Glancing Angle X-ray Diffraction (GAXRD) Technique

Glancing Angle X-ray X ray Diffraction (GAXRD)
• In the x-ray diffraction pattern of thin films
deposited on a substrate, contribution from
substrate to the diffraction can sometimes
overshadow the contributions from thin film.
• GAXRD is used to record the diffraction pattern of
thin films, with minimum contribution from
substrate. substrate
• Non-destructive surface sensitive technique
Technique
• Parallel, monochromatic X-ray beam falls on a sample
surface at a fixed angle of incidence (α I ) and diffraction
profile is recorded by detector only scan.
X-ray Tube with
Göbel mirror
α
Film
Substrate
Soller Slits
2θ
1
Detector
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Basics
• When the angle of incidence (α I ) of X-ray beam
decreases, so that the refractive index is less
than unity, total external reflection of X-rays
occurs below the critical angle of incidence α C .
The diffracted and scattered signals at the angle
2θ arise mainly from the limited depth from the
surface.
ε
ε’
k
E
Scattering
α
z
E”
α
α’
E’
k’
k”
x
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Scattering
Consider two homogenous slabs which have a
plane interface at z=0. Let the medium above have
the permittivity ε and below ε”. Suppose a plane,
linearly polarized electromagnetic wave
ur ur ur ur
ikr ( . −ϖ
t)
E = Ee
falls upon the interface from above
r r
k = wave vetor; r = position vector;
ϖ = 2Πν
ν = frequency of the radiation
Scattering contd..
There will be reflected wave
ur ur
ur ur
( )
".
"
above the interface and a refracted wave
ik r t −ϖ
E" = E e
ur ur ur ur
ik ( '. r−ϖt) E
'= E 'e
below the interface. α is the angle of incidence and α’ the
angle of refraction
Maxwell’s equations and the boundary conditions at z = 0
determine the refracted and reflected waves in terms of
parameters of the wave k and E.
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Scattering contd..
Collecting the relevant formulas for our purpose we get
1 2
ε '
k'= ⎡ ⎤
⎢⎣ε⎥⎦ k
ε cosα = ε'cosα' Equation (1) is required by the Maxwell’s equations and (2) is
Snell’s law (since the indices of refraction equal the square roots
of the respective permittivity
Amplitude of the refracted wave is linearly related to the amplitude
of the incident , and is written as
ur ur
E ' =Φ.
E
where φ is a 2 nd rank tensor whose components can be expressed as
Scattering contd..
2 ( εε −ε
1
2 2 α )
+ ( − )
2 ' cos
Φ = xx
ε'sinα εε'ε cos α
1
2 2 2
Eqs. (3) & (4), when expressed in more elementary terms
for the separate cases of parallel and perpendicular
polarization, are sometimes known as Fresnel’s equations
(1)
(2)
(3)
Φ yy =
2 εsinα 1
2 2
εsinα + ( ε'−εcosα) 2εsinα Φ = zz
1
2 2 2
ε'sinα + ( εε'−ε cos α)
Φ =Φ =Φ =Φ =Φ =Φ = 0
xy yx xz zx yz zy
(4)
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Glancing Angle
For high energy x-rays, the refractive index of material is
slightly less than unity, as a result of which material is for Xrays
less refractive than it is for a vacuum. As the angle of
incidence is reduced below a critical angle, X-rays undergo
total external reflection. Hence under these conditions, ε’< ε,
and therefore α< α C , and is expressed as
α = cos
C
1
2
−1 ⎡ε'⎤ ⎢⎣ε⎥⎦ Under these conditions, α’ becomes imaginary and the
refracted wave propagates parallel to the interface while
being exponentially damped below the interface.
Glancing Angle
When α≤αC , the component of k’ become
k'x= kcosα
k ' = 0
y
1
2
⎡ 2 ε '⎤
k' =−ik cos α z ⎢
−
⎣ ε ⎥⎦
(6)
Imaginary value of k z ’ provides damping, which is an
essential feature of Glancing Angle geometry.
(5)
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Penetration Depth
• Penetration depth defined as the length at which the field
has fallen by a factor e-1 and can be expressed as
1
l ≡
Im k 'z 1
=
1
2 2
k ( cos α −ε'/
ε)
λ
≅
2 2
2Π
α −α
1
2
(7)
( c )
• However this term does not include absorption effects. On
including absorption effect, equation (7) is modified as
where
{
λ
2 2 2 2 2 2
l = ⎡ ( α − α ) + 4β + α −α
⎤ 2
C C
4Π
⎢⎣ ⎥⎦
Penetration Depth
µλ
β =
4Π
}12 −
(8)
µ is the linear mass absorption coefficient
λ is the the wavelength of the x-rays
• In the range of α=0 and when no absorption occurs equation
(8) is expressed as
λ
l =
4Πα C
• For α> α C; the depth penetration depth is determined mainly
by the photo absorption of the medium and is expressed as
⎛sinα ⎞
l ≅ ⎜
µ
⎟
⎝ ⎠
(9)
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Penetration Depth Vs Angle of Incidence
This figure shows penetration depth as a function of
incident angle for Si 3 N 4 for CuK α (λ=0.154 nm) radiation
Si 3 N 4
α=0.5 o
Penetration Depth Vs. Energy
Penetration Depth (µm)
Photon Energy (eV)
• Spikes in the curve occur at energies corresponding to the
characteristic absorption energies of the material.
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Useful Applications
• By reducing the angle of incidence, hence
penetration of x-rays in to the specimen, the
contribution of substrate to the diffraction
pattern can be minimized.
• Alternatively, a specimen can be probed
through its thickness by varying the angle of
incidence.
• (a) As deposited 20 nm Ir metal
film deposited on Si wafer. XRD
curve for αα=0.5 =0.5 o and 1.0 o shows
the peaks for cubic iridium metal
phase represented by (+)
• (b) Ir film annealed at 873K for
1hr. XRD curve for αα=0.5 =0.5o shows
the presence of the dominating
IrO 2 phase (*). As αα was
increased to 1.0°, 1.0 the contribution
from the underlying layer of Ir
metal increased and the Ir peaks
dominated the XRD curve. The
results indicate the presence of an
overlying oxidized layer of Ir
metal
GAXRD: Example
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Experimental Issues
• The diffraction peaks appear to shift to higher angles at
small glancing angles.
• For ααC, shift due to refraction caused by
penetration of x-ray beam is approximately given as
( ) ( ) 1
θ α α α
∆ 2 − −
C
2 2 2
• In order to estimate crystallite size from FWHM, somewhat
higher glancing angle than the α C for film should be used.
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