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

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X-Ray Reflectivity Measurement

(From Chapter 10 of Textbook 2)

X-ray is another light source to be used to perform

reflectivity measurements.

Refractive index of materials (: X-ray):

re: classical electron radius = 2.818 × 10-15 m-1

e: electron density of the materials

x: absorption coefficient

1

Definition in typical optics:

n1sin1 = n2sin2

In X-ray optics: n1cos1 = n2cos2

 > 1  n <1,

2

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-

X-ray reflectivity from thin films:

Single layer:

Path difference = BCD

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:

Si on Ta

Slope = a

/180

b

use

So that the horizontal axis is linear

Fresnelreflectivity: 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

2

Incident

beam

k1

1

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.

E3x

E1x

E1

E3

1

3

H1y

H3y

E2x

H2y

E2

2

(horizontal field)

(scalar)

&

(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

http://en.wikipedia.org/wiki/Image:Fresnel2.png

Rs: s-polarization; TE mode

Rp: p-polarization; TM mode

Another good reference (chapter 7)

http://www.ece.rutgers.edu/~orfanidi/ewa/

In X-ray arrangement n1 = 1, change cos  sin

1

2

cos1/n2

all angles are small; sin1 ~ 1.

Snell’s law obey  cos1 = n2 cos2.

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!

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

X-ray reflection for multilayers

L. G. Parratt, “Surface studies of solids by total reflection of x-rays”, Phys. Rev. 95 359 (1954).

y

z

Electric vector of the incident beam:

Reflected beam:

Refracted beam:

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

From first boundary condition

From second boundary condition

Shape of reflection curve: two media

The Fresnel coefficient for reflection

Page 10

A, B are real value

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

0

n

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)

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!

Interface roughness

z

Probability density

Integration

Refractive index

Same roughness & refractive index profile