Nanotechnology and Nanoelectronics - Materials, Devices, Measurement Techniques Part 5 doc

catch up with each other again, a linearly polarized beam develops, which can be brought to extinction with an analyzer.. The direction of the linearly polarized beam to the reference pl

split up into a parallel and a perpendicular fraction The two fractions have a phase

difference ' = 0 After reflection the new phase difference ' of the two fractions

transforms a linear oscillation into an elliptical oscillation of the form:

ǻ B

A ǻ y x B

y

A

2

2

2

2

sin cos 2



A and B are the amplitudes of the two oscillation directions in the previous

coordi-nate system (Fig 4.35) It should be noted that the phase angle ' between the

components of the ellipse applies to the selected coordinate system only By

ro-tating the coordinate system, the phase changes In particular, ' = S / 2 holds if the

large semiaxis and the new x' axis coincide Such a coordinate transformation

takes place if the elliptically polarized beam is exposed to a quarter wave retarder

Its two polarization directions are put in the directions of the ellipse axis so that

the fast beam corresponds to the lagging component of the ellipse (angle of

rota-tion - of the quarter wave retarder from the x-direcrota-tion) Since the two beams

Fig 4.33 Geometric and material definitions in the ellipsometer experiment

Fig 4.34 Ellipsometer [71]

catch up with each other again, a linearly polarized beam develops, which can be

brought to extinction with an analyzer The direction of the linearly polarized

beam to the reference plane is K K (rather: K + S / 2) is measured with the

ana-lyzer, - is known as the direction of rotation of the quarter wave retarder The

criterion of the correct determination of K and - is the extinction of the beam

behind the analyzer Thus, the axes ratio of the ellipse is given as

) tan(

The knowledge of this angle enables us to determine the ratio of the oscillation

components B / A = tan \ in the reference plane system:

-\ cos2 cos2

2

For the phase difference ' in the reference plane system, we find

G 2 tan

2 tan

What has been achieved so far? Originally, a linearly polarized beam with the

components A0 (parallel to the reference or incidence plane) and B0 (perpendicular

to it) is produced in the polarizer This beam has the phase difference '0 = 0

(oth-erwise it would not be linearly polarized) and the amplitude ratio \0 = B0/ A0,

which is known from the rotation of the polarizer against the reference plane by

the angle \0 A particularly simple solution is obtained in the case of a 45°

rota-tion; A0 = B0, or tan \0 = 1

This method enables us to determine how the amplitude ratio and the phase

dif-ference have changed due to reflection If a theoretical statement can be made on

the \-' change resulting from the reflection at a thin film of a thickness d and

refractive index n2 [i.e., tan \ / tan \0 = f(d, n2) and ('  '0 = f(d, n2)], then the

determination of the layer thickness and refractive index should be possible by

forming the inverse functions d(\, ') and n2(\, ')

Fig 4.35 Oscillation ellipse after reflection

The theory of Fresnel and Neumann offers a solution Its knowledge, however,

is not required for the operation of the ellipsometer, as will be shown below For historical reasons, the procedure is done somewhat differently without changing the basic idea First, the compensator is usually set behind the reflecting surface Therefore, the sequence is polarizer–sample–compensator–analyzer Sec-ond, one does not rotate the compensator but the polarizer (for the compensator, a direction of r45° of the fast axis is maintained against the reference plane) Thus,

by varying the amplitude ratio B0/ A0, the ellipse created after reflection is rotated

If the position of the compensator is adjusted correctly, the elliptical oscillation is brought back into a linear form If the angles \ and 'are determined in this way, then these two values can be brought into a set of curves where ' is presented as a

function of \ The parameters are film thickness (in units of wavelength or as phase difference 2Sn2d / O) along a '-\ curve and the refractive index which has a

fixed value for each '-\ curve An example of such a curve set is found Fig 4.36 These curves are calculated according to the Fresnel Neumann theory For each wavelength, angle of inclination of the two polarization levers against the sample and path difference of the compensator, a new record of curves must be calcu-lated In our graph, many specializations have taken place For instance, the quar-ter wave retarders can be replaced by a compensator of any path difference The analysis must then be corrected accordingly Moreover, we have not treated the other pairs of solutions for the compensator and analyzer angle Furthermore, an accurate discussion about the determination of the direction of rotation of the ellipse must be done This is connected with the question about the sign of the directions of rotation during the measurement and the trigonometric functions At last, we neglected the question of how the absorption influences the measurement

It should be noted that the ellipsometer can easily be automated by a micro-processor controller Of course, a technique is required that enables us to

(from curve to curve); normal angle of the ellipsometer levers M1 = 70°, n3 = 4.05 (silicon),

extinction coefficient of the silicon k3 = 0.028, O = 546.1 nm [71]

mine the two angles of rotation of analyzer and polarizer simultaneously Such

devices are commercially available

A completely different approach consists of measuring the ellipse by rotating

the analyzer photometrically (with the multiplier) Such devices have been

con-structed as well

In the above discussion, a homogeneous layer is assumed whose refractive

in-dex is constant with depth Two values ' and\ are measured, which deliver the

thickness and the refractive index However, our assumption can be invalid if, for

instance, one or more films are deposited on the first one, or if the conditions of

the depositions are changed so that a refractive index profile is produced

There-fore, an improved model and a sophisticated instrumentation is required Instead

of using a single ellipsometer wavelength, the whole available spectrum can be

used This method is known as spectroscopic ellipsometry, which requires new

approaches in data evaluation On the basis of well known technological data such

as film thickness, film material, and profiles, a model of the layer structure can be

set up and the spectral '-\ curves can be determined The model is subsequently

improved by fitting the resulting '-\ curves to the measured curves The

knowl-edge on some layers is so good that the derived profile even delivers physical

models of the film As an example, the ratio of amorphous to microcrystalline

fractions can be determined during growth of amorphous films The surface

roughness of the substrate can be determined, and the transition layers between

substrate and film or between the films can be resolved to within Ångströms

In spectroscopic ellipsometry, the components of the dielectric function H rather

than ' and \ are plotted The transformation of ' and \ to H is given by

¸¸

·

¨¨

§





2 2 2

1 2 1

2

1

) cos 2 sin 1 (

) sin 2 sin 2 (cos tan

1 sin

ǻ

ǻ

\

\

\ M

M

2 1 2 1 2

2

) cos 2 sin 1

(

sin 4 sin tan sin

ǻ

ǻ

\

\ M M

H

An example of the dielectric function (H2) depending on the crystalline state of Si

is shown in Fig 4.37 An evaluation of the dielectric function and the

transfor-mation to a layer model is depicted in Fig 4.38

Profilometer

A thin film, whose thickness is to be measured, is grown on a substrate Therefore,

the sample is partly covered with wax or photoresist so that a sharp edge between

the covered and untreated surface is given The untreated surface is etched in acid,

which removes the film but does not attack the substrate (selective etching)

Con-sequently, the edge deepens in the sample Now, the cover (wax) is removed The

edge’s depth is measured with a needle, which is moved across the edge and

which is sensitive to changes in the surface (Fig 4.39)

The vertical position of the needle is checked with a piezoconverter (with step

heights larger than 2 nm) or an inductive converter (with step heights larger than

some micrometers) These machines are called Talysurf or Talystep An example

of a measurement is shown in Fig 4.40 When small edge differences are to be measured, the major difficulty in the handling of the device is the leveling of the unetched surface serving as measuring reference If this plane and the needle’s path do not correspond, the noise during the high measurement amplification will prevent a reasonable determination of the height In the meantime, however, self-adjusting versions are available

Scanning Tunneling Microscopy (STM), Atomic Force Microscopy (AFM)

x STM The surface of a conducting material is covered with an electron cloud, whose density reduces with increasing distance from the surface

Fig 4.37 Dielectric functions for amorphous, polycrystalline, and

monocrystalline Si [72]

Fig 4.38 Layer sequence derived from spectroscopic ellipsometry [72]

If a metal tip is within a close distance to this surface, a current flows between the tip and the surface The current flow begins from a distance

of about 1 nm, and it decreases by a factor of 10 for every reduction of

0.1 nm in the distance This phenomenon can be used for an x-y

presen-tation of the roughness When moving the tip laterally, a constant current

is maintained by following the distance of the tip The necessary adjust-ment is a measure of the roughness The fitting is done with piezoelectric actors These piezoelements can displace the tip with a minimum incre-ment of 107 mm / V

x AFM Basically, this system consists of a cantilever with a tip, a devia-tion sensor, a piezoactor, and a feedback control (Fig 4.41) If the tip is within a small distance to the surface, an atomic force develops between the tip and the surface so that the cantilever is bent upwards A regulator keeps a constant force to the surface of the sample The input signal for the regulator is laser light, which is reflected by the cantilever and which

is sensitive to its position

Fig 4.39 Thickness measurement with a needle

Fig 4.40 Measuring an edge with an inductive needle [73] Quartz deposition on glass

substrate Test ridges are produced by removing the mask Vertical magnification 1,000,000 fold, one small division represents 2 nm Horizontal magnification 200fold, one small division represents 0.025 mm Thickness of the deposited layer (mean value) approximately

26 nm (25.9 mm on the diagram)

Two images of the surfaces of CVD diamond films after thermochemical

pol-ishing (Fig 4.42) are shown as examples of such a measurement With AFM it is

possible to determine a surface roughness in the nanometer range

4.2.2 Crystallinity

The crystalline state of a material is best investigated with a diffraction experiment

(Fig 4.43) A monochromatic x-ray or electron beam impinging on a crystal with

the three primitive axes a&, b&, and c& is assumed An electron beam of energy E

can be considered as a wave with the wavelength

[eV]

12 Å]

[

E

The wave vector of the incoming wave is k& with k = 2S / O The wave vector ' k

of the scattered wave has the same wavelength O

The phase difference between the incoming beam serving as a reference and the

outgoing beam is

a k a k a

k& & & '& &

Now, all scattered amplitudes from every lattice point must be added together The

vector a& is generalized to a vector

c o b n a

m& & &

&





The total amplitude is proportional to

¦

o m

k c o b a m i k

i

A

,

) (

e

e & & & &

&

&

&

&

U

U

¦

¦

¦ei m a&'k& ei b&'k& ei o c&'k& (4.28)

Fig 4.41 Atomic force microscope (schematic) [74]

The intensity, I, of the scattered beam is proportional to |A2| Every sum of the

right hand side of Eq 4.28 can be written in the form

) (

) ( 1

0

) (

e 1

e 1 e

k a i

k a M i M

m

k a

m

i

&

&

&

&

&

&

'



'





'







In order to get the total intensity, we multiply Eq 4.29 (and the other two sums)

by their conjugate complexes This delivers

Fig 4.42 (a) SEM images of an as-grown CVD diamond film of optical grade—average

surface roughness of 30 µm (profilometer measurement), (b) AFM image of the same

surface after thermochemical polishing—average surface roughness of 1.3 nm [75]

25

50

0 nm

] 2 / ) [(

sin

] 2 / ) ( [

sin

2

2

1

k a

k a M

I & &

&

&

'

The graph of Eq 4.30 is a curve with sharp maxima (“lines”) The maxima occur

for

q

k

where q is an integer This is one of Laue’s equations The other two are

r

k

s

k

The curve sketching of Eq 4.31a shows, for instance, that the maximum height for

I1 is v M2, while the width is v 2S / M I1 is proportional to its height (v M2)

times its width (v 1 / M), i.e., proportional to M Thus, the intensity of the central

reflex, I, is proportional to M 3 in three dimensions

We define a new set of vectors A&, B&, C&, which satisfy the relations

0

0

ʌ

2

c

A

b

A

a

A

&

&

&

&

&

&

0

ʌ 2 0

c B

b B

a B

&

&

&

&

&

&

ʌ 2 0 0

c C

b C

a C

&

&

&

&

&

&

(4.32)

If these vectors are additionally normalized in the form

c b

a

a c

B

c b

a

c b

A

&

&

&

&

&

&

&

&

&

&

&

&

u u

u u

ʌ

2

ʌ

2

, ʌ

2

c b

a

b a

C & & &

&

&

&

u

u

(4.33) then every vector

C s B r A

q

k& & & &

with integer numbers q, r, and s is a solution of the Laue equations The vectors

A&, B&, C& define the fundamental vectors of the reciprocal lattice

It should be noted that only the contribution of the lattice structure to the

dif-fraction pattern has been regarded up to now Of course, the so-called atomic

scattering factors must be considered for the calculation of the expected

intensi-ties, i.e., the scattering ability per atom If two sources of irradiation are compared

regarding the investigation of thin films, it turns out that a thin film of a few

na-nometers diffracts an electron beam so that useful information can be obtained

while the same film is not suitable for x-ray diffraction

X-ray diffraction (XRD) There are several ways to utilize the Laue equations

One of them is white light irradiation, i.e., a broad x-ray spectrum is illuminated

on the crystal to be examined The crystal (or rather all sets of lattice planes)

inter-acts with the light of the wavelength (i.e., k' ) that fulfills the Laue equations &

This method offers some advantages such as a quick determination of the

crys-tal symmetry and orientation As a disadvantage, the lattice constant cannot be

determined

Vice versa, a monochromatic beam can be used, which is exposed on the

pow-der of a crystal to be examined Therefore, there is always a great number of

crystallites (powder grains) in the correct orientation for a given wavelength so

that the Laue equations are fulfilled again The important information is the

inten-sity as a function of the diffraction angle

For crystalline materials, the wafer is usually rotated (e.g., by an angle T)

Si-multaneously, the detector is rotated by an angle 2T (Bragg-Brentano

diffracto-meter)

An efficient version of XRD is x-ray topography The fundamental idea

con-sists of aligning the crystal which to be examined in such a way that a reflection is

measured under a certain angle A perfect crystal should maintain the diffraction

intensity if the beam (or rather the crystal) is shifted laterally Every imperfection

of the crystal violates the diffraction equations Usually, the structure is created in

such a way that a first reference crystal is carefully adjusted so that a sharp

mono-chromatic beam is produced This one is in turn directed toward the crystal, which

should be measured, shifted perpendicularly to the beam An example is given in

Fig 4.44

Additionally, the system can be improved by “rocking” the crystal

perpendicu-larly to the plane of incidence (Fig 4.45) As an advantage of this rocking setup,

extremely small deviations in the lattice constant can be picked up If, for instance,

heteroepitaxial films are deposited, two (a doublet line) instead of only one signals

are sometimes found This means that the film still differs from the substrate

Fig 4.44 (a) Wafer before heat treatment and (b) after formation of dislocations by

oxidation at 1200 °C Examined with x-ray topography [76]

(a) (b)