1. X-Ray Reflectivity Measurement
(From Chapter 10 of Textbook 2)

2. X-ray is another light source to be used to perform
reflectivity measurements.
Refractive index of materials (: X-ray):
re: classical electron radius = × m-1
e: electron density of the materials
x: absorption coefficient

3. Definition in typical optics: n1sin1 = n2sin2
In X-ray optics: n1cos1 = n2cos2
 > 1  n <1, 2

4. Critical angle for total reflection
n1cos1 = n2cos2, n1= 1; n2=1- ;
1 = c; 2 = 0 
cosc =1-  sinc =
 and c <<1 
 ~ 10-5 – 10-6; and c ~ 0.1o – 0.5o 1 c
1-

6. X-ray reflectivity from thin films:
Single layer:
Path difference = BCD

7. Snell’s law in X-ray optics: n1cos1 = n2cos2
cos1 = n2cos2=(1-)cos2.
1- 2
cos1
When 1 , 2, and  << 1
Ignore 
Constructive interference:

8. Si on Ta
Slope = a
/180 b
use
So that the horizontal axis is linear

9. Fresnel reflectivity: classical problem of reflection of an
EM wave at an interface – continuity of electric field and
magnetic field at the interface
Refracted
beam
Reflected
beam k3 3
Reflection and Refraction:
• Random polarized beam
travel in two homogeneous,
isotropic, nondispersive, and
nonmagnetic media (n1 and n2).
Snell’s law: k2 x
Incident
beam k1 1 2 y n1 n2
and
Continuity can be written for two different cases:
(a) TE (transverse electric) polarization: electric field
is  to the plane of incidence.

10. (horizontal field) (scalar) & E1x E3x E1 E3 1 3 H1y H3y E2x H2y E22
(horizontal field)
(scalar)
&

11. (b) TM (transverse magnetic) polarization: magnetic
field is  to the plane of incidence.
E1y
E3y E1 E3 1 3
H1x
H3x
E2y
H2x E2 2

12. Another good reference (chapter 7)
Rs: s-polarization; TE mode
Rp: p-polarization; TM mode
Another good reference (chapter 7)

13. In X-ray arrangement n1 = 1, change cos  sin2
cos1/n2
all angles are small; sin1 ~ 1.
Snell’s law obey  cos1 = n2 cos2.

14. in term of
Effect of surface roughness is similar to Debye-Waller
factor
The result can be extended to multilayer. The treatment is
the same as usual optics except definition of geometry!

15. One can see that the roughness plays a major role at high wave vector transfers and that the power law regime differs from the Fresnel reflectivity at low wave vector transfers

16. X-ray reflection for multilayers
L. G. Parratt, “Surface studies of solids by total reflection of x-rays”, Phys. Rev (1954). y z
Electric vector of the incident beam:
Reflected beam:
Refracted beam:

17. Boundary conditions for the wave vector at the
interface between two media:
frequencies must be equal on either side of the interface: 1 = 2 , n1 1 = n22  n2k1 = n1k2;
wave vector components parallel to the interface are equal

18. From first boundary condition
From second boundary condition

19. Shape of reflection curve: two media
The Fresnel coefficient for reflection
Page 10
A, B are real value

20. From Snell’s law
Page 4

22. N layers of homogeneous media
Thickness of nth layer:
medium 1: air or vacuum
an : the amplitude factor for half the perpendicular depth
n-1 n

23. The continuity of the tangential components of the electric vectors for the n-1, n boundary
(1)
The continuity of the tangential components of the magnetic field for the n-1, n boundary
(2)
Solve (1) and (2); (1)fn-1+(2), (1)fn-1-(2)

25. For N layers, starting at the bottom medium (N+1 layer: substrate)
Also, a1 = 1 (air or vacuum)
Finally, the reflectivity of the system is
For rough interfaces:
Can be calculated numerically!

26. Example of two layers with roughness
Au on Si substrate

27. Same roughness & refractive index profile
Interface roughness z
Probability density
Integration
Refractive index
Same roughness & refractive index profile

28. Félix Jiménez‐Villacorta