XX--ray Reflectometryray ReflectometryXX--ray Reflectometryray Reflectometry

Determination of layer properties such asDetermination of layer properties such as Li id hLi id hDetermination of layer properties such as Determination of layer properties such as

thickness, thickness,

density,density,independently of independently of each othereach other

Liquid, amorphous, Liquid, amorphous, polypoly--crystalline, crystalline, epitaxial or any epitaxial or any combination preparedcombination prepareddensity,density,

roughnessroughnesseach othereach other combination, prepared combination, prepared

on glass, metals, semion glass, metals, semi--conductors or fluidsconductors or fluids

NonNon--destructive and nondestructive and non--contact method for thickness contact method for thickness determination between 2determination between 2--200nm with a precision of about 0.1 nm200nm with a precision of about 0.1 nm

The method uses a natural property of every matter: the refraction index for X-rays is smaller than unity. This means that the effect of total external reflection is observed for X-rays This effect is similar to the effect of totalreflection is observed for X rays. This effect is similar to the effect of total reflection, which is known from visible light (e.g. a diver tries to watch the beach from below the water line). The difference is that for X-rays this effect

is only observed at very low incident angles, below 1°.

Transmission and reflection

At the boundary of theAt the boundary of the medium, part of the light will be reflected at angle θr while the other part will bethe other part will be transmitted through the sample at angle θt. Snell's law requires that all three beams be in thethat all three beams be in the plane of incidence. The plane of incidence is defined as that l h h hplane which contains the input

beam, the output beam, and the direction normal to the

Schematic showing the incident, reflected, and transmitted light.

sample surface. T and R are defined as the ratio of the light intensity being transmitted or reflected over the incident light intensity on the sample.

xx--ray reflectometryray reflectometryxx ray reflectometryray reflectometryreflection at a ideally flat surface theory

n1 Θ1Θ1

y

n2 < n1 Θ2

n1 cos Θ1 = n2 cos Θ2 n1 cos ΘC = n2

thin filmsinterference

n1 ΘΘ

d = λ/2ΔΘinterference

1

n3

Θ1Θ1

Θ1n2

film densityfilm densityxx--ray reflectometryray reflectometry film densityfilm densityThe complex refractive index in the x-ray region is slightly less h 1 d i i b

xx ray reflectometryray reflectometry

than 1 and is given byβδ in +−=1~

dispersion absorptiondispersion absorptionFor frequencies far greater than the resonance frequencies, υ0, of the atom δ can be given by the expression Bohr atomic radius electron density

e nrne λλδ2

022

y

( ) ee n

cm⋅==

πλ

πεδ

2220

20⋅= nZn atome

ρ⋅=A

Nn Aatom ( )

ρπλλ

πεδ ⋅⋅==

ANZr

cmne Ae

222

202

20

2

( )πε 0

film densityfilm densityxx--ray reflectometryray reflectometry

We consider reflection at an interface between air, nair = 1 and another

film densityfilm densityxx ray reflectometryray reflectometry

material, n1 = 1 -δ . For incident angles below a critical angle, θc, (θ < θc), total reflection occurs. By applying Snell’s law and small angle approximations, the critical angle, θc can be expressed as

θ

θδ =− c2

cos1

( )λ

θ

′+

−≈

fZ

c

2

21

( ) ρπλδθ ⋅

′+=≈⇔

AfZNr

Ac02

Film densityFilm densityFilm densityFilm density

( )λ ′+ fZr 2 ( ) ρπλδθ ⋅

+=≈

AfZNr

Ac02

b f ll i hb f ll i hAbrupt fall in the Abrupt fall in the reflected intensityreflected intensity

Film thicknessFilm thicknessFilm thicknessFilm thickness

θθ > cθθ >The XThe X--ray beam ray beam penetrates inside the filmpenetrates inside the film

Reflection therefore occurs at the top and the bottom surfaces of the film. The interference between the

dθθ

λ−

≈+1

12 interference between the

rays reflected from the top and the bottom of the film surfaces results in

mm θθ +12

interference fringes which does not depend on the frequency like in the case of optical spectroscopy but areoptical spectroscopy but are angle dependent.

xx ray reflectometryray reflectometrysurface roughness

xx--ray reflectometryray reflectometryg

increasing surface roughness leads to a f d ffaster decay of reflectivity

xx--ray reflectometryray reflectometryxx ray reflectometryray reflectometrysetup & substrates

setup• fixed x-ray source (Cu Kα)

Θ 2Θ t ( l /d t t )substrates•silicon wafers (best)• Θ-2Θ geometry (sample/detector)

• monochromator (graphite)• detector: NaI scintillation counter• knife edge to align sample

•silicon wafers (best)•glass, especially float glas•polymers (if flat enough)

e edge to a g sa p e

xx--ray reflectometryray reflectometryexample100

0 8

xx--ray reflectometryray reflectometry

10-2

10-1

0 2

0,4

0,6

0,8

Si

PolystyrolLuft

ρ el [e

- Å-3]

10-3

10

-100 0 100 200 300 400 500 6000,0

0,2

R

Siρ

z [A]

1x10-5

1x10-4

0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,610-6

1x10

data analysis: • a model is used to simulate the measured curve on the basis of the Fresnel

θ [°]

equations• parameter: electron densities, roughnesses of interfaces, thickness

xx--ray reflectometryray reflectometrysummary

xx--ray reflectometryray reflectometry

advantages• direct measurement of geometric thickness (no refractive index needed)• roughness data over large area• roughness data over large area• relatively easy to align• well-suited for standard substrates (silicon wafers, glass)

disadvantages• long measurement times (hours)• theoretical modelling of curve needs experience (many parameters that

influence each other)• cannot be combined with „cells“ (e.g. for solutions)

ellipsometryellipsometryellipsometryellipsometry

principleprinciplep

s

ns- and p-light is reflected

differentlyn1differently

• phase: determined by Δ• amplitude: determined by Ψ

n2

n3

n3 < n2< n1

ellipsometryellipsometryp yp y

Plane polarized waves that are in the plane of incidence pare referred as p-waves and plane polarized waves perpendicular to the plane of incidence as s-waves

A linearly polarized input beam is converted to an elliptically

l i d fl t d b Fpolarized reflected beam. For any angle of incidence greater than 0° and less than 90°, p-polarized light and s-polarized

ill b fl t d diff tl

• Polarized light is reflected at an oblique angle to a surface

will be reflected differently

• The change to or from a generally elliptical polarization is measured.• From these measurements, the complex index of refraction and/or the

thickness of the material can be obtained.thickness of the material can be obtained.

Theory

Polarized lightIf the electric field vectors describing the propagation of electromagnetic waves are all orientedIf the electric field vectors describing the propagation of electromagnetic waves are all oriented in one plane, the light is referred to as polarized or, more completely, linearly polarized light

Of primary interest in ellipsometry is the fact that when linearly polarized light makes a reflection on a surface, there is a shift of the phases of both the components (parallel and perpendicular to the plane of incidence). The shift is, in general, not the same for both, hence an elliptically polarized wave results.

a) Two linearly polarized beams, in phase, add to another linearly polarized beam. b) Two linearly polarized beams, out of phase by 90o; the resultant beam is circularly polarized. If the phase difference is

h h l h ll ll l dother than 90o, light is elliptically polarized

Theory

Reflection of light on a surfacePlane polarized waves that are in the plane of incidence are referred as p-waves and plane polarized waves perpendicular to the plane ofperpendicular to the plane of incidence as s-waves

The FresnelFresnel reflection coefficient r is the ratio of the amplitude of the reflected wave to the amplitude of the incident wave for a single interface. The Fresnel reflection coefficients are different for p and s waves:different for p and s waves:

2112 cos~cos~ φφ NNr p − 2211 cos~cos~ φφ NNr s −

2112

211212 cos~cos~ φφ NN

r p

+=

2211

221112 cos~cos~ φφ NN

r+

=

Theory The complex index of refraction When light passes through a medium,

~

g p g ,it interacts with it; it is slowed-down and absorbed by matter

iknN −=υcn =

where n is the index of refraction, k is called the “extinction coefficient”, kis defined through the absorption coefficient α. In an absorbing medium:

λzeI)z(I α−= 0

απ

λ4

=kand

Determines how fast the amplitude of the

dFor dielectric material k=0 (no of the light is absorbed)

wave decreases

Theory

211212 ~~

cos~cos~

φφφφ

NNNNr p −

= 221112 ~~

cos~cos~

φφφφ

NNNNr s

+−

= 22 sspp rr == RR reflectance2112 coscos φφ NN + 2211 coscos φφ NN + r,r == RR

If both media are dielectric in nature (k=0):

reflectance

2⎞⎛ angleBrewster2

1n

tan =φ

2

2

2

212

2

212 1

111

11

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

==+−

=+−

=nn

,nn

r,nn

r spsp RRo436321

angleBrewster

21

11

.,n,n

,n

tan

⇒==

=φ

Theory

Reflection and transmission with multiple interfacespThe resultant reflected wave returning to medium 1 is made up of the light reflected directly from the first interface plus all of the transmissions from the light p gapproaching the first interface from medium 2

The addition of the infinite series of partial waves leads to the resultant and has been derived by Azzam[i]. The ratio of the amplitude of the resultant reflected wave to the amplitude of the incident wave is given by:

)2exp(2312 βirrRpp

p −+=

)2exp(2312 βirrRss

s −+= 2 φπβ cosN~d

⎟⎞

⎜⎛

Total reflection coefficient

[i] R M A Azzam N M Bashara “Ellipsometry and polarized light” North-Holland Publishing Co Amsterdam

)2exp(1 2312 βirrR pp −+ )2exp(1 2312 βirr

R ss −+ 222 φλ

πβ cosN⎟⎠

⎜⎝

=

Film phase thickness[i] R.M.A. Azzam, N.M Bashara, Ellipsometry and polarized light , North-Holland Publishing Co., Amsterdam, 1977, p.283, 287

Ellipsometry definitionsLet us define δ1 as the phase difference between the parallel and perpendicular components of the

incident wave. Also, let us define δ2 as the phase difference between the same components but for the 2outgoing wave. Delta, Δ, is then defined as:

21 δδ −=ΔDelta (Del) is the change in phase difference between parallel and perpendicular components of the incident wave that occurs upon reflection.

Without regards to phase, the amplitude of both components of the beam may change upon reflection

pR Ψ is then the angle whose tangent is the ratio of the magnitudes of

sR=Ψtan

g g gthe total reflection coefficients

These definitions lead us to the These definitions lead us to the fundamental equation of ellipsometryfundamental equation of ellipsometry

pRsR

Ri =ΔΨ )exp(tan Now, in principle, given Ñ1, Ñ3 , φ1(angle of incidence) and λ,

and if Δ and Ψ are measured, one could calculate d, the thickness of a film and Ñ2, its index of refraction. These are tedious calculations, but are easily done by a computery y p

Null ellipsometryp y

We choose our polarizer orientation suchorientation such that the relative phase shift from R fl ti i j tReflection is just cancelled by the phase shift from the retarder. The angular setting of the QWP and the analyzer could be used

to determine the phase shift and attenuation ration. This is conceptual use of these elements. Actual practice is somewhat different

We know that the relative phase shifts have cancelled if we can null the signal with the analyzer

different

null the signal with the analyzer

ellipsometryellipsometryellipsometryellipsometry

summary

advantages• simple setup (easy to understand & align)• swollen layer thicknesses can be measured• swollen layer thicknesses can be measured

(there is almost no other way to do this)• segment density profile can be derived• model free data analysis by FT• model-free data analysis by FT

disadvantagedisadvantage• Fitting procedure