X-ray optics in Langmuir-Blodgett films

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Submitted on 1 Jan 1987

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X-ray optics in Langmuir-Blodgett films

F. Rieutord, J.J. Benattar, Louis Bosio, P. Robin, C. Blot, R. de Kouchkovsky

To cite this version:

X-ray optics

in

_{Langmuir-Blodgett}

films

F. Rieutord

_(*),

J. J. Benattar

_(*),

L. Bosio

_(**),

P. Robin

_(***),

C. Blot

_(*)

and R. de

_Kouchkovsky

_(*)

(*)

DPhG/SPSRM 2014 CEN

_Saclay

91191 Gif-sur-Yvette Cedex, France

(**)

Laboratoire «

_Physique

_des

_liquides

et électrochimie »

ESPCI,

10 rue

_Vauquelin,

75231

_Paris,

France

(***)

THOMSON CSF - Laboratoire central de

recherche,

Domaine de

Corbeville,

BP

10,

91401

_{Orsay Cedex,}

France

(Reçu

le 20 octobre 1986,

accepté

le 25 novembre

₁₉₈₆₎

Résumé. 2014 Les milieux stratifiés et

plus

particulièrement

les films de

_{Langmuir-Blodgett}

_produisent

d’interessants effets aux incidences rasantes sur les courbes _{de réflectivité des rayons X. Nous} avons étudié la

projection

de la densité

_{électronique}

sur la normale aux couches dans des films L.B. d’acide

_béhénique

_pour

différentes

_séquences

de

_dépôt,

afin de corréler leur structure aux

_figures

d’interférence

_{particulières}

se

produisant

autour des

_pics

de

_Bragg

_(00l).

En utilisant le formalisme de

_l’optique

et la théorie

_cinématique

des rayons

X,

nous avons

_analysé

ces différents effets d’interférence. Par

ailleurs,

nous avons mis en évidence une

nouvelle structure non inclinée de ces

_systèmes.

Abstract. 2014 Stratified media and

especially Langmuir-Blodgett

films

_give

rise to

_interesting

features of the X-ray

reflectivity

curves at

_grazing

incidences. We have studied the

_projection

of the electronic

_{density along}

the normal to the

_layers

of

_{differently sequenced}

L.B. Films of behenic

_acid,

in order to correlate their structure

with the

_particular

interference patterns

occurring

around

_(00l)

_Bragg

_{peaks. Using}

both

_optical

formalism and kinematical

_theory

for

_X-rays,

we have

_analysed

these various interference

_phenomena.

Moreover, we found a new untilted structure of these systems.

Classification

Physics

Abstracts 61.10 -

68.90

1. Introduction.

X-ray

scattering techniques

under

_grazing

incidence have

_proved

to be

_powerful

tools for

_{investigating}

the surfaces and the structure of thin films

_[1].

In the case of films that are _{made up of} a few

_layers

deposited

over a thick

_substrate,

the electronic

density

profile

along

the normal to the

_layers

can be studied

_{using reflectivity experiments.}

It has been

previously

shown

_[2]

that the whole

_reflectivity

curve

displays

various

_interesting

features due to the both finite size effects and substrate influence. The aim of this paper is to

_{analyse theoretically}

and

experimen-tally

the interference effects which are visible on the

reflectivity

pattern

of stratified thin films and thus to

show how information about the film

_deposition

can be

_directly

obtained.

Our

_experiments

were carried out over thin

or-ganic

films

_{deposited by}

the classical

Langmuir-Blodgett (L.B.) technique [3, 4].

This

_technique

has

recently

been the

_subject

of a renewed interest since it allows the construction of various

_systems,

_having

both fundamental and

_{technological}

interests.

_(For

a

review see Refs.

_[3-5].)

The data were taken on standard behenic acid

_layers

_(C22H44O2)

but the results we mention

_apply

to _{any L.B. film} or _any

regular

stratified film as for

_example

composition-modulated

_amorphous

thin films

_[6].

The interference effects we

_study

here in

_detail,

and which have

_already

been observed on L.B.

multilayers

by

a few

_workers,

_mainly

consist of

subsidiary

maxima between main

_Bragg

_peaks.

The first observation of these effects was carried out

_by

Bisset and Iball

_[7].

Their

_{interpretation}

of the

_X-ray

diffraction data was

_{only qualitative}

and based on the

_analogy

with a finite size

_{optical grating.}

More

recently,

an

_{investigation}

of fine interference

struc-ture

_{in L.B. films of manganese} stearate was per-formed

_by

Pomerantz et al.

_[2]

who used both

_X-rays

[2]

and neutrons

_[8].

Their data

_processing

involved a

_{general optical}

formalism

_[9],

well suited for

computations

but which obscured the

_{physical origins}

of the observed

_patterns.

_Apart

from these

studies,

several structure determinations

_using

standard methods of

_{crystallography (i.e.}

Fourier

_transforms)

have been

_performed

on L.B. films

_including

a

_large

680

number of

_layers

which

_prevent

the observation of finite size effects

_[10].

In this _paper, we used both methods

(optical

formalism and Fourier

_transform)

to

_analyse

interfer-ence effects. Fourier transform method is less gener-al but enables

_analytic

_{calculations and easy} discrimi-nation between the

_origins

of the

_pattern

features. Thus we could

_separate

in the

reflectivity

curve

Kiessig fringes [11]

and finite-size

_{Bragg peak}

con-tributions and

_study

the _way

_they

interfere in connection with the film

_deposition

_sequence.

The observation of such interferences was made

possible only

because of the _{very small} thicknesses of the film and the

_{high regularity}

of the stratification. In our case,

using

a

_high

resolution

device,

we found that these effects were most visible on

_{samples including}

about 30

_layers

_{corresponding}

to an overall thickness of 1 000

A

_typically.

2.

_{Experimental.}

2.1 SAMPLE PREPARATION. - The L.B.

_technique

for film

_deposition

has been

_extensively

reviewed in many

places [3-5].

Let us

_only

recall that the films are

_{prepared by}

_{transferring floating organic}

mono-layers

onto solid substrate. In the most common

deposition

mode

_{(Y type) amphiphilic}

molecules

(i.e.

molecules with

_hydrophilic

head and

_aliphatic

tail)

stack in a tail-to-tail

_{configuration}

_{(see Fig. 1).}

The orientation of the first

_{layer strongly depends}

on the treatment of the substrate. The substrates we used were

_optically

_polished

circular

₍₁₀₀₎

silicon disks

₍₅₀

mm in diameter and 3 mm

_thick).

When cleaned with

_propanol,

these were found to be

hydrophilic

so that no

_deposition

was observed

during

the first immersion. If cleaned with a dilute

fluorhydric

acid

_solution,

silicon wafers become

hydrophobic

and

_aliphatic

tails of

_monolayer

bind first to the substrate. Since the

_deposition

_sequence ends

_by

an emersion of the

_sample,

the last

_layer

is

Fig.

1. - Schematic

representation

of the four

_possible

deposition

sequences for Y-type L.B. films.

normally

hydrophobic (from

the

_aliphatic

tails of the

molecules)

(Figs. 1-1, 1-3).

Attempts

were made however to _{end the sequence with}

_hydrophilic

heads

upwards (Figs. 1-2, 1-4),

by

decompressing

the

float-ing monolayer

before

_removing

the substrate.

Unfor-tunately, they

result so _{far in poor}

_quality

_samples

which did not exhibit the

_designed

features. This is

certainly

due to the

_departure

of the last

_deposited

layer

during

the

_{decompression,}

since the

hyd-rophilic

heads were then oriented towards water.

Two series of

_samples

_{corresponding}

to

_figures

1-1 and 1-3 were built with different numbers of

_layers.

The

_deposition

conditions were the same for all the

samples ;

behenic acid was

_spread

on the surface as a solution of

10- 3 moll

in chloroform. The

_subphase

was

_triple

distilled water

_(Millipore).

The

_monolayer

was

_kept

at a constant surface pressure of 35

mN. m-1

_{during deposition}

and the rate of transfer was 0.3 cm/min. We will later

_only

mention the results obtained on the 26

_layer

and 27

_layer

samples,

since

_apart

from variations due to

_layer

number,

all observed features were

_{qualitatively}

the same in a series.

2.2 X-RAY TECHNIQUES. -

_Reflectivity

measure-ments were carried out

_using

the

_experimental

device which is

_represented

in

_figure

2 and described in detail in reference

_[12].

In our

_geometry

_{81 = 02}

and the wavevector transfer

_(Q

=

kf -

ki)

is

parallel

to the z-axis

_{perpendicular}

to the substrate

_plane

(see Fig. 3).

Thus we

_investigate

here

_only

the

Fig. , 2.

-

Experimental

device for

_reflectivity

measurements ; the screw mechanism driven

_by

the motor M, acts on

₈₁

₁ and

_{02 ;}

the motor m moves

₀₂

via

SA-detector.

Fig.

3. -

Geometry

for

_X-ray

_{reflectivity.}

The

_scattering

projection

of the three dimensional electron

_density

distribution onto the z-axis.

The

_X-ray

source in this device is a conventional tube. The Cr

_Ka

radiation is obtained from a double monochromator

_equipped

with two flat

_LiF(200)

crystals

and a

_divergence

slit

_SD

which both isolates the

_Ka

_line

and collimates the incident beam. The width of

_SD

was 80 _um and the

_resulting

_angular

broadness 0.45 mrad FWHM. The

_intensity

of the monochromated beam is monitored

_by

an ionization chamber while the reflected beam is detected

_by

a scintillation counter

_placed

behind the

_analysis

slit

SA.

This

_apparatus

was

_employed

for small

_angle

measurements

_only

_{(0-50 mrad).}

At

_higher

_angles,

the

_patterns

were recorded

_by

means of classical

(0 -

2

₀₎

_goniometer

_using

V-filtered Cr K radi-ation. In the last case, the absolute values of the

reflectivity

were obtained

_by

_calibrating

on an

over-lapping

range of

angles

scanned

_using

the two

apparatuses.

The

reflectance,

defined as the ratio

I /Io

of reflected to incident

intensities,

could be measured down to

10-7.

3. Methods for

_reflectivity

_{computations.}

For

_{X-ray wavelengths,}

the refractive index n of

matter is

_{given by :}

where

e2

A is the

_{wavelength, re}

=

e 2

the classical electron

mc

radius,

p the

density,

M the atomic mass,

f

+

A f ’

+

_{i A f "}

the

_complex

atomic

_{scattering factor}

the linear

_absorption

coefficient.

_Propagation

of X-rays

through

matter can thus be described

_using

the laws of

_optics.

In

_particular,

the

_reflectivity

of a

plane

diopter

is

_given

_by

the Fresnel formulae. Note that since the real

_part

of the index is less than

_unity,

total reflection will occur at an air/matter interface if the incident

_angle

is smaller than a critical

_angle

oc =

-,,/2-5.

We shall see now how to deal with a medium in which the index varies as a function of the z-coordinate

_only

_(so-called

« stratified »

medium).

3.1 MATRIX FORMALISM. -

_Reflectivity

calcula-tions for

_angles

_{including grazing}

incidences

_require

a _very

_general

method

_taking

into account

_multiple

reflection effects. Such a method is available from

multilayer

film

_optics

_{[13, 14].}

The most convenient way to

_compute

the

_reflectivity

curve is to treat the

system

as a succession of

_homogeneous

laminae. Each lamina is characterized

_by

a transfer matrix

Mi

depending

on three

_{parameters :}

its thickness

dj,

the refractive

_{index nj}

and the incident

_angle

_0;

Mi

is

_{given by :}

where

2 irti d,

sin 0

_{and pj}

_{= nj}

sin 0. The last

_expression

is valid for

_{s-polarization}

but in the range of

angles

we

_consider,

the

_reflectivity

for the two

_{polarizations}

s and _p are about the same.

The whole medium is then

_{represented by}

a characteristic matrix which is the

_product

of the matrices of the

_constituting

laminae. Note that for

layered

systems

such as L.B.

films,

a natural inter-mediate

_stage

of this calculation is the evaluation of the matrix of one

_layer.

The reflection coefficient is

given

in terms of the matrix element

_{mij :}

where p, =

n, sin

0,

po = _no sin 0 and _{no, n,} are the indices of air and substrate

_{respectively.}

The final

_reflectivity

is then :

It should be noted that

_{angular dependence}

of the atomic

_scattering

factor

_(i.e.

the

_index)

can be

_easily

included in this

_general

formalism. The method or

equivalent computation

schemes

_[9]

have been used

previously

in many

reflectivity

studies

_{using X-rays}

[15, 16]

or neutrons

_[8,

_17,

_18].

3.2 FOURIER TRANSFORMS. - The other method

for

_reflectivity

calculations derives from kinematical

theory

of diffraction. The calculation of the reflected

amplitude

is

_performed

_by

_adding

all the

_spherical

wavelets diffracted

_by

each small volume element

d3r

of the

_sample

_{[19] :}

where

_Ao

is the

_amplitude

of the incident wave at the

origin

0 ;

rl, r2, r are the distances between the element of volume

d3r

and,

respectively

the source,

the detector and the

_origin

0.

_{Pel (r)}

is the electronic

density

in the volume

d3r.

On account of the distances involved in

_{reflectivity experiments,}

the

plane

wave

approximation (Fraunhofer

approxi-mation)

does not suffice. The

_problem

must be treated

_by

Fresnel diffraction. For a stratified

682

where

Note that such a calculation assumes that each element of volume sees the same incident wave and that the diffracted wave is not diffracted

_again

(absorption

and

_dynamical

effects are

_neglected).

Hence,

the

_expression

is valid

_only

_{for q >}

_{qc =} 4

sin

Oc

.

A

4.

_Experimental

results.

The

_reflectivity

_patterns

of all the

_samples

we studied are _very similar and the main differences concern the interference

_{phenomena occurring}

around the first

_Bragg

_peak.

Next section will be devoted to the

_study

of these interferences. The aim of this section is to

_show,

_through

the

_study

of a

typical

sample

(27

layers

deposited

over an

hyd-rophilic

substrate),

how the different

_parts

of the

reflectivity provide

the information used to build a model for the electronic

_density

_profile.

Three consecutive

_part

of a

_reflectivity

_pattern

are rep-resented in

_figure

4.

4.1 GRAZING ANGLES. - The first

part

of this

reflectivity

curve

(Fig. 4a)

_ranges between 0 and 10 mrad. For these

_angles,

the _{film may be} con-sidered

_homogeneous

with a mean refractive index nF = 1 -

8f - Bf.

Measurements of the two critical

angles

for film and substrate

_(Ocf

=

J2 8f,

Oc

=

J2 8s

respectively)

and the

_shape

of the

_reflectivity

curve at low

_angle,

which

_depends

_mainly

on

absorp-tion before

_Ocs,

allow determination of film and substrate mean

_complex

indices.

_Fringes

of

_equal

inclination

_{(Kiessig fringes)}

are

_already

visible but their

_shape

is much altered

_by

the instrumental resolution. This can be observed when

_compared

to

the theoretical curve without convolution. The latter

displays

a narrow first minimum as well as a

sup-plementary fringe.

This

_shape

of the

_pattern

is due

to

_multiple

diffraction effects which become

import-ant in this

_angular

_range.

4.2 INTERFERENCE STRUCTURE AROUND FIRST

BRAGG PEAKS. - The second

part

of the

_reflectivity

pattern

(Fig. 4b)

is the most

_interesting

since it exhibits _many

_striking

features. Let us first consider the

_{(001) Bragg}

_peak

surrounded

_by

_subsidiary

maxima.

One can notice that the

_intensity

of the

_subsidiary

maxima is

_high

before the first

_peak

_{and very low} after it. This cannot be

_explained

_{by regular damping}

with

_increasing

_incidence,

since there is about one order in

_magnitude

between the intensities. The

Fig.

4. -

X-ray

reflectivity

from a

_27-layer

film. The dots are

_{experimental points.}

The solid line

corresponds

to the theoretical curve

_{computed using}

the model of

_figure

_5,

after convolution

_{by experimental}

resolution.

_{a) grazing}

angles

- The dashed line is the theoretical unconvoluted

curve.

_b)

interference _pattern around first and second

Bragg

peaks.

Note the shift of the

_position

of the

₍₀₀₁₎

peak

from its theoretical value

_{(dashed line). c) higher}

orders

_Bragg

_peaks.

main

₍₀₀₁₎

_Bragg

_peak

is

_clearly

shifted toward small

Actually,

these

_particular

effects result from inter-ference

_phenomena

between waves

_coming

from

two

_origins.

_Firstly,

the waves reflected at the two

interfaces air/film and film/substrate which would

give

rise,

if

alone,

to

_Kiessig

_{fringes. Secondly,}

waves

coming

from the finite-size stratified structure which

_{yield Bragg peaks}

with

_secondary

maxima.

Hence,

subsidiaries are a

_resulting

_pattern

between

Kiessig

fringes

and

_{secondary Bragg peak}

maxima. The

_phase

difference between the two beams

de-pends

on the relative

_position

of interfaces and stratification

_(i.e.

the film

_boundaries).

Note that these interference effects which will be described further in detail in section

5,

can take miscellaneous forms. For instance the

_dip

in the

_reflectivity

curve close to the

₍₀₀₂₎

_Bragg

_{peak position}

is due to

destructive interferences.

4.3 KIESSIG FRINGES. - Far from main

Bragg

peaks,

subsidiaries are

_genuine

Kiessig fringes [11].

These are

_fringes

of

_equal

inclination which result from the interference between the beams reflected

at the air-film and film-substrate interfaces. The

study

of these

_fringes

_provides

information relative

to the film and its two interfaces. The

_position

of the maxima of the

_fringes

is

_given

_by

the standard

equation

where t is the total thickness of the

film,

and p

is an

integer

(interference order).

The term

is the classical corrective term for refraction. Measurement of the

_position

of the

_fringes

allows an

accurate determination of the total thickness of the film. In our case we found t = 815 ± 5

A.

Supplementary

information can be obtained from the

_analysis

of the

_fringes

intensities. This kind of

investigations

has been

_{performed by}

Croce and Nevot on thin metallic films

_{[20, 21].}

Let us recall that

_rough

interfaces result in additional

_damping

of the

_reflectivity

with

_increasing

_angles.

If a Gaussian distribution is assumed for the

_height

variations of the

_interface,

the

_density

_profile

will be

_represented

by

an error function and the

reflectivity

will have an additional Gaussian attenuation.

For a

homogeneous

film limited

by

two

_rough

interfaces the

_intensity

of the

_fringes

on the

reflectivi-ty

curve is

_given,

from

_{equation (1),}

_by

where

_(ZÃF)

and

_z]FS)

are the mean

_squared

roughnesses

and t is the distance between the interfaces.

_Thus,

measurements of both mean level and

_amplitude

of the

_fringes

far from a

_{Bragg peak}

enable a determination of these values. We find

(ZÃF) 1/2

= 20

A

and

(zbs)

1/2

= 5

A.

The

roughness

of the air-film interface is rather

_important

and indicates that the last

_layer

should be

_poorly

filled. This fact

is

in

_agreement

with a

_{previously expected}

mechanism

_{[22, 23]}

and with our observations made

by

electron

_microscopy

on

_{replicas [24]. Micrographs}

taken on these

samples

look rather

blurred,

on

account of the disorder of the last

_layer.

These features are

_probably

dependent

also on the film

structure which we shall examine now.

4.4 STRUCTURE ALONG THE NORMAL TO THE LAYERS. -

Bilayer

spacing

can be

_easily

obtained from the last

_part

of the

_pattern

_{(Fig. 4c).}

Here,

only

main

_Bragg

_peaks

are

_visible,

subsidiaries

_being

drowned in the

_{background scattering.}

Interlamellar

periodicity

obtained from

_Bragg

_peaks

_up to

₍₀₀₇₎

was found to be :

_d001

= 60.0 ± 0.1

A.

This result is

surprising

since it

_corresponds

_precisely

to twice the

length

of a behenic acid

_molecule,

thus

_involving

untilted

_disposition

of molecules within the unit cell. To our

_knowledge,

structures observed so far in L.B. films of behenic acid were similar to known bulk

_crystalline

structures of

_long

normal chain

carboxylic

acid ;

all these forms

_(A,

B or

C)

involve non zero tilt

_angles

_{[25, 26].}

Note

_that,

since the film thickness

₍₈₁₅

_A)

_corresponds

to 27 times the

_layer

thickness

_(60.0

_Å/2),

then all the

_layers

have the same structure.

The electronic

_density

_profile

within a unit cell can be obtained from the

_analysis

of the

_Bragg

_peak

intensities

_(i.e.

the structure

_factor).

The first feature which can _{be noticed is the very weak}

_amplitude

of even order

_Bragg

_peaks.

This results from both excess and lack of electrons due to

_hydrophilic

heads

(-COOH groups)

and interchain

_spacing

respect-ively, separated

by

half a

_{period [2, 27].}

It should be

pointed

out that the structure factor decreases

strongly

and

_only

seven orders can be observed. For a

_comparison

we observed _up to 24 orders in films of C-form structure under similar conditions. This means that the electronic

_{density profile}

within the unit cell is rather smooth. This fact may indicate that this

_system

should be a

_stacking

of two dimensional

layers weakly

correlated.

Another

_argument

for that was

_suggested

_by

observation of the surface

_by

means of electron

microscopy

on

_replicas.

_Contrary

to

_multilayers

of C-form behenic

_acid,

their surfaces do not exhibit

multistep

defects

_(extra

islands of

_holes)

_[24]

but look rather blurred.

Using

the formalism

_given

in section

_3.1,

the

684

profile

8 (z)

is known. The

_reflectivity

curve calcu-lated

_using

the

_{8 (z)}

_profile

_given

in

_figure

5

_(solid

line in

_Fig.

₄₎

is found to fit the

_experimental

data very

well,

after convolution

by

the instrumental

resolution.

The fitted values found for 8

_{(z )}

are close

to the theoretical _ones, calculated

_by

_taking

into

account the steric

_hindrance,

atomic

_composition

and standard distances between atoms. The

absorp-tion for this kind of

_layers

is _{very small}

(A - 2 x 10-8 )

and can be

_neglected

for

_angles

greater

than the critical

_angle.

_Concerning

the

_layer

structure, one can _assume, since the molecules are

not

_tilted,

that their lateral

_stacking

within a

_layer

is

hexagonal.

Fig.

5. - Model of the index

profile

for the

_27-layer

sample along

the z-axis. The mean value of the real _part of the index of the film is

_5f

= 7.6 _x

10-6

and the

correspond-ing

imaginary

part 8

f = 2 x

10-g.

For the silicon

sub-strate :

_{6, = 17.1 x 10-6}

and

_{Bs = 0. 8}

x

10-6.

5. Calculation of

the

interference structure.

The

_previous

section showed that

_particular

interfer-ence features are visible at small

_angles

on the

reflectivity

curve. The main drawback of the

_optical

matrix formalism

_{(Sect. 3.1)}

is

_that,

_concerning

our

systems,

it does not show the

_origins

of the different features on the

_reflectivity

_pattern.

_Fortunately,

for

angles

greater

than a few

_Oc,

_multiple

diffraction effects are weak

_{(reflectivities}

are

_small)

so that kinematical

_{approximation (i.e. single scattering)}

is valid

_{(Sect. 3.2).}

Analytic expressions

can therefore be

obtained,

showing

clearly

the effects we aim to discuss. We shall assume that the

_systems

are

_composed

of identical

_{monolayers. For}

the sake of

_clarity,

calcula-tions will be made in all cases with the same

structure within the unit cell and with

_step-like

interfaces. The four cases of

_figure

1 will be revie-wed.

5.1 EVEN NUMBER OF LAYERS. - The easiest

case deals with films

_including

even numbers of

_deposited

layers

since these are

_periodic

_systems

with an

integer

number

_(N)

of

_bilayers.

The index

_profile

is

represented by

a

_periodic

function of

_{period d}

(bilayer spacing)

between z = 0 and z = - Nd.

Setting

ð(z)= (ð(z)-ðf)+ðf

where

_ðf

_f is the mean value of the index within the

_film,

one

_gets,

performing

the Fourier transform

_{(1) :}

where

_{F (q )}

is defined as :

(F

is

_proportional

to the structure factor for the unit

cell.)

It is assumed from the

_deposition

mode that the cell is

_{centrosymmetric}

so that F is real. F is

_positive

in case 3 of

_figure

1 and

_negative

in case 2.

The leftmost two terms in the brackets in

equation

(3)

are the

_{Kiessig fringes}

contribution. The

_expression

is the same as

_equation

(2)

where the

roughnesses

have been taken to be

_equal

to zero. The third term

_corresponds

to

_Bragg

_peaks

with

secondary

maxima. Their

_intensity

is

_proportional

to

the

_squared

structure factor. The

_expression

is

exactly

the same as for a finite size

_{optical grating.}

The

_rightmost

term is the interference term

be-tween the two

_{previous phenomena.}

In the

_present

case, this term is small since

₍₂

_{8f -}

_8s)

is small : the mean

_density

of the film

_{(8 f}

=

7.6)

is close to half

the

_density

of silicon

_{(8 =17.1).}

Figure

6 shows a

_plot

of

_separate

contributions of

Kiessig fringes

and of

₍₀₀₁₎

_Bragg

_peak,

then the result of all the terms of

_{equation (3).}

Due to

_the

small interference

_term,

intensities

_merely

add and there is

_only

a

_slight

difference between the cases

(F

0)

and

_{(F > 0).}

Note that

_{Kiessig fringes}

have the same

_spacing

as

_{secondary Bragg peaks,}

but are shifted

_by

half a

_{period. Fringes}

on both sides of the

peaks

have about the same

_amplitude,

and no

significant

shift of the main

_Bragg

_peak

can be detected.

5.2 ODD NUMBER OF LAYERS. - The calculations

are in this case somewhat more tedious since the

Fig. 1).

The

_simplest

_way to tackle this

_problem

is to

consider the

system

as a succession of

_{(N) bilayers}

and an additional

_monolayer

_(the

last

_layer

in the

deposition

sequence for

instance).

This last

layer

can be

_{replaced by}

a

_homogeneous

lamina

_having

the

mean index of the film : this includes its contribution

to the

_phase

difference between the reflected waves,

but

_neglects

its small contribution to

_Bragg

_peak

intensity.

Under this

_assumption,

the Fourier

trans-form

becomes :’

with the same notation as in

(3).

The value of

qd

_{corresponding}

to a

_{(00 )}

_Bragg

peak

is l7T. It can be seen that the last interference

term is the sum of two

_opposite

terms for even 1 and the difference of these terms for odd 1. As the absolute values of these terms are close to each

other,

the interference term is small in one case

(even 0

but

_large

in the other

_{(odd 1).}

Figure

7 shows the interference effect around the

(001)

Bragg

peak.

In this case,

fringes

and

_secondary

maxima

coincide,

so that the interference has a _very

large

effect on the

_intensity

of subsidiaries. The

sign

change

of the

_large

interference term from one side of the

_Bragg

_peak

to the other results in a _very

strong

damping

of the

_fringes.

The ratio of their

intensity

before and after the

_peak

is

_actually

about 100. The calculations

_predict

that this

_striking

effect should be inverted in the case of a

_deposition

over an

hydrophobic

substrate

_{(Fig. 1-4)}

since this case

fol-Fig.

6. -

Interference between

_{Kiessig fringes}

and

sub-sidiary

maxima of the

₍₀₀₁₎

_Bragg

_peak

for an even

number of

_layers.

The data used in the calculation

correspond

to behenic acid,

assuming

no

_roughnesses

nor

absorption.

Numbers 2 and 3 refer to cases 2 and 3 in

figure

1.

Fig.

7. - Interference structure around

(001)

for odd number of

_layers.

Numbers 1 and 4 refer to cases 1 and 4 in

figure

1.

lows from the

_previous

one

_by

_solely

_changing

the

term F into - F.

The unusual

_shape

for the main

_{Bragg peak}

is also

directly explained

by

this interference _{process :} a

fringe

is added on the low

_angle

side of the main

peak

and subtracted on the other side. The

ampli-tude of the

_displacement

of the

_peak

maxima

depends

on the relative

_heights

of

_{Kiessig fringes}

and

_Bragg

_peaks

_(i.e.

on

_{N, F,}

_SF,

_6J.

In our

example

in

_{figure 7,}

the shift of the maxima is

ð.(J(J

= 4 % and the

_{intensity change by}

a factor of

t7

two. Note that we also

_computed

the interference

pattern

of

_figures

6-7

_using

matrix formalism.

_Apart

from a small

_angular

shift due to the

refraction,

no difference with the

_plot

of

_equations

₍₃₎

and

₍₄₎

was detected.

5.3 EXPERIMENTAL INTERFERENCE PATTERN.

-Figure

8 shows the

_{experimental reflectivity}

data around the

₍₀₀₁₎

_Bragg

_peak

for different _sequences of

_{deposition. Figure}

8a

_displays

the

_reflectivity

for a

686

Fig.

8. -

Experimental

results for the interference struc-ture around

₍₀₀₁₎

_Bragg

_peak.

_a)

_26-layer

_sample

on

hydrophobic

substrate

_(case

3 in

_{Fig. 1), b) Sample}

plan-ned to be

_27-layer

on

_hydrophobic

substrate

_(case

4 in

Fig. 1).

The last

layer actually

went off so that the sequence is the same as for the

_{26-layer sample.}

_c)

27-layer sample

on

_hydrophilic

substrate - _case 1 in

figure

1.

in

_{Fig. 1).}

The main features that we mentioned in section 5.1 are observed : there is no

_special

_damping

of the subsidiaries from one side of the

(001)

_Bragg

peak

to the other one and no

_important

shift of the

position

from the theoretical value. It should be noted however that the intensities of both

_Kiessig

fringes

and

_{Bragg peaks}

are much smaller for the

26-layer

than for the

_{27-layer sample.}

This

_originates

in the _{substrate treatment, which increases its}

rough-ness and

_changes

the

_quality

of

_layer

_deposition.

Apart

from this overall

_damping,

the structure factor remains the same.

Figure

8b

_displays

the same

_angular

_{range for} a

sample

which was

_designed

to include

_{27 layers}

deposited

over an

_hydrophobic

substrate

_(i.e.

case 4 in

_{Fig. 1).}

The curve does not show the

_expected

feature but has a similar

_shape

as the

_previous

one. This indicates that the last

_layer

went off

_during

the final

_{decompression.}

This fact is also

_{supported by}

a

measurement of the film thickness which indicates less than

_{27 layers.}

For the same _reasons, similar

problems

were encountered on

_samples

correspond-ing to

the case 2 in

figure

1.

(Even

number of

_layers

on

_{hydrophilic substrates.)}

The

_patterns

we obtained were

_typical

of odd number

_samples

on

_hydrophilic

substrates

_(case

1 of

_{Fig. 1).}

_Figure

8c

_reproduces

an

example

of such

_patterns

taken from

_figure

4b

(27

layer

sample) ;

all the features described in section 5.2 and

_plotted

in

_figure

7 are visible. 6. Conclusion.

Interference

_{phenomena play a major}

role in

_X-ray

reflectivity

studies of

_layered

thin films. We saw that these effects are

_large

in the small

_angle

_part

of the

reflectivity

curve and that

_{they strongly}

alter classical diffraction features. These

effects,

which must be taken into account in structure factor

determination,

may be used to

_investigate

the boundaries of the film : the

_deposition

_sequence, i.e. the orientation of the first and the last

_layer

_{may be}

_{straightforwardly}

determined. The method is

_actually

more

_{general :}

we illustrate here

_only

cases

_differing

from each other

_by

_{the presence of} one additional

_layer

but it is clear that other cases

_involving

a

_layer

with tilted

molecules,

a half filled

_layer

and so on are

_possible.

All will

_change

the

_phase

difference between the waves

_coming

from the boundaries and those

coming

from the structure

_{itself ;}

thus the interference

pattern

will be modified and one can take

advantage

of this fact to set one with

_respect

to the other.

This kind of

_{investigation}

can be

_{performed ideally}

on L.B. films since the L.B.

_technique

allows the

deposition

of a

_given

number of

_{highly regular}

layers.

Our results obtained on behenic acid enabled us to describe a new untilted form for this molecule as well as to confirm various observations on L.B. films

_(i.e.

_mainly

the existence of a disordered last

layer).

Further

_experiments

on new

_samples

built under different conditions

_(especially

to

_prevent

the

departure

of the last

_layer

when

_hydrophilic)

will be carried out. _{Moreover this method may be}

_applied

to the

_study

of

_overturning

mechanism and

_X-type

layers.

X-ray

reflectivity study

of many other

systems,

such as

_composition

modulated

_amorphous

thin films or MBE grown

_layers

[28],

_{may also be very}

interesting,

since a _very

_large

_variety

of artificial

structures

may be

designed

with these

_techniques.

Acknowledgments.

We wish to thank E.

_Chastaing

for technical assist-ance and M.

_Dupeyrat

for

_helpful

discussions.

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