07_high-resolution xrd

8/7/2019 07_High-Resolution XRD

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High-Resolution XRD



What is thin film/layer?

Material so thin that its characteristics are dominated primarily by twodimensional effects and are mostly different than its bulk propertiesSource: semiconductorglossary.com

Material which dimension in the out-of-plane direction is much smallerthan in the in-plane direction.

A thin layer of something on a surface

Source: encarta.msn.com



Epitaxial Layer

A single crystal layer that has been deposited or grown on a crystallinesubstrate having the same structural arrangement.Source: photonics.com

A crystalline layer of a particular orientation on top of another crystal,where the orientation is determined by the underlying crystal.

Homoepitaxial layer

the layer and substrate are the same material and possess the same lattice parameters.

Heteroepitaxial layerthe layer material is different than the substrate and usually has different lattice parameters.



Accessible Information to X-rayDiffraction

Definition for structural types

Structure Type Definition

Perfect epitaxial Single crystal in perfect registry with the substrate that is also perfect.

Nearly perfect epitaxialSingle crystal in nearly perfect registry with the substrate that is alsonearly perfect.

Textured epitaxialLayer orientation is close to registry with the substrate in both in-plane and out-of-plane directions. Layer consists of mosaic blocks.

Textured polycrystallineCrystalline grains are preferentially oriented out-of-plane but randomin-plane. Grain size distribution.

Perfect polycrystalline Randomly oriented crystallites similar in size and shape.

Amorphous Strong interatomic bonds but no long range order.

P.F. Fewster “X-ray Scattering from Semiconductors”



Accessible Information to X-ray Diffraction

Defects that are common in epilayer structures



Crystalline state of the layers:

Epitaxial (coherent with the substrate, relaxed)

Polycrystalline (random orientation, preferred orientation)

Amorphous

Crystalline quality

Strain state (fully or partially strained, fully relaxed)

Defect structure Chemical composition

Thickness

Surface and/or interface roughness

What we want to know about thin films?



Accessible Information to X-rayDiffraction

Structural parameters that characterize various material types

× – parameters that have meaning? – parameters that could have meaning

Thickness Composition Relaxation DistortionCrystalline

sizeOrientation Defects

Perfect epitaxy × × ×Nearly perfect epitaxy × × ? ? ? × ×Textured epitaxy × × × × × × ×Textured polycrystalline × × ? × × × ?

Perfect polycrystalline × × × × ?

Amorphous × ×



Accessible Information to X-ray Diffraction



(000)

(00l)

(100)

(10l)

(200)

(20l)

Tetragonal: aII

L= a

S, a⊥

L> a

S

Cubic

Strained Layer

Tetragonaldistortion



Cubic

Cubic

Cubic

Tetragonal

ReciprocalSpace

(000) (000)

(00l) (00l)

(hkl) (hkl)

aL > aS

Perfect Layers: Relaxed and Strained



Mismatch

Consider two materials with the same space group, same atomicarrangements, but slightly different lattice parameters and elasticparameters.

aL

aL=aS

aS

aS

aS

aS

cL

aL

Beforedeposition

Afterdeposition

0

0

L

z

L

z

L

z zz

d

d d −==⊥ ε ε



Beforedeposition

After

depositionRL

R

LL

RL

R

LL

zz d

d d

a

aa −=

−==

⊥⊥

⊥ ε ε

Tetragonal Distortion

Lattice mismatch betweencubic lattice parameters:

S

S

R

L

a

aa

a

a −=

∆

Lattice mismatch induces lattice strain:

R

La

R

La

S aS a

Lc

S L aa =

S aS a

RLa

R

La

S aS a

0≠∆φ

0=∆φ

Relaxed

Strained, coherent, pseudomorphic

Partially relaxed



Mismatch

S

S

R

L

a

aam

−=

Si(004)

SiGe(004)

True lattice mismatch is:

F. C. Frank and J. H. van der Merwe, Proc. R. Soc. London, Ser. A 198 , 216 (1949).



Mismatch

Single layer AlGaAs on GaAs substrate.

Specimen angle (arc sec)

Intensity (cps)

δθ



Mismatch

The peak separation between substrate and layer is related to the change of interplanar spacing normal to the substrate through the equation:

θ δθ δ

cot−=d

d

If it is 00l reflection then the “experimental x-ray mismatch”:

d

d

a

am

δ δ ==*

True lattice mismatch is:

s

s

R

l

a

aam

−=

then

+

−=

ν

ν

1

1*mm

ν – Poisson ratio 2

*

3

1

mm ≈

≈ν



CompositionVegard's law a law stating that the lattice parameters of substitutional solid solutions vary linearly

between the values for the components and where the composition is expressed inatomic percentage.

BABAaxxaa

xx

)1(1

−+=−

For SiGe:

( ) 112007.0)1(2

1

−−+−+=−

xaxxaaSiGeSiGe

xx



Substrate misorientation

Substrates are often specified at some angle from (001) or (111).

This may need to be verified.

2

21 ω ω ϕ

−=

Rotation of the surface plane through 180o between measurements

If we do measurements at 0o and 180o to get ϕ0 and at 90o and 270o to get ϕ90, thenmaximum ϕmax is given by:

=

+=

0

90max

90

2

0

2

max

tan

tanarctan

tantanarctan

ϕ

ϕ φ

ϕ ϕ ϕ



Layer TiltIf the layer is tilted relative to the substrate then this will result in a shift of thelayer peak relative to that of the substrate.

This is not connected with the composition.

( )2

1800 δθ δθ δθ

+= in-plane sample rotation

We do not know direction of the maximum tilt, φ 0

( )

( ) 00180

0090

00

cos180cos

sin90cos

cos

φ β φ β

φ β φ β

φ β

−=+=∆

−=+=∆=∆

then0

900tan ∆

∆

=φ

True splitting mismatch:

β – tilt angle, α – rotation angleDisplacement of a layer peak: α β cos



Relaxation

The problems occur when the elastic parameters are incapable of accommodating the distortions necessary for perfect epitaxy.



Relaxation

Before we assumed that our layer is completely coherent. In this case it is enough to measure misfit only along (00 l ) direction.

For partially relaxed layers we need to measure misfit parallel to theinterface as well as perpendicular.

For this we need an asymmetric reflection (e.g. 224, 113).



Relaxation

The effect of tilt on the peak splitting is reversed if the specimen is

rotated by 180o about its surface normal.The splitting due to mismatch will not be affected by such rotation.

We can make grazing incidence or grazing exit measurements toseparate the tilt from the true splitting.



Relaxation

The resulting measured splittings are now different between these twogeometries:

δφ δθ θ +=∆ i – grazing incidence

– grazing exitδφ δθ θ −=∆ e

We need to know the lattice parameter of the layer parallel and perpendicular tothe substrate: al, bl, and cl. From these we may calculate the relaxation and thefully relaxed lattice parameter as.

Consider al = bl (tetragonal distortion).

δφ φ φ

δθ θ θ

+=

+=

sl

sl

φ – angle between reflecting plane and the surface



Relaxation

θ λ sin2

1

2

2

2

22

2

d

c

l

a

k h

d l l hkl

=

++

=

Using interplanar spacing equation and Bragg law:

we obtain cell constants for the layer (001 oriented):

2

22

sin2

cossin2

l

k hl a

l c

l

l

l l

l

+=

=

θ

λ

φ θ

λ



Relaxation

( )

2

222

222

cottan

tancosec

l a

l k hcl

ck ha

s

l

−

+−∆

++−

+=

φ θ θ

θ φ

If we know that (001) planes of layer and substrate remain parallel, wecan derive the relaxation from just symmetric and the grazingincidence reflections (no grazing exit).

Important in quality control to save measurement time.

The symmetric reflection provides cl.

valid if δθ and δφ are small.



Determination of Thickness

Interference fringes observedin the scattering pattern, dueto different optical paths of the x-rays, are related to thethickness of the layers.

( )( )

( )

1

21

21

21

cos2~

sinsin2 ω ω

λ

ω ω

λ

∆

−

−

−=

nnnnt



Determination of Thickness

Interference fringesobserved in thescattering pattern,due to differentoptical paths of thex-rays, are related to

the thickness of thelayers.

( )

( )21

21

sinsin2 ω ω

λ

−

−=

nnt

Substrate Layer Separation

S-peak: L-peak: Separation:Omega(°) 34.5649 Omega(°) 33.9748 Omega(°) 0.590172Theta(°) 69.1298 2Theta(°) 67.9495 2Theta(°) 1.18034

Layer Thickness

Mean fringe period (°): 0.09368Mean thickness (um): 0.113 ± 0.003

2Theta/Omega (°) Fringe Period (°) Thickness (um)_____________________________________________________________________________

66.22698 - 66.32140 0.09442 0.11163766.32140 - 66.41430 0.09290 0.11352866.41430 - 66.50568 0.09138 0.11548166.50568 - 66.59858 0.09290 0.113648

66.59858 - 66.69300 0.09442 0.11187866.69300 - 66.78327 0.09027 0.117079



Area Homogeneity

Whatever the crystal growers claim, epitaxial layers are not uniformacross their area!

1% consistency is good.

3×3 grid

9×9 gridSurface mesh plot showing the variation of In

content in an InAlAs layer on GaAs



Relaxed SiGe on Si(001)

Shape of the RLP mightprovide much moreinformation



Heteroepitaxy Compressive stress

L

S

Heteroepitaxy Tensile stress

Homoepitaxy d-spacing variation

Heteroepitaxy Mosaicity

Finitethicknesseffect

cL < aS

Real RLP shapes



(000)

(00l)

(hkl)

Partially Relaxed + Mosaicity

(000)

(00l)

(hkl)

Partially Relaxed



Reciprocal Space (revisited)

The Ewald sphere construction for a set of planes at the correctBragg angle

θ λ θ λ

sin22

1

2

1sin

1 *

hkl

hkl

hkl d d

=→=== dOC

*

hkl dOB =




Bragg’s law expressed in vector notation.

Vectors (s – s0) and d*hkl are parallel and the ratio of the moduli is λ.

( ) *

hkl d

ss0 =

−

λ



(000)

(00l)

(hkl)

SymmetricalScan

AsymmetricalScan

(000)

(00l)

(hkl)

(00l) scan

(h00) scan

(h00)

(-hkl)

(00l) scan

Relaxed Layer Strained Layer

Scan Directions

Incident

beam

Diffracted

beam Scatteringvector

hkl

hkl

d

1sin2 * ===− dss 0

λ

θ

λ λ

0ss −

λ

s

λ

0sθθθ λ sin2 hkl d =

Reciprocal Lattice Point



Sample Surface

(00l)

Symmetrical Scanθ - 2θ scan

θθ

2θ

(hkl)

Asymmetrical Scanω - 2θ scan

αα = θ − ω

ω

2θ

Scan Directions



(00l)

(hkl)

Symmetricalω - 2θ scan

Asymmetricalω - 2θ scan

Sample Surface

2θ scan

Scan directions

ω scanω scan



(000)

(00l)

ω direction

ω-2θ direction

Defined by receivingoptics (e.g. slits)

Mosaicity



(000)

(00l)

ω direction

ω-2θ direction

Symmetrical Scan

receivingslit

analyzercrystal

mosaicity

receivingslit

analyzercrystal

d-spacing variation



With receiving slitWith channel analyzer

(002)SrTiO3

(220)SrRuO3



Reciprocal Space



Reciprocal Space

ω-scan is in the direction of an arc centered on the origin2θ-scan is an arc along Ewald sphere circumferenceω-2θ scan is always strait line pointing away from the origin of the reciprocal space



High-Resolution Diffractometry

The geometry of the Bragg-Brentano diffractometer.




Schematic of high resolution double-axis instrument




The geometry of the double crystal diffractometer.




Bartels



Incident Beam:

X-ray Hybrid Monochromator

High Resolution Geometry

Mirror

Sample

F

θ = 0.5o

θ < 19 arcsec

Parabola

Ge(220) Crystal

07_high-resolution xrd

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