Chapter 11 Xray Reflectivity

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Chapter 11 X-ray Reflectivity
10.1 Reflection of Electromagnetic wave from a
Surface
X-ray is a kind of electromagnetic wave
n2
qt
qi
n1
qr
E
B
( E perpendicular to the plane of incidence)
Snells Law
Boundary conditions normal component of D B
should be continuous
tangential component of E H should be
continuous
Electric field perpendicular to the plane
Electric field parallel to the plane
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• Fresnel Reflectivity

n1 1 vacuum set nn2
Same for the electric field parallel to the plane
10.2 Index of Refraction X-ray
Free electron approximation
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Atomic electrons
10.3 Fresnel reflectivity (in x-ray regime)
z
n1
r
i
x
t
n1-dib
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n lt 1
Total external reflection
Critical Angle

• Fresnel Reflectivity

6
1
log
10-4
q/qc
5
7

10.4 X-ray reflection from a surface Born
Approximation
Volume integral --gt Surface integral Gauss
Theorem
Two dimensional Fourier transform of this term
Similar to the thermal diffuse scattering
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Height fluctuation functions
dW
Lorentz factor
i)
Lorentz factor
ii)
Illuminated Area
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Reflectivity
10.5 Surface Roughness Height Fluctuations
z(x,y)
(x,y)
(x,y)
Height Function
Either thermal average or configurational average
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Height fluctuation function
Height-Height correlation function
For many isotropic solids, self affined
Small h jagged surface h close to 1
smooth surface
h determines the texture of the roughness
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10.6 Examples
A. Smooth Surface
qx
qz
Only specular reflection
A. Self-Affined rough surface
specular
diffuse
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qx
qz
Transverse
Longitudinal
10.7 Reflectivity from a thin film on a substrate
X-ray
v1
r1
v2
r2
The x-rays reflected from the surface interfere
with the x-ray reflected from the interface
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r1
v1
r2
v2
1
3
2
Apply the same scheme as we have applied for a
single surface
1.
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Specular Part
Diffuse Part
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Comment 1. There is scattering from both the
surface and the interface 2. Interface with
smallest roughness contribute most. 3.
Interference fringes in the specular reflectivity
gives thickness 4. Intensity oscillation in the
diffuse scattering indicates that two interfaces
are correlated.
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Au/GaAs
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Fe/Silicon
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Silicon Homoepitaxy
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10.8 Parratts Formalism
For a smooth surface, we can calculate the
reflectivity exactly, resulting the Fresnel
reflectivity
T1
R1
T2
For Multiple interfaces
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Distorted Wave Born Approximation
(DWBA) (Vineyard et al, S.K. Sinha et al.)
k2
k1
Beyond the Born Approximation 1. Distorted Wave
Born Approximation (DWBA) 2. Nevot Croces
Approximation
3. Yoneda Wings (Angels wings) Enhancement of
the intensity at the critical angles
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10.9 Distorted Wave Born Approximation
(DWBA) (Vineyard et al, S.K. Sinha et al.)
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Diffuse Scattering
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Intensity Enhancement at the Critical angle
Specular
Yoneda Wing
Typical transverse scan
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10.10 Grazing Incident X-ray Scattering (GIXS)
transmittance
log m
2
Attenuation length
10nm
ac
ac
a. b near or below the critical angle
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Grazing Incident X-ray Scattering (GIXS)
Surface sensitivity increase since the x-ray
penetration length is only of order 100 angstrom
(Surface sensitive) Enhancement of intensity due
to the increase of the x-ray transmission coeffici
ent at the critical angle For lateral structure,
the kinematic approximation still works. Many
surface reconstruction have been studied using
the grazing incident X-ray scattering