High-Resolution X-ray Diffraction

C-563

Overview• Need for High Resolution XRD

• What we require to carry out HR-XRD studies

• Types of HR-XRD systems

• duMond Diagrams and typical setup

• Applications of HR-XRD

• Suggested readings

High Resolution XHigh Resolution X--ray Diffractionray Diffraction

• Why High Resolution X-ray Diffraction (HR-XRD)?

• How to achieve HR-XRD?

• Applications and information from HR-XRD

Why HRWhy HR--XRD?XRD?

• Divergence (δθ) is expressed as

where “h” is the source size, “s” the slit size and “a” the source-specimen distance. For a typical case where h=0.4mm, s= 1mm and a= 500mm, δθ~500 arc seconds. However, the width of the rocking curve for highly perfect crystals is a few arc seconds.

s h

h s

aδθ +

=

δθ Source

a

HR-XRD

∆ω

Si

SiGe

Few 100s arc secL

og (

Inte

nsit

y)

ω 2θ

Source Detector

ExampleExample

Rocking curve for In0.2Ga0.8As/Rocking curve for In0.2Ga0.8As/GaAsGaAs multilayer with powder multilayer with powder diffractometer diffractometer (low resolution) and High Resolution XRD system(low resolution) and High Resolution XRD system

What’s required for HRWhat’s required for HR--XRD?XRD?

• The spectral width of X-ray characteristic line is approximately δλδλ / / λλ~10~10--44,, which rises to 1010--33 if both Kα1and Kα2 are diffracted by the sample.

• Therefore, for HR-XRD measurements we need to limit the divergence and wavelength spread of the beam incident upon the specimen.

•• Collimated and to monochromaticCollimated and to monochromatic beam is obtained with the help of Beam ConditionersBeam Conditioners.

Our RequirementOur Requirement: A system with high angular resolution : A system with high angular resolution and sufficient monochromatic source with adequate and sufficient monochromatic source with adequate scattered intensity.scattered intensity.

What we need for HRWhat we need for HR--XRD?XRD?

• X-ray Source

• Beam conditioners on incident and possibly diffracted beam side

• A sample stage which is capable of movement in different directions, tilt and rotation for precise placement of sample.

• Detector

In the following slides we will describe XIn the following slides we will describe X--ray ray sources and beamsources and beam--conditionersconditioners.

XX--ray Sourcesray SourcesTypical X-ray sources used are:

•• Sealed XSealed X--ray tube sources in laboratoriesray tube sources in laboratories–Cu X-ray source is preferred for most applications due to high intensity and convenient wavelength for most inter-planar spacings.–CuKα is actually a doublet comprising of CuKα1(λ=1.540562Å) and CuKα2 (λ=1.54439Å) with intensity of Kα2 being half of Kα1. –Beam conditioners are used to separate Kα1 and Kα2

•• Synchrotron radiation sourcesSynchrotron radiation sources–Built on national or international level

• Provides continuous radiation spectrum• High intensity and brightness is obtained• High degree of polarization is achieved• Pulsed time structure ~MHz

High Resolution SystemsHigh Resolution Systems

DoubleDouble--Axis HR-XRD system

TripleTriple--Axis HR-XRD

DoubleDouble--Axis HRAxis HR--XRD SystemXRD System

First AxisFirst Axis :Adjustment of the Beam Conditioner (comprising of optical elements and slits)

Second AxisSecond Axis : Scan of the specimen through Bragg Angle

DifferentialDifferential movement of the two axistwo axis makes the measurement and determine the precision and accuracy of the instrument

Beam ConditionerBeam Conditioner

RotationRotation

TiltTiltSampleSample

DetectorDetector

Rocking CurveRocking Curve

TripleTriple--Axis HRAxis HR--XRD SystemXRD System

Provides detailed and finer information, but at the expense of Provides detailed and finer information, but at the expense of intensity, hence increased time for acquisitionintensity, hence increased time for acquisitionFirst AxisFirst Axis :Adjustment of the Beam Conditioner (comprising

of optical elements and slits)Second AxisSecond Axis :Scan of the specimen through Bragg AngleThird AxisThird Axis :Adjustment of the Analyzer (comprising

of optical elements and slits)DifferentialDifferential movement of the three axisthree axis makes the measurement and determine the precision and accuracy of the instrument

Beam ConditionerBeam Conditioner

RotationRotation

TiltTiltSampleSample

DetectorDetector

Rocking CurveRocking Curve

Analyzer Analyzer

duMond duMond ((dMdM) Diagram) Diagram

• Is used in analyzing various diffracting elements and decide between available choices

• A highly useful qualitative and semi- quantitative tool used for visualization and understanding of multiple diffracting elements

Principle:Principle: Wavelengths and angles diffracted by various crystals obey the Bragg’s law and a plot of wavelength versus incident angle would show the diffraction geometry of the crystal reflection

duMond DiagramduMond Diagram• A plane monochromatic wave is represented by a single point on the

wavelength axis and will diffract at a single point

• An aperture before the crystal imposes certain angles that may be accepted by the crystal; this information is derived from the horizontal axis

Figure shows the Figure shows the dMdM diagram for the angle at whichdiagram for the angle at which SiSi (220) reflection will diffract (220) reflection will diffract and a 2and a 2oo angular aperture required to easily allow it to pass.angular aperture required to easily allow it to pass.

duMond DiagramduMond Diagram• In practice the source is polychromatic and divergent. The following

figure shows the dM diagram for the Si(220) reflection and the angles at which CuKα1and CuKα2 lines will diffract. An aperture of 0.06o is required to exclude CuKα2 from the subsequent beam

duMond Diagram: Addition of 2duMond Diagram: Addition of 2ndnd ReflectionReflection

Let us add another reflection. Notation used for multiple reflections is as follows:

• 1st reflection is given by notation “+n+n”• The next reflection is represented by “nn”,, if the planar

spacing is identical for the previous reflection (same material, same spacing) and “mm” if it is different

• Reflection is given the “++” if it deflects the beam in the same sense as does the first crystal and “--” if the opposite sense.

duMond Diagram: duMond Diagram: Dispersive and NonDispersive and Non--Dispersive SettingDispersive Setting

• λ(θ) plot for the firstfirst crystal is drawn starting from the origin. This defines the direction of the incident beam and hence zero for angular rotations of the crystals

• With the beam entering from the left and specimen beneath the beam, the crystal diffracts when rotated anti-clock wise through the Bragg Angle.

• λ(θ) plot for the secondsecond crystal is also drawn starting from the origin. However there are two possibilities:

NonNon--Dispersive (+n, Dispersive (+n, --n) Settingn) Setting• If the second reflection is “––nn”, then the plots superimpose and

no rotation is necessary for diffraction from the 2nd crystal to occur, i.e. the crystals remain parallel.

The (+n,(+n,--n)n) setting for Si 220 reflection with CuKα (a) Real space geometry, (b) duMond diagram when the crystals are parallel and second crystal diffractsparallel and second crystal diffracts (the plots superimpose),

Sour

ce

Range of angles

Range of wavelengths

(a)

(b)

NonNon--Dispersive (+n, Dispersive (+n, --n) Settingn) Setting

• Small rotation (δθ) causes the curves to separate and diffraction stops.

• This is called non-dispersive setting since the 2nd crystal accepts the whole spread reflected by the first monochromator.

• Kα1and Kα2 are diffracted simultaneously.

•• Although the intensity in this configuration is high, but the beAlthough the intensity in this configuration is high, but the beam incident am incident on sample is on sample is notnot monochromaticmonochromatic

Sour

ceRange of angles

δθ

NonNon--Dispersive (+n, Dispersive (+n, --m) Settingm) Setting

• If the second reflection is “––mm”, then the plots diverge and second crystal second crystal doesn'tdoesn'tdiffract. diffract.

• The (+n,(+n,--m)m) setting for InP (004) Si (004) reflection with CuKα

– (a) Real space geometry,

– (b) duMond diagram when the crystals are parallel and second crystal parallel and second crystal doesn'tdoesn't diffractsdiffracts,

Sour

ce

Range of angles

(a)

(b)

NonNon--Dispersive (+n, Dispersive (+n, --m) Settingm) Setting

• Diffraction from the second crystal occurs when its curve on theduMond diagram overlaps part of curve of 1st crystal.

• The (+n,(+n,--m)m) setting for InP (004) Si (004) reflection with CuKα(a) Real space geometry (b) duMond diagram when 22ndnd crystal is rotated anticrystal is rotated anti--clockwise by clockwise by

2.8972.897oo and thus diffracts•• This configuration yields poorly monochromatic radiationThis configuration yields poorly monochromatic radiation

Sour

ceRange of angles

(a)

2.897o

Kα1

Dispersive Setting (+n, +n)Dispersive Setting (+n, +n)

• If the 2nd reflection is “+n+n”, then the plot goes in the opposite direction as that of the 1st crystal. Hence the 2nd

crystal does not diffract.• The (+n, +n)(+n, +n) setting for Si 220 reflection with CuKα

– duMond diagram when the crystals are parallel and parallel and second crystal second crystal doesn'tdoesn't diffractdiffract

Dispersive Setting (+n, +n)Dispersive Setting (+n, +n)

• Diffraction from the second crystal occurs when its curve on the duMond diagram intersects part of curve of 1st crystal.

• 2nd crystal needs to be rotated by 2θB for diffraction to occur.

•• This configuration gives excellent monochromatic radiation but pThis configuration gives excellent monochromatic radiation but poor oor intensityintensity

Improved Non-Dispersive Setting: Multiple Reflections in Channel Cut

• A “channel”“channel” is grooved into a perfect crystal block aligned parallel to a lattice plane

• Incident beam is diffracted two diffracted two or more timesor more times before leaving the crystal

• Multiple reflections cause the reduction in the tails to occur reduction in the tails to occur (tails add to the noise rather than (tails add to the noise rather than to the signal) without affecting to the signal) without affecting the signal maxima,the signal maxima, thus beam obtained has a better signal to noise ratio. The rocking curves produced from 1,2,3,4 The rocking curves produced from 1,2,3,4

successive reflections in a channelsuccessive reflections in a channel--cut Si 220 cut Si 220 crystal for crystal for CuKCuKαα raditionradition. Inset shows little . Inset shows little intensity is lost near peaks.intensity is lost near peaks.

duMondduMond--HartHart--Bartels (Bartels (dHBdHB) Design) Design

• dHB design comprise four reflecting crystals in (+n,-n,-n,+n) arrangement

• If the gap between the crystals are equal, the beam is not deviated in height on exit.

• In practice it is hard to align four independent crystals. This is overcome by using two channel cuts in (+,+) settings.

• This design combines the low tail intensity of a channel cut (+,-) setting and improved wavelength separation of the (+,+) setting.

• This is the most efficient way to isolate a single Kα1line.

Ge Ge 220 and 440 channel cut crystals 220 and 440 channel cut crystals for generating monochromatic beams. for generating monochromatic beams. 440 is used for better resolution at the 440 is used for better resolution at the expense of intensityexpense of intensity

Typical Experiment SetupTypical Experiment Setup

XX--ray Sourceray Source

44--Bounce MonochromatorBounce Monochromator((dHB dHB Design)Design)

AnalyzerAnalyzer

DetectorDetector

Specimen on a stage Specimen on a stage capable of rotation and tiltcapable of rotation and tilt

KKαα11 beam with beam with divergence ~0.0035divergence ~0.0035oo

Applications of HRApplications of HR--XRDXRD• Lattice mismatch• Thickness• Layer tilt• Alloy composition• Curvature• Mosaic layers• Dislocations• Homogeneity• Exit

Recording the Rocking CurveRecording the Rocking Curve

• Alignment of the sample cradle and diffraction machine is verified

• Output of beam conditioners is ascertained

• Sample is placed at the center of the cradle.

• A suitable substrate peak is selected and aligned for maximum intensity by rotation and tilt.

• A rocking curve with suitable step size and counting time is recorded over a suitable angular range across the substrate peak.

Rocking Curve Rocking Curve SiGe layer on Si substrateSiGe layer on Si substrate

Si 004Si 004

SiGeSiGe

Theta (degrees)

Inte

nsity

(co

unts

)

Information from HR-XRD

•Broadening may increase with beam size up to mosaic cell size

• No shift of peak with beam position on the sample

Broadens peakMosaic Spread

•Broadening invariant with beam size

•No shift of peak with beam position on the sample

Broadens peakDislocation content

Changes sign with sample rotationSplitting of layer and substrate peak

Misorientation

Invariant with sample rotationSplitting of layer and substrate peak

Mismatch

Distinguishing FeaturesDistinguishing FeaturesEffect on Effect on

Rocking CurveRocking Curve

Material Material ParameterParameter

The following table provides an overview of the information thatThe following table provides an overview of the information that can can be derived from rocking curves obtained from HRbe derived from rocking curves obtained from HR--XRD experiments XRD experiments

Effect of substrate and epilayer parameters upon the rocking curves

Information from HR-XRD Contd..

•Individual characteristics may be mappedEffects vary with position on sample

Inhomogeneity

•Integrated intensity increases with thickness up to a limit

•Fringe period is controlled by the thickness

•Effects intensity of peaks

•Introduces interference fringes

Thickness

Different effect on symmetrical and asymmetrical reflection

Changes the splittingRelaxation

•Broadening increases linearly with beam size

•Peak shifts systematically with beam position on sample

Broadens peakCurvature

Distinguishing FeaturesDistinguishing FeaturesEffect on Effect on

Rocking CurveRocking Curve

Material Material ParameterParameter

Effect of substrate and epilayer parameters upon the rocking curves

Lattice Mismatch• Lattice mismatch between the substrate and layer

leads to the splitting of the rocking curve.

SiGe

Si

Theta

Lattice mismatch=0.00484

Lattice mismatch

Lattice parameters in (a) pseudomorphic and (b) partially relaxed single-layer heterostructure

(a)

aLII aL∞

aS

aS

aL⊥ ∆φ

aS

aL∞

aS

∆φ=0

(b)

Lattice mismatch• Lattice mismatch leads to the angular separation between

the layer and substrate Bragg peaks. • In general, the angular separation (∆η) has following

component∆ηi= ∆θ0+ ∆θB+ ∆φwhere

1. ∆θ0 is the contribution from the different amount of X-ray refraction at the air-layer and layer-substrate interface

2. ∆θB is the Bragg angle difference due to lattice mismatch3. ∆φ is difference between the inclination angles of the

substrate and layer diffracting planes wrt to the sample surface. This occurs as a results of the lattice distortion of the layer.

Lattice Mismatch: Symmetric diffraction

• For a symmetric reflection, lattice mismatchlattice mismatch normal to the surface ((∆d/d)⊥) is calculated by taking a derivative of Bragg’s law and is expressed as

( )sin

1sin

B

B B

dd

θθ θ

⊥∆ = − + ∆

∆d :Change in the interplanar spacing “d”

∆θ :Angle difference measured in the rocking curve from substrate to the layer peak

θB :Bragg Angle

tanB B

dd

θ θ⊥∆ ∆ = −

More accurately it is expressed as

Relaxed Lattice MismatchSymmetric diffraction

•• True or relaxed mismatchTrue or relaxed mismatch (∆a/a) is defined with respect to relaxed lattice parameter for substrate (aS) and layer (aL) and is expressed as

• Assuming that lattice mismatch parallel to the surface ((∆d/d)II) to be zero,

L S

S

a aa d d dP

a a d d d

⊥ −∆ ∆ ∆ ∆ = = − +

a dP

a d

⊥∆ ∆ =

Relaxed Lattice Mismatch Contd.Symmetric diffraction

• Here P is a constant dependent on the surface orientation and on the elastic moduli. For a cubic lattice P is given by

( )

( )

11

11 12

11 44 11 12

11 12

12 44 11 12

11 12

( ) (001)2

0.5 2( ) (011)

2

1.5 2( ) (111)

2

( ) 0.5

Cmno P

C C

C C C Cmno P

C C

C C C Cmno P

C C

mno others P

= =+

+ − += =

+

+ − += =

+= =

Lattice mismatch: Asymmetric diffraction

• If the contribution from refraction is ignored, then contribution from ∆θ and ∆φ can be separated by measuring angular separation ∆ηi

between a pair of complementary diffraction geometry.

αi

φθB

θBαf

αi

θB

θBφ

(a)

(b)

Schematic for asymmetricasymmetric Bragg diffraction (a) φ + and (b) φ- set-up

Lattice MismatchLattice MismatchAsymmetric ReflectionAsymmetric Reflection

• In the φ- setup;

αi = θB- φ and αf = θB+ φ.

The angular separation in this case is expressed as

∆η+ = ∆θB+ ∆φ• In the φ+ setup the angular separation is expressed as

∆η- = ∆θB- ∆φ

Hence

2

2

B

η ηθ

η ηφ

+ −

+ −

∆ + ∆ ∆ = ∆ − ∆ ∆ =

Lattice MismatchLattice MismatchAsymmetric Reflection Contd..Asymmetric Reflection Contd..

• The lattice mismatch normal and parallel to surface can then be expressed

( ) ( )

( ) ( )

sin cos1

sin cos

sin sin1

sin sin

B

B B B B

B

B B B B

dd

dd

θ φθ θ θ θ

θ φθ θ θ θ

⊥∆ = − + ∆ + ∆

∆ = − + ∆ + ∆

tan cot

cot cot

B B B

B B

dd

dd

φ θ θ θ

φ φ θ θ

⊥∆ = ∆ − ∆

∆ = −∆ − ∆

More accurately it is expressed as

Asymmetric reflections: SiGe(224)±

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

79.5 79.6 79.7 79.8

2Theta (Degrees)

Inte

nsi

ty (

CP

S)

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

8 8.5 9 9.5 10 10.5

Inte

nsity

(C

PS

)

2Theta (Degrees)

SiGe(224)+ SiGe(224)-

Asymmetric Reflection:SiGe on Si

Delta Theta

(∆d/d)⊥=0.00621

(∆d/d)||=-0.000018

SiGe(224)-

SiGe(224)+

Layer tilt or misorientation• The relative tilt of the layer

with respect to the substrate also leads to the shift of the layer peak relative to that of the substrate

• The tilt angle β and the tilt direction with respect to a particular main axis of the surface can be determined by recording

diffraction curves i.e. (hkl) reflection at four possible different azimuthalorientations wrt the surface normal

as

as

ap

av

β

∆φ1∆φ2

( ),( ), ( ), ( )hkl hkl hkl hkl

Layer tilt calculation

is the tilt relative to (hkl) plane

• ∆ϕ1 = Difference in the inclination of the diffracting lattice planes of the layer and substrate wrt to the surface measured at (hkl) i.e. ψ=0o

• ∆ϕ2 = Difference in the inclination of the diffracting lattice planes of the layer and substrate wrt to the surface measured at (hkl) i.e. ψ=180o

•• Similar measurements are needed for Similar measurements are needed for ψψ=90=90oo and and ψψ=270=270oo

1 2 1sin 2

φ φβφ

∆ − ∆= −

Layer tilt measurements

(026) rocking curve of a [GaInAs/GaAs]38 multilayer grown on GaAs (001) measured for four different azimuthal orientation. The layer is tilted ~ 1000 arcsec towards [110] and 300 arcsec towards [110]

ThicknessThickness

Thickness of the layer is estimated from the thickness fringes in the rocking curve.

∆η = angle difference of the period of the thickness fringes

λ : Wavelength

θB : Bragg Angle

φ : Bragg plane inclination angle

( )( )( )

sin

sin 2B

B

tθ φ

η θ+

=∆

Layer thickness: SiGe layer on Si

Layer Thickness=170.9 nm

Delta Theta (Degrees)

Si(004)

Alloy CompositionAlloy Composition•• Vegard’s LawVegard’s Law

The lattice parameter of a solid solution will be given by a linear dependence of lattice parameter on composition following a line drawn between the values for the pure constituents.

• For alloys with one degree of freedom: AXB1-X (A,B can either be element (Si, Ge,..)or compound (GaAs, InAs..) composition

• For alloys with two degree of freedom : AXB1-XCYD1-Y

Alloy composition is estimated by iterative method involving bandgap/ photoluminescence measurements

layer B

A B

a aX

a a

−=

−

Alloy Composition: Alloy Composition: SiGeSiGe layer on layer on Si Si

Curvature• To measure the curvature of a

specimen, it is translated a distance “x” in its plane along a diameter . The shift δθ in the absolute position of the Bragg peak is recorded and the radius of curvature is calculated using the relation shown. “s” is the beam diameter.

• For better accuracy, number of points are measured on the wafer

s

δθ

sR

δθ=

Curvature • If the sample is rotated

by an angle “α” in that case

s = x cosα + y sinα• A linear relation is

performed on θspecimenand s to get “R”

P(x,y)

α

0

1specimen s

Rθ θ = +

Mosaic layer• A mosaic layermosaic layer consists of

randomly shaped mosaic blocks (blocks withmisoriented crystalline planes)

• Crystal lattice of each block is assumed to be perfect but it is rotated by a random rotation random rotation vector “vector “ξξ”” with respect to the averaged lattice.

• Presence of mosaic defect causes the broadening of the broadening of the rocking curvesrocking curves

Mosaic SpreadCalculated double crystal reflection curves for a 3 µmZnTe layer for

(a) Various block radius “R” values R=0.1 µm (—); R=0.2 µm(----) and R= 0.3 µm (….)

(b) Various values of root mean square (r.m.s.) misorientation (∆ ) of the mosaic block. ∆= 50 arcsec (—); ∆= 100 arcsec (----) and ∆= 200 arcsec(….)

∆= 100 arcsec

R=0.2 µm

Holy etal.; J. Appl. Phys.74 (1993) 1736

Dislocations

• Dislocations in crystals are lattice defects that destroy the long range order of the crystal structure and cause diffraction broadening

• Dislocations are present in two regions:

(a) Dislocations at interface

(b) Dislocations in epilayer

Dislocations at interface• Interface dislocations are created when a

layer with high mismatch relax to accommodate the strain. A network of dislocations is observed at the interface

• Interface dislocations give a specified relaxation of strain between substrate and the epilayer, which gives quantifiable shifts in the position of peaks in asymmetric reflections

Dislocations in Epilayer• Dislocations in epilayer are generated during growth

process.

• These dislocations do not shift the rocking curve, but broaden the rocking curve and add to diffuse scattering

• A simple relation between the dislocation density “ρ”(cm-2) and broadening of the rocking curve “β” (radians) is expressed as

2

29bβρ =

Where b is the Burgers vector for the dislocation in cm

A crystal vector which denotes the amount and direction of atomic displacement which will occur within a crystal when dislocation moves.

Dislocations in Epilayer• GaAs (004) rocking curve

for 1.5 µm GaAs/Si(001) sample. The FWHM of the rocking curve is 475 arc sec (0.132o)

• For GaAs with 60o

dislocations, b= 4.0 Å. This leads to a value of dislocation density of 3.7x 108 cm-2

Nor

mal

ized

Int

ensi

tyDelta Theta

J.E. Ayer etal. J. Cryst. Growth 125 (1992) 329

Homogeneity

• Epitaxial layers are not grown uniform across their area

• Hence a layer parameter must be mapped across the film area to record its inhomogeneity.

• This is accomplished using automated stage and data collection systems

Homogeneity

Variation of InIn content in InAlAsInAlAs layer on GaAsGaAs

Suggested Readings

• High Resolution X-ray Scattering from Thin Films and Multilayers (V. Holy, U. Pietsch, T. Baumbach) Springer Tracts in Modern Physics

• High Resolution X-Ray Diffractometry and Topography D. K. Bowen, B. K. Tanner, Taylor and Francis