X Ray Reflectivity

7/27/2019 X Ray Reflectivity

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ReflectionReflection

Specular or mirror like reflection from

smooth surfaces Diffuse reflection from rough surfaces

Specular XSpecular X--ray Reflectivityray Reflectivity

A non-destructive, routine technique, used for estimationofdensity, thickness and roughnessdensity, thickness and roughness of thin film structures(single layer and multi-layered)

Based on total external reflection of Xtotal external reflection of X--raysrays from surfacesand interfaces

Can be used with amorphous, crystalline and liquidsamples

Used for typical layer thickness between 5 and 400 nmand surface roughness from 0 to 20

This technique does not work effectively if there is noThis technique does not work effectively if there is nodifference between the electron density of different layersdifference between the electron density of different layers

or layer and substrateor layer and substrate


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Incident Angle (degrees)

No

rmalizedReflectivity(ArbitraryUnits)

I

I F

F

Source Detector

Sample

Knife-edge

Typical Measurement Setup


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Alignment

Detector

(Scintillation)

Slit

Sample

ZSlit

X-ray

Tube

(Cu) Tilt ()

Rotation () not needed

0o0o

Experimental SetupExperimental Setup

According to I-4 law of Fresnel reflectivity, the intensityleaving a smooth surface decreases very rapidly onincreasing the angle of incidence

Since XRR requires recording reflected intensity over 5-6orders of magnitude, highly intense X-ray source anddetector with low noise are needed

In order to measure the angles accurately, thus to minimizeerror in the results, the rotational axis of the sample circle(-circle) has to be aligned exactly with the sample surface.This is accomplished in following steps:


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Sample AlignmentSample Alignment Lateral movement and rocking sample across the primarybeam (-scan) are iterated until the maximum intensity of-scans equals half the intensity of the primary beam,compared with the intensity measured without thesample. With this - axis lies on the sample surface andthis surface is parallel to the primary beam direction

The angular position of the sample after the adjustment,

however, may not coincide with the zero point of the-circle. This is caused by various surface treatments orby the miscut of the sample surface with respect to anycrystallographic main axis. The following step is requiredto redefine the -scale.

Detector(Scintillation)

Slit

Sample

Z Slit

X-ray Tube

(Cu)

Sample Alignment: RedefiningSample Alignment: Redefining --scalescale To redefine the -scale we choose an angle of incidence

(I) in the range 0< I< C; where C is the critical angleof incidence for total external reflection of X-rays.Typically this angle is chosen to be 0.2o.

Angular position of the specularly reflected beam ismeasured on the detector circle 2. The sample surface isalso corrected for any tilt (-scan). If 2 2I; scale isreadjusted by (2/2- I). This procedure may be repeatedfor different values of I to improve the precision of thesample alignment

This completes the sample alignment. This procedure isThis completes the sample alignment. This procedure is

repeated for every samplerepeated for every sample.


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Reflectivity Measurement: SetupReflectivity Measurement: Setup

Reflectivity experiments are optimized in such a way thatthe specular reflectivity and large features characterizingthe sample (Braggs peak, Kiessig oscillations) appear upto a large value ofI (~2o).

Higher angular resolution is needed to separate specularspecular from diffuse scattering events. This can beachieved by decreasing angular divergence of incident

beam (I) and angular acceptance of detector (F). Inpractice low I and (F) can be obtained by usingnarrow slits at incident beam and detector siderespectively. However, narrow slits decrease the intensitythus increasing the experimental time.

To obtain high resolution and good scattered intensity, thefollowing steps are taken

Reflectivity Measurement: SetupReflectivity Measurement: Setup

As a trade-off, higher values I and F (hence widerslits) are used but the irradiated sample area is reduced toachieve sufficient angular resolution.

In practice, this is achieved by using a knife-edge very

close to the axis of sample rotation i.e. to the samplesurface.

Under these conditions only those beams leaving thesample surface directly below the knife-edge arrive atdetector.

I

I F

F

Source Detector

Sample

Knife-edge


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Reflectivity MeasurementReflectivity Measurement

Different ranges of the reflected curve are selected andrecorded under various conditions of angular resolution andcounting time. While the measurements nearC are carriedout with highest resolution, angular resolution can berelaxed at higher angles. Typically measurements are setup

to have at least 1500 counts for the maxima of eachoscillation.

The specular reflectivity is recorded while running -2scan, where is the angular position of sample circle and2 is the angular position of detector. In this scan, I andangle of exit (F) are changed simultaneously and I = F.

High resolution measurements with Four-bounce

monochromator are typically for films thicker than ~0.5m

Transmission and Reflection of XTransmission and Reflection of X--raysrays

IT

R

( ) i k rOr E e=r rur r

( )/i k r

t f OE r E e=

r uruur r

( ) ri k rr r OE r E e= ur ruur r

n

n0


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Basic Equations: Density of Single layerBasic Equations: Density of Single layer

e = electron density (Z electrons/ atom) = wavelength of X-ray =linear absorption coefficient for energies far from X-ray

threshold re= classical electron radius =e2/mc2

At X-ray frequencies, the refractive index can be expressed as

n= 1--i (1)Where 2

2

e er

=

4

=

(() and) and (() ~ 10) ~ 10--66 and describes the dispersion andand describes the dispersion andabsorption termsabsorption terms

Basic Equations: Density of Single layerBasic Equations: Density of Single layer

For specular X-ray reflectivity I = R; angle of incidenceis equal to angle of exit.

Since the real part of index of refraction of materials if lessthan unity (index of refraction for vacuum) the material isfor X-rays less refractive than it is for vacuum.

According for Snell-Descartes law

0

cos

cosI

T

n

n

= (2)


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Basic Equations: Density of Single layerBasic Equations: Density of Single layer There is a critical angle of incidence C for which the X-rays are

totally reflected at the interface, hence T =0. Neglecting theabsorption in this case, we find

Expanding the cosine for small angles

where Z is the atomic number, A is the mass number and NA is

Avogadros constant

cos1

cosI

T

n

=

32 1.64*10 *c m =

em

AN Z

=

CC is determined at R(is determined at R(II) =R) =Rmaxmax/2/2

(3)

Basic Equation:Thickness of Single LayerBasic Equation:Thickness of Single Layer

Reflectivity of a single layer deposited on a semi-infinitesubstrate can be expressed as

0

0

221 2

21 21

z

z

ik t

ik t

r r eR

r r e

+=

+Where rr1,21,2 are the Fresnel reflectivity coefficients of the freesurface and the substrate interface respectively, kk0z0z, is thevertical component of the wave vector of the beam transmittedthrough the layer and tt is the layer thickness.

Intensity maxima occurs wheneverexp(exp(--2ik2ik0z0z

t) =1t) =1

i.e. at angle positionsi.e. at angle positions imim

(4)


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Basic Equation:Thickness of Single layerBasic Equation:Thickness of Single layer

Alternatively, for intensity maxima, the path differencebetween the reflected waves should be an integral multipleof the incident wavelength

2 2Im2 sin sin Ct m = (5)

Where m is an integer

In most cases, the angle of incidence are small, hence (5) can beexpressed as

2

2 2 2

2Im Cm

t

=

(6)

Thickness CalculationThickness Calculation

Squares of the positions ofthe intensity maximaversus squares of theKiessig fringe order m

is plotted The slope of the linear

dependence is used toestimate layer thicknesst while intercept of theline at m=0 is used todetermine C.


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Accuracy in EstimationAccuracy in Estimation

Accuracy in density is defined as

2 I

C

=

Where I is the step width of the goniometer

Accuracy in thickness is defined as

max

1I

C

t

t m

=

Where mmax is the largest fringe order that is detected in thereflectivity curve with an accuracy of one-half of a fringe period

Effect of RoughnessEffect of Roughness

In real world surfaces/ thin film structuresare not perfectly smooth and posses surfaceand/or interface roughness

While the presence of surface roughnessdecreases the specular intensity of thewhole curve progressively, interfaceroughness gives rise to progressive dampingof the Kiessig fringes.


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Reflectivity curves for films withdifferent thickness

1.E-06

1.E-04

1.E-02

1.E+00

0 1 2 3 4

Incident Angle (Degrees)

Normalizedreflectedintensity

5.nm

10.nm

50.nm

100.nm

Ir metal film on Si

substrate at E=8049

Reflectivity curves for materials withdifferent densities

1.E-06

1.E-04

1.E-02

1.E+00

0 0.5 1 1.5 2

Incident Angle (Degree)

Log(Normalizedreflec

tedintensity

Si=2.33

Ti=4.51

Ta=16.65

Ir=22.65

30 nm films on Si

substrate at E= 8049 eV


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e ect v ty curves or ms w tdifferent film surface roughness

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

0 1 2 3 4



0 nm

0.2 nm

0.5 nm

1 nm

1.5 nm

Reflectivity curves for films withdifferent substrate density

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

0 1 2 3 4



Si = 2.33

SiO2= 2.65

Ti = 4.51

20 nm Si on

various substrates


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Simulated Results:Ir metal on Si substrate

Results are simulatedstarting with simpleststructure; addingcomplexity as needed

An attempt is madeto have minimumdifference betweensimulated andexperimental results.

Red curve= simulated results;Grey curve = experimental results

Suggested Readings

REFSIM Version 1.2, Users Manual(Bruker AXS)

High Resolution X-ray Scattering from ThinFilms and Multilayers (V. Holy, U. Pietsch,T. Baumbach) Springer Tracts in ModernPhysics


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Website of Interest

http://cindy.lbl.gov/optical_constants

X-ray specular reflectivity curves for singleand multiple layers

Estimation of depth penetration as afunction of incident angle or energy

Documents

X Ray Reflectivity