X-ray Diffraction (HRXRD) Ashish Garg Materials Science & Engineering IIT Kanpur ashishg@iitk.ac.in
Contents Reciprocal Lattice Ewald Sphere Stereographic projections Diffraction methods Reciprocal Space Mapping and Rocking curve analysis References: B.D. Cullity, Elements of X-ray Diffraction Paul Fewster, X-ray scattering in semiconductors, Imperial College Press
Reciprocal Lattice Essential for understanding wave propagation (diffraction of x-rays or electron beam in crystals) and quantum mechanical properties of electron in periodic structures. Basically, the waves are analyzed in reciprocal space using Fourier transform. In the reciprocal lattice, the crystal and its orientation are represented in 3 dimensions by a lattice of reciprocal lattice points where each point represents a set of parallel crystal planes (hkl).
Reciprocal Lattice Every periodic structure has two lattices associated with it. The first is the real space lattice, and this describes the periodic structure. The second is the reciprocal lattice, and this determines how the periodic structure interacts with waves
Periodic Functions Any periodic function can be expressed in terms of its periodic Fourier components (harmonics). Example of periodic potential U (r) = k U k exp(i2π K.r) where U k is the coefficient of the potential, and r is a real position vector However only values of K are allowed which are reciprocal lattice vectors, G.
Proof U (r) = G U G exp(i2πg.r) For a periodic lattice, U(r) = U(r + R) (for any periodic function, f(r)=f(r+r)), where R is a lattice vector U G exp(i2πg.r) = U G exp(i2πg.(r + R)) G G exp(ig.r) = 1
Definitions "Direct" lattice vector R in terms of orthogonal or non-orthogonal primitive unit cell vectors a i is, R= n a i = na + n a + na i 1 1 2 2 3 3 Reciprocal lattice vectors, G, are defined by the following condition: e ig.r =1 Leading to G.R = 2πn (n: integer)
Reciprocal Lattice Vectors The reciprocal lattice vector G in terms of primitive lattice vectors, b i, of reciprocal space G = m i b i = m 1 b 1 + m 2 b 2 + m 3 b 3 b 1 = 2π ( a 2 a ) 3 a 1.( a 2 a ) 3 b 2 = 2π ( a a ) 3 1 a 1.( a 2 a ) 3 b 3 = 2π ( a a ) 1 2 a 1.( a 2 a ) 3 V p =a 1.(a 2 xa 3 )=a 2.(a 3 xa 1 )=a 3.(a 1 xa 2 )
The real lattice basis vectors and the reciprocal lattice basis vectors satisfy the following relation: b i.a j = 2πδ ij δ ij is the Kronecker delta, which takes the value 1 when i is equal to j, and 0 otherwise. Any reciprocal lattice vector G can be defined in terms of vectors b i as G = hb 1 + kb 2 + lb 3 The length of a reciprocal lattice vector is the reciprocal of the spacing of the corresponding real lattice plane * G hkl = G! * hkl = 1 d hkl
The real lattice a! 2 a! 1! * b 2! * b 1 The reciprocal lattice Note that vectors in reciprocal space are perpendicular to planes in real space (as constructed!) But do not measure distances from the figure! Courtesy: Prof. Anandh Subramaniam, IITK
Real Lattice Decoration of the lattice with motif Purely Geometrical Construction Real Crystal Decoration of the lattice with Intensities Structure factor calculation Ewald Sphere construction Reciprocal Lattice Reciprocal Crystal Selection of some spots/intensities from the reciprocal crystal Diffraction Pattern Courtesy: Prof. Anandh Subramaniam, IITK
SC Examples of 3D Reciprocal Lattices weighed in with scattering power ( F 2 ) + Single sphere motif Selection rule: All (hkl) allowed In simple cubic crystals there are No missing reflections 001 011 Lattice = SC 101 111 000 010 = 100 110 SC crystal Reciprocal Crystal = SC SC lattice with Intensities as the motif at each reciprocal lattice point Courtesy: Prof. Anandh Subramaniam, IITK
BCC Selection rule BCC: (h+k+l) even allowed In BCC 100, 111, 210, etc. go missing BCC crystal 202 x 002 x x 222 x 022 Important note: The 100, 111, 210, etc. points in the reciprocal lattice exist (as the corresponding real lattice planes exist), however the intensity decorating these points is zero. x x 101 000 x 110 011 x x x x 020 100 missing reflection (F = 0) 200 x 220 Weighing factor for each point motif 2 F = 4 f 2 Reciprocal Crystal = FCC FCC lattice with Intensities as the motif Courtesy: Prof. Anandh Subramaniam, IITK
FCC 202 002 222 022 111 Lattice = FCC 200 000 020 100 missing reflection (F = 0) 220 110 missing reflection (F = 0) Weighing factor for each point motif 2 F =16 f 2 Reciprocal Crystal = BCC BCC lattice with Intensities as the motif Courtesy: Prof. Anandh Subramaniam, IITK
Bragg s Law: Real Space nλ=2d.sinθ n: Order of reflection d: Plane spacing = a h + k + l 2 2 2 θ: Bragg Angle in out 2θ Path difference must be integral multiples of the wavelength θ in =θ out
X-ray Diffraction in Reciprocal Space Suppose an incident wave with wave vector k O of magnitude 1/λ impinges on two electrons A and B, where r is given by: r = n 1 a + n 2 b + n 3 c a, b and c: unit cell translations n i are integers Must be integer An Bm = λ ( r.k O r.k ) H = λr.δk (u a + v b + w c).δk = integer
Laue Equations a.δk = h b.δk = k c.δk = l Δk = G a h b k b k c l c l a h.δk = 0.Δk = 0.Δk = 0 G must be perpendicular to plane (hkl) d hkl = a.g h G = 1 G
Diffraction Condition Ewald Sphere k H -k o =G Ewald Sphere Paul Peter Ewald (1888-1985) H Plane hkl For a diffracted beam with vector k to be present after diffraction from plane (hkl), the vector k o +G hkl MUST lie on the Ewald sphere. k H sinθ+ k o sinθ = G 1 d = 2sinθ λ
Diffraction Condition in Reciprocal Space Laue Equations G.(a + b + c) = 2π (h + k + l) Δk.(a + b + c) = 2π (h + k + l) Δk = k o k H = G ( k H +G) 2 2 = k o k H 2 + 2k H.G +G 2 = k o 2 (k o ) 2 = (k H ) 2 and G = -G 2k.G = G 2 Diffraction Condition
Bragg s Law 2k.G = G 2 => Diffraction Condition k=2π/λ and d=2π/ G Bragg s Law Concept of Limiting Sphere λ 2d as sinθ varies between -1 and +1 1 d 2 λ G 2 k Diffraction can occur from only those planes whose reciprocal lattice vectors lie within the Ewald sphere of radius 2 k.
Example Consider a reciprocal lattice of an orthorhombic crystal with lattice spacing being 3 nm -1 for (100) planes and 2 nm -1 for (010) planes. It is drawn such that the points in the plane belong to [001] zone. If an x-ray beam with wave number 4 nm -1 is incident on the crystal, then what should be its direction to result in a diffraction from (120) planes. Assume that x-ray beam lies within the plane of the figure. De Graef and McHenry, Structure of Materials
Example
Change in wavelength
Stereographic projections
Stereographic Projection A stereographic projection depicts the angles between plane normals and can be useful as a 2-D representation of planes and their orientation relationship to each other.
Poles Poles of a cubic crystal Angle between planes Ref: BD Cullity, Elements of X-ray Diffraction
Stereographic Projection Ref: BD Cullity, Elements of X-ray Diffraction
Projection of great and small circles Ref: BD Cullity, Elements of X-ray Diffraction
Standard Cubic Projections (001) (011) Ref: BD Cullity, Elements of X-ray Diffraction
(001) Projection of a cubic crystal Ref: BD Cullity, Elements of X-ray Diffraction
Diffraction vs Reflection Diffraction occurs from subsurface atoms where as reflection is a surface phenomenon Diffraction occurs at only specific angles where as reflection can occur at any angle???? Despite this we call diffracting planes as reflecting planes
Diffraction Methods Method Wavelength Angle Specimen Laue Variable Fixed Single Crystal Rotating Crystal Fixed Variable (in part) Single Crystal Powder Fixed Variable Powder
Laue Method The Laue method is mainly used to determine the orientation of large single crystals. White radiation is reflected from, or transmitted through, a fixed crystal. Zone axis Zone axis Reflection Transmission Used for determining (a) Crystal orientation (using Greninger chart for back-reflection patterns and the Leonhardt chart for transmission patterns) and (b) Crystal perfection from the size and shape of the spots. If the crystal has been bent or twisted in anyway, the spots become distorted and smeared out.
Laue Method and Stereographic Projection Ref: BD Cullity, Elements of X-ray Diffraction
Use of stereographic projections
Rotating Crystal Method Pattern from Quartz (Hexagonal) Uses monochromatic radiation for single crystals Cullity, Elements of XRD Used for determining crystal structures of unknown single crystals
Four Circle Diffractometer For single crystals
Two Circle Diffractometer For polycrystalline Materials
Powder Diffractometer
Diffraction under nonideal conditions Effect of crystallite size (BD Cullity, Elements of X-ray Diffraction)
Willamson-Hall Method Size contribution to broadening β c = kλ (t: crystallite size) t cosθ Strain contribution to broadening β s = Cε tanθ (C: constant) β net = β measured(size+strain+instrumental) β instrumental β net = Cε tanθ + kλ t cosθ β net cosθ = Cε sinθ + Kλ t Kλ/t W.H.Hall (Acta Metall. 1, 22-31 (1953))
Lattice Strain d o No Strain 2θ Uniform Strain Δθ α Δd α strain 2θ Non-uniform Strain Δd Broadeing b = Δ 2θ = 2 tanθ d 2θ
Diffraction from a variety of materials Angle, 2θ( ) (From Elements of X-ray Diffraction, B.D. Cullity)
Modern Diffractometers Typically Diffractometer Operates only in one circle Panalytical X pert
Four Circle Diffractometer Panalytical MRD Pro
High Resolution XRD
Diffractometer components
Bragg s Law and Reciprocal Space
Alignment
Alignment Before a scattering measurement is performed the azimuth containing the reciprocal lattice points of interest must be located in the same plane as the incident and scattered beams (coplanar arrangement).
Example: (0001) oriented GaN layers
Accessible reflections Forbidden reflections due to systematic absences Geometry related constraints Maximum Diffractometer angle k H Q k o Not accessible (2θ<ω) Not accessible (ω<0)
For GaN
Tasks
Thin Films
Single Crystal Diffraction In single crystals the sample and detector need to be aligned to the diffraction condition. Symmetric Scan ω ω = q 2θ 2 2θ Asymmetric Scan Grazing Incidence (-) To get a precise and robust lattice parameter need to fit many peaks and refine move sample each time ω q 2θ
Single Crystal Diffraction In single crystals the sample and detector need to be aligned to the diffraction condition. q z Si q x Symmetric Scan Asymmetric Scan ω = 2θ 2 Grazing Incidence (-) ω q 2θ ω q 2θ
Single Crystal Diffraction Double Axis Triple Axis θ-θ 0 (µrad) Angular acceptance is very high. Only accepts parallel beams and gives energy discrimination. Removes height errors
Single Crystal Diffraction Double axis Triple axis What one sees in reciprocal space depends on the detector resolution
Tilts and Mosaic WARNING: Cannot distinguish in a Double axis rocking curve A mosaic crystal broadens the peak which should be constant in θ In-plane periodicities within the coherence length (a couple of microns) will also cause a broadening of the peak in q x (c.f. particle size) but will be constant in q x
Epitaxial Layers NiSb(~50nm)/GaAs J. Aldous et al J. Cryst. Growth 357 (2012) 1-8
Scanning options There are many different possibilities for exploring diffraction space by scanning either or both the instrument and sample. Both a change in 2θ, namely δ2θ, or a change in ω, namely δω, result in a change in the diffraction vector Q, namely δq.
A scan is simply a series of steps in δq, where an intensity value is measured for each step. The results for a scan can be graphically presented as plots of diffraction vector against intensity (in diffraction space ), or angle (e.g. the 2theta angle) against intensity (in angular space ).
Scan results The shape and size of the recorded profile is dependent both on the size and shape of the reciprocal lattice spot, the size and shape of the instrument probe and the direction in which they scan across one another. In the case of d-spacing measurements, it is the position or centroid of the scanned peak that is of primary importance.
Scan units Scans can be performed or displayed in angular units or diffraction space units. The diffraction space coordinates, (Q x, Q z ), are expressed with reference to the angular positions as follows: Relationship between the diffraction vector and the angles of incidence and detection R is typically taken as 1/λ or 2π/λ. The length of the diffraction vector Q is given by:
Symmetric vs Asymmetric scans Symmetric scan: (002), (004) diffraction Symmetric scans can give us information regarding c lattice constant (assuming da/a = 0 for accuracy), and the length of a period and total length of superlattice layers It can also give us information about the quality of the material Two basic types of symmetric scans: Omega-two theta scan Omega scan or rocking curve Low angle of incidence Asymmetric scan: (102), (104), (114) etc. Asymmetric scans are useful for measuring a lattice constant and reciprocal space maps used to determine crystalline quality High angle of incidence
2Theta/Omega Scan The 2θ/ω scan or radial scan. The step size for the 2θ scan, δ2θ, has twice the value of the change in incident beam angle ω, i.e. δ2θ = 2δω. In diffraction space this results in the diffraction vector scanning radially outwards from the origin.
2Theta/Omega Scan A special case of radial scan in thin films is the symmetric scan in which ω = θ and the scan is then perpendicular to the sample surface. Bragg diffraction will occur if there are crystal planes parallel to the surface and when θ = θ Bragg. The intensity of peaks usually proportional to the thickness of the layers The ω-2θ scans give the composition of the respective layers in the entire structure
2Theta/Omega Offset Scan For other scans also known as offset scans, ω θ and the difference between the two values, ω-θ, is called the offset (o).
2Theta/Omega vs Omega map A 2-axis reciprocal space map can be obtained when 2θ/ω scans are repeated for a sequence of offset values. The difference from one offset value to the next δo, is called the omega step size. The total difference between the largest and smallest offset values is called the omega range.
Omega Rocking Curve The source is fixed, the detector is fixed but the sample is rocked around the Bragg peak slightly The FWHM of the rocking curve is an important indicator of the material crystalline quality
Putting them together ω ω scan ω/2θ scan ω-2θ
Crystal Quality For symmetric scans q is always perpendicular to the sample surface. q z is varied to measure c lattice constant For obtaining reciprocal space maps both the q z and q x are varied. AlGaN in (a) and (b) are both pseudomorphic, but (b) has much worse quality. AlGaN in (c) is not pseudomorphic as it does not have the same q
Measurement of plane spacing from RCM Uses in strain calculations and estimating structural distortions In a 2-D reciprocal space map it is important that the plane containing the reciprocal lattice spots of interest is coincident with the diffraction plane. The reciprocal space map contains Bragg spots that correlate with reciprocal lattice spots of the crystal.
Calculation of d-spacing (reference to 000) This method assumes that 2θ and ω zero positions are precisely calibrated d-spacings of 11-20 and 0004 planes of GaN Correct only if the sample and goniometer are precisely aligned and if there is no tilt of the GaN unit cell with respect to the surface. i.e. ω = 0 (0001) GaN.
In the presence of an off-cut
Rocking curve and d-spacing The peak on the rocking curve corresponds to the centroid of the Bragg peak in reciprocal space. To achieve this it is necessary for the crystal layers to have almost no defects so that their peaks are not too broad. For well-behaved materials, such as single crystal semiconductor device structures which do not exhibit large loss of symmetry and whose crystal orientations have a simple relationship with the surface, rocking curves are extremely useful. When the layers contain many defects, the peaks can appear asymmetric and it may happen that the whole peak is not being captured in the rocking curve in which case it would be necessary to collect a reciprocal space map.
d-spacing from a symmetrical rocking curve where there is no layer tilt The peak splitting between the layer reflection and the substrate reflection, Δω, in the rocking curve in this simple case is entirely due to the difference in plane spacing Δd and therefore Δω = Δθ.
d-spacing from two symmetrical rocking curves where there is layer tilt
If there is any tilt of the layer unit cell with respect to the substrate unit cell then the measured Δω value at azimuth φ=0 (Δω 0 ) will contain a component of tilt, Δα, i.e. If the rocking curve measurement is repeated for the azimuth, φ+180, then the sense of α is reversed whilst Δθ remains the same:
d-spacing from two asymmetrical rocking curves where there is layer tilt Measurements of symmetric rocking curves provide values for plane spacing perpendicular to the sample surface, dz. Calculations of composition and strain require that plane spacings parallel to the interface dx are also measured. A measure of dx can be obtained using planes that are inclined.
Furthermore, if Δα has been obtained from a pair of symmetric rocking curves, then Δφ can be obtained from
The above analysis can be repeated for an azimuth at 90 to obtain d y.
Summary In short, one can use HRXRD analysis for in-depth analysis Strain Composition Layer thickness Mismatch Epitaxy References: Paul Fewster, Reciprocal Space Mapping, Critical Reviews in Solid State and Materials Sciences, 22:2, 69-110 (1997) Paul Fewster, X-ray Scattering from Semiconductors, Imperial College Press P. Kidd, XRD of gallium nitride and related compounds: strain, composition and layer thickness