Reflection

**Reflection**

• Specular or mirror like reflection from

smooth surfaces

• Diffuse reflection from rough surfaces

Specular X-ray X ray Reflectivity

• A non-destructive, routine technique, used for estimation

of density, thickness and roughness of thin film structures

(single layer and multi-layered)

• Based on total external reflection of X-rays X rays from surfaces

and interfaces

• Can be used with amorphous, crystalline and liquid

samples

• Used for typical layer thickness between 5 Å and 400 nm

and surface roughness from 0 to 20 Å

• This technique does not work effectively if there is no

difference between the electron density of different layers

or layer and substrate

Normalized Reflectivity (Arbitrary Units)

∆α I

Incident Angle (degrees)

Typical Measurement Setup

Source Detector

Knife-edge

α I α F

Sample

∆α F

Alignment

Slit

X-ray

Tube 0 Sample

(Cu) Tilt (χ)

o

Z

ω

Rotation (φ) not needed

Slit

0 o

Experimental Setup

Detector

(Scintillation)

• According to α -4

I law of Fresnel reflectivity, the intensity

leaving a smooth surface decreases very rapidly on

increasing the angle of incidence

• Since XRR requires recording reflected intensity over 5-6

orders of magnitude, highly intense X-ray source and

detector with low noise are needed

• In order to measure the angles accurately, thus to minimize

error in the results, the rotational axis of the sample circle

(ω-circle) has to be aligned exactly with the sample surface.

This is accomplished in following steps:

Sample Alignment

• Lateral movement and rocking sample across the primary

beam (ω-scan) are iterated until the maximum intensity of

ω-scans equals half the intensity of the primary beam,

compared with the intensity measured without the

sample. With this ω- axis lies on the sample surface and

this surface is parallel to the primary beam direction

• The angular position of the sample after the adjustment,

however, may not coincide with the zero point of the

ω-circle. This is caused by various surface treatments or

by the miscut of the sample surface with respect to any

crystallographic main axis. The following step is required

to redefine the ω-scale.

X-ray Tube

(Cu)

Slit

Z

Sample

ω

Slit

Detector

(Scintillation)

Sample Alignment: Redefining ω-scale scale

• To redefine the ω-scale we choose an angle of incidence

(α I ) in the range 0< α I < α C; where α C is the critical angle

of incidence for total external reflection of X-rays.

Typically this angle is chosen to be 0.2 o .

• Angular position of the specularly reflected beam is

measured on the detector circle 2θ. The sample surface is

also corrected for any tilt (χ-scan). If 2θ ≠2α I; ω scale is

readjusted by (2θ/2- α I). This procedure may be repeated

for different values of α I to improve the precision of the

sample alignment

This completes the sample alignment. This procedure is

repeated for every sample.

sample

Reflectivity Measurement: Setup

• Reflectivity experiments are optimized in such a way that

the specular reflectivity and large features characterizing

the sample (Bragg’s peak, Kiessig oscillations) appear up

to a large value of α I (~2 o ).

• Higher angular resolution is needed to separate specular

specular from diffuse scattering events. This can be

achieved by decreasing angular divergence of incident

beam (∆α I ) and angular acceptance of detector (∆α F ). In

practice low ∆α I and (∆α F) can be obtained by using

narrow slits at incident beam and detector side

respectively. However, narrow slits decrease the intensity

thus increasing the experimental time.

• To obtain high resolution and good scattered intensity, the

following steps are taken

Reflectivity Measurement: Setup

• As a trade-off, higher values ∆αI and ∆αF (hence wider

slits) are used but the irradiated sample area is reduced to

achieve sufficient angular resolution.

• In practice, this is achieved by using a knife-edge very

close to the axis of sample rotation i.e. to the sample

surface.

• Under these conditions only those beams leaving the

sample surface directly below the knife-edge arrive at

detector.

Source Detector

Knife-edge

∆α I

α I α F

Sample

∆α F

Reflectivity Measurement

• Different ranges of the reflected curve are selected and

recorded under various conditions of angular resolution and

counting time. While the measurements near αC are carried

out with highest resolution, angular resolution can be

relaxed at higher angles. Typically measurements are setup

to have at least 1500 counts for the maxima of each

oscillation.

• The specular reflectivity is recorded while running ω-2θ

scan, where ω is the angular position of sample circle and

2θ is the angular position of detector. In this scan, αI and

angle of exit (αF ) are changed simultaneously and αI = αF .

• High resolution measurements with Four-bounce

monochromator are typically for films thicker than ~0.5µm

Transmission and **Reflection** of X-rays X rays

ur r

E r

r r

ikr

= E e

uur r

E r

ur r

ik r r =Φ E e

n 0

n

( )

α I

O

( )

( )

r r O

α R

α T

uur r

E r =Φ E e

t f O

r ur

/

ikr

Basic Equations: Density of Single layer

• At X-ray frequencies, the refractive index can be expressed as

n= 1-δ-iβ (1)

Where 2

e e r ρ λ

⎡ ⎤

δ = ⎢ ⎥

⎣ 2Π

⎦

⎛ µλ ⎞

β = ⎜ ⎟

⎝4Π ⎠

δ(λ) ) and β(λ) ) ~ 10 -6 and describes the dispersion and

absorption terms

• ρ e = electron density (Z electrons/ atom)

• λ= wavelength of X-ray

• µ=linear absorption coefficient for energies far from X-ray

threshold

• r e = classical electron radius =e 2 /mc 2

Basic Equations: Density of Single layer

• For specular X-ray reflectivity α I = α R ; angle of incidence

is equal to angle of exit.

• Since the real part of index of refraction of materials if less

than unity (index of refraction for vacuum) the material is

for X-rays less refractive than it is for vacuum.

• According for Snell-Descartes law

cosα

I

cos

T

n

n

α = (2)

0

Basic Equations: Density of Single layer

• There is a critical angle of incidence α C for which the X-rays are

totally reflected at the interface, hence α T =0. Neglecting the

absorption in this case, we find

cosα

I

n ≅1− δ =

cosα

• Expanding the cosine for small angles

α δ ρ λ

−3

c ≈ 2 = 1.64*10 m *

ρe

A

ρ m =

N Z

where Z is the atomic number, A is the mass number and N A is

Avogadro’s constant

αα C is determined at R( R(αα I) ) =R max/2 max/2

A

Basic Equation:Thickness of Single Layer

• Reflectivity of a single layer deposited on a semi-infinite

substrate can be expressed as

r1+ re 2 R =

1+

rre

1 2

0 z

0 z

T

−2ik

t

−2ik

t

Where r1,2 1,2 are the Fresnel reflectivity coefficients of the free

surface and the substrate interface respectively, k0z, 0z is the

vertical component of the wave vector of the beam transmitted

through the layer and t is the layer thickness.

•Intensity maxima occurs whenever exp(-2ik exp( 2ik0zt) 0zt)

=1

i.e. at angle positions αim im

2

(4)

(3)

Basic Equation:Thickness of Single layer

• Alternatively, for intensity maxima, the path difference

between the reflected waves should be an integral multiple

of the incident wavelength

2 2

2 sin Im sin C

t α − α = mλ

(5)

Where “m” is an integer

In most cases, the angle of incidence are small, hence (5) can be

expressed as

2 2 2

Im C

2

m

⎛ λ ⎞

α − α = ⎜ ⎟

⎝ t ⎠

Thickness Calculation

2

(6)

• Squares of the positions of

the intensity maxima

versus squares of the

•

Kiessig fringe order “m”

is plotted

The slope of the linear

dependence is used to

estimate layer thickness

“t” while intercept of the

line at m=0 is used to

determine αC.

Accuracy in Estimation

• Accuracy in density is defined as

∆ ρ ⎛δα⎞ I = 2⎜

⎟

ρ ⎝ αC⎠

Where δα I is the step width of the goniometer

•Accuracy in thickness is defined as

∆ t ⎛δα ⎞ I 1

= ⎜ ⎟≈

t ⎝ αC

⎠ m

Where m max is the largest fringe order that is detected in the

reflectivity curve with an accuracy of one-half of a fringe period

max

Effect of Roughness

• In real world surfaces/ thin film structures

are not perfectly smooth and posses surface

and/or interface roughness

• While the presence of surface roughness

decreases the specular intensity of the

whole curve progressively, interface

roughness gives rise to progressive damping

of the Kiessig fringes.

Log (Normalized reflected intensity)

Normalized reflected intensity

Reflectivity curves for films with

different thickness

1.E+00

1.E-02

1.E-04

1.E-06

Ir metal film on Si

substrate at E=8049

5.nm

10.nm

50.nm

100.nm

0 1 2

Incident Angle (Degrees)

3 4

Reflectivity curves for materials with

different densities

1.E+00

1.E-02

1.E-04

1.E-06

30 nm films on Si

substrate at E= 8049 eV

Si=2.33

Ti=4.51

Ta=16.65

Ir=22.65

0 0.5 1

Incident Angle (Degree)

1.5 2

Normalized reflected intensity

Normalized reflected intensity

Reflectivity curves for films with

different film surface roughness

1.E+00

1.E-02

1.E-04

1.E-06

1.E-08

0 nm

0.2 nm

0.5 nm

1 nm

1.5 nm

0 1 2

Incident Angle (Degrees)

3 4

Reflectivity curves for films with

different substrate density

1.E+00

1.E-02

1.E-04

1.E-06

1.E-08

20 nm Si on

various substrates

Si = 2.33

SiO2= 2.65

Ti = 4.51

0 1 2

Incident Angle (Degrees)

3 4

Simulated Results:

Ir metal on Si substrate

Red curve= simulated results;

Grey curve = experimental results

Suggested Readings

• Results are simulated

starting with simplest

structure; adding

•

complexity as needed

An attempt is made

to have minimum

difference between

simulated and

experimental results.

• REFSIM Version 1.2, User’s Manual

(Bruker AXS)

• High Resolution X-ray Scattering from Thin

Films and Multilayers (V. Holy, U. Pietsch,

T. Baumbach) Springer Tracts in Modern

Physics

Website of Interest

http://cindy.lbl.gov/optical_constants

• X-ray specular reflectivity curves for single

and multiple layers

• Estimation of depth penetration as a

function of incident angle or energy