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Real time growth control by Neutron Diffraction Reflection High Energy Electron Diffraction (RHEED) • λ= h/p = h/(2mE)1/2 mass much larger than electron ⇒ λ≈1Å → 80 meV Thermal energy kT at room T: 25 meV called "cold" or "thermal' neutrons • Don't interact with electrons. Scattered by nuclei

Growth start • Better to resolve light atoms with small number of electrons, e.g. Hydrogen (BaTiO ) (BaTiO ) (BaTiO ) 3 8 3 8 3 8 (SrTiO ) (SrTiO ) (SrTiO ) 3 4 3 4 3 4 • Distinguish between isotopes (x-rays don't) • Good to study lattice vibrations

Ti shutter open Disadvantages: • Need to use nuclear reactors as sources; much weaker intensity compared to x-rays – need to use large crystals RHEED Intensity (arb. un.) Growth end Ba shutter open Sr shutter open • Harder to detect 0 200 400 600 800 1000 1200 1400 Lecture 2 Andrei Sirenko, NJITTime (sec.) 43 Lecture 2 Andrei Sirenko, NJIT 44 110 azimuth 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(∆k ⋅r) f = 4πr 2n(r) dr a ∫ 0 ∆k ⋅r Atomic factor decreases with increasing scattering angle

™ Structure factor 2πi(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 2 Andrei Sirenko, NJIT 45