Experimental Analysis of River Patterns in Silicon Brittle Fractures

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Experimental Analysis of River Patterns in Silicon Brittle Fractures

Joël Chevrier

To cite this version:

Joël Chevrier. Experimental Analysis of River Patterns in Silicon Brittle Fractures. Journal de

Physique I, EDP Sciences, 1995, 5 (6), pp.675-683. �10.1051/jp1:1995159�. �jpa-00247093�

(2)

J.

Phys.

I France 5

(1995)

675-683 JUNE 1995, PAGE 675

Classification

Pbysics

Abstracts

46.3ùNz 62.2ùMk

Experimental Analysis ^of ^River Patterns in Silicon Brittle Fractures

J.

Chevrier(*)

Institut für

Grenzflàchenforschung

und

Vakuumphysik, Forschungszentrum Jülich,

52428

Jülich, Germany

(Received

17

January 1995,

received in final form 27

February

1995,

accepted

7 March

1995)

Abstract.

During propagation

of brittle cracks in silicon

single crystal,

_steps

parallel

ta trie direction of the crack and

separated by

_terraces, _are often created. As trie crack further

propagates, these steps

gradually

coalesce. This

produces

trie characteristic surface

morphology usually

identified _as

a river pattern. An

experimental analysis

of these _river patterns _appearing

during

fractures of silicon

single crystals

at room temperature ^is

presented.

The coalescence of the steps and the

coarsening

of the terrace widths reveal _an effective interaction between the

propagating

steps. With _an interaction between steps

supposed

to be deterrnined

by

the size of the terraces, the observed river patterns are

numerically reproduced.

The fracture _process _is _one of trie most

important

^behavior of solids observed in

everyday

life. In many

circumstances, preventing

cracks to appear and to

propagate

ⁱⁿ ^materials ^is

an absolute

requirement. Therefore,

_numerous studies deal with this

problem,

_as shown in reference

iii.

One of trie

major concepts

is trie distinction between trie brittle fracture and trie ductile fracture. Trie brittle fracture is

thought

to create two _new surfaces and _to leave

trie bulk structure without _new

defects,

whereas trie ductile fracture _is associated with trie

plastic

deformation of trie sohds and trie _appearance of defects like dislocations in trie bulk.

Trie brittle fracture is then

expected

to be

essent1ally

_a surface eflect. Trie

opening

of trie crack with trie creation of _new surfaces is trie main mechanism which

dissipates

trie elastic

energy of trie

applied

stress. This bas been

pointed

out and

developed by

Grillith

[2].

^In ^bis calculation of trie

equilibrium

of _a

crack,

trie _energy balance between trie surface _energy and trie stored elastic _energy is

clearly

stated

(see

also Ref.

[3]).

^A ^second

important point

_concerns trie

subsequent propagation

^of _a crack. Under trie

application

of _an externat stress

large enough

to induce _a

crack,

_a brittle fracture should

propagate

^with an

increasing velocity.

It bas been realized

long

_ago that a

speed

limit _must be encountered and that trie

specific

effects due to a

crack

velocity comparable

to trie

speed

of sound should be taken into account to describe trie

(*)

_on leave from: CRMC2 CNRS

Marseille,

Campus de Luminy, Case 913, 13288 Marseille Cedex 9,

France

©

Les Editions de

Physique

1995

(3)

676 JOURNAL DE

PHYSIQUE

5'°6

Stress

Si sampie

Fig.

l. Schematic representation of the method used _to induce _a fracture _m silicon

single crystals by bending.

The silicon

single crystal

is hold between two metal blocks. A _stress is

applied

_on trie free

part which induces trie fracture.

crack

patin [4, 5].

^Trie ^surface

morphology produced by

trie brittle fracture bas indeed been

investigated long

_ago both

experimentally _[6,

_7] and

theoretically

_[8].

However, despite significant results,

the

empirical description

of the surface

morphology

shows that _a detailed

understanding

of the

propagation

^of a brittle fracture and of trie _mor-

phology

^of trie created

surfaces,

is not well-established. One

major example

is trie

description

of

a

commonly

observed surface

morphology

issued from trie brittle fracture with trie1&>ell-known

"mirror-mist-hackle" scenario

[6, 7].

^This ^refers ^to ^trie

general change

^of trie surface from _an

apparent optical

mirror state to _an

extremely rough

surface with charactenstic

lengths

in the

micrometer range or above. This often

reported

observation shows the evolution towards _an

important

^surface

roughness

with the appearance of

increasingly larger

characteristic

lengths during

the crack

propagation.

These observations have

recently triggered

new

experimental

work

[9].

^Trie

propagation

of _a brittle crack bas been

specifically mvestigated

in order to

identify

trie basic mechanisms which induce trie appearance of this

increasing roughness

of trie created surface and at trie _saine time trie unstable behavior of trie crack

velocity.

These

results [9]

strongly suggest

trie existence of _a

dynamical mstability dunng

trie

propagation

of _a brittle fracture. This conclusion _is further reinforced

by

recent numencal studies which

reproduce

the essential

experimental

features

[loi.

The existence of such _an

instability

_is also discussed in recent theoretical

descriptions iii].

The fracture of silicon at _room

temperature

is

usually presented

as a

prototype

of the brittle fracture

(see

for

example

Refs.

[12,13]).

In this covalent

matenal, cleavages along

the

(ii1)

and trie

(110)

atomic

planes

_are often observed.

Also,

it bas been

repeatedly reported [14,15]

that

cleavage

or fracture of _a silicon

single crystal

induced

by bending

_as descnbed in

Figure

1,

can

produce

trie

typical

surface

morphology

shown in

Figure

2.

Shortly

after the initiation of trie

crack,

its

propagation produces

a mirror surface

(such

a surface _is

thought

to bave _a residual

roughness

_m trie nanometer range

[14,15]) During

further

propagation,

_one observes the simultaneous _appearance of _many steps ^which run m

the direction

parallel

to the crack

propagation [14,15]. During

the

subsequent development

of the

crack,

this set of

steps

^and

terraces,

which forms _a

"zigzag"

crack

front,

evolves to form what is called in the fracture literature _a river

pattern.

The mechanism which

produces

this

pattern

is

quite

clear: two

(or occasionally more) steps jour

to form _a

single higher

_step. In

Figure 2,

_we present a

particularly well-developed

and ramified

example

of such _a pattem:

trie steps appear white and

they

_are

separated by

flat terraces _in black. Trie ~vord terrace

(4)

N°6 EXPERIMENTAL ANALYSIS OF RIVER PATTERN 677

a

b

10jm 100~m

~

'ei.=ùl=-

_~

Fig.

2.

a)

Observation

by Scanmng

Electron

Microscopy

of the silicon surface

produced

at room

temperature

by

_a fracture induced

by bending

_as shown _m

Figure

1. The X-axis shows the direction of the fracture

propagation.

^The Y-axis is

parallel

to trie crack front. Trie black fines indicate the

part ^of ^the pattern which is simulated in

Figure

5.

b)

Observation of the _same

sample by Optical Microscopy

in

polarized light.

used here does _not

imply

^that ^these terraces _are

atomically

^flat well-defined terraces. At trie resolution of trie SEM and of trie

Optical Microscope, they experimentally

_appear

flat,

1-e, with _no observable _contrast. Beside trie "mirror-mist-hackle" structure, trie river

pattern

is a

well-organized

structure

developed by

trie brittle fracture. Trie observation of these river

patterns

is

by

no means limited to trie aforementioned references and to silicon. This is _a

generic

structure often observed in trie brittle fracture of other materials

including glass.

It is sometimes

reported

that these _structures _can be

empirically

used to

identify

the local direction of the crack

propagation [16].

^Gilman [17] ^has

experimentally investigated

^the formation of the

initial steps at trie

beginning

of _a river pattern. ^He ^bas

provided experimental

evidence that trie

appearing

steps

during

crack

propagation

in LiF

single crystals ongmated

at _screw dislocations.

However, although analysis

and

descriptions

bave been

reported by

dilferent authors

[18],

trie river

pattern

structure does not appear to be

completely

and

satisfactonly

understood.

Several basic

experimental points

are still _open. In trie _case of _a silicon

single crystal

with

no

pre-existent dislocation,

trie creation of _numerous

steps

^which

simultaneously

_appear

along

trie crack front _over

macroscopic

distances much

larger

than _one milhmeter

certainly

calls for further

experiments [19].

^{As trie} ^set ^of

steps

and terraces is

created,

trie moving front

invariably

(5)

678 JOURNAL DE

PHYSIQUE

I N°6

evolves towards _a similar but coarsened structure with

increasing

characteristic

lengths (1.e.,

trie

step heights

and trie terrace widths

increase).

Mechanisms _at trie

origin

of this behavior bave _not

received,

to our

knowledge,

_a

satisfactory expenmental explanation.

In this

article,

_we

experimentally

describe trie river

patterns

observed after trie fracture of silicon

single crystals.

Trie river

patterns

^show ^that ^trie ^crak ^front

presents

a scale invanant

structure

produced by

trie

propagation

^of

interacting steps.

An

experimental

evidence for _an

effective interaction between trie steps ^will ^be ^deduced ^from ^trie ^curved

patin

of

joining

steps.

On trie basis of this

expenmental

result and

taking

into account _an interaction related to trie terrace widths between

steps,

we shall describe how it is

possible

to

numerically _generate

river

patterns.

In

particular,

this simulation

reproduces

trie observed

step

structures and trie

coarsening

of trie characteristic

lengths.

We bave used silicon

single crystals

which _are 2 _mm thick

(111)

silicon wafers 2 inches in diameter. One face of trie

sample

_was

optically polished.

Trie

experiments

bave been

performed

as described in

Figure

ⁱⁿ air at ambient

temperature.

Trie

samples

_were bent in such _a _way that trie fracture bas

always

been initiated in this

polished

face. No further surface

preparation

bas been made

(especially

_no initial notch made _on _purpose to localize trie

fracture).

Before trie

fracture,

trie silicon wafers

presented

_a circular

shape

with _no indication of

crystallographic

orientation. No

attempt

bas been made to determine trie orientation of trie

original crystals

and then to initiate cracks

along

well-defined

cleavage

directions. This lack of orientation control does not _seem to

impinge

_on trie

reproducibility

of trie

experimental

results here

presented.

However,

_a

precise

control _may become necessary for further

quantitative

measurements like trie

precise investigation

of trie transition between trie mirror and trie faceted

regions.

Dur observations _are very much

comparable

to trie

morphologies

observed after fracture in _vacuum

by bending [14,15].

This also shows that

working

in air does not _seem to introduce noticeable

changes. Optical Microscopy

and

Scanning

Electron

Microscopy (SEM)

bave been used to

analyze

trie surface

morphology. Using

_an interferometric

technique

with

polarized hght,

the

perpendicular

resolution of the

Optical Microscopy

is

improved

to about few nanometers. In order to increase trie contrast between trie steps ^and ^trie terraces in electron

microscopy

^and thus to

specifically

reveal trie structure of trie river pattern, we bave used _a

large angle

of

incidence between trie

steps

and trie electron beam in SEM

investigations. Figure

2a bas been

acquired

in those conditions. A mechanical

profilometer

bas been used to

quantitatively analyze

trie surface

morphology (Fig. 3).

Trie lateral resolution of this instrument is hmited to trie micrometer range, whereas _a

perpendicular

resolution around 10 _nm _can be achieved.

Furthermore,

Laue

diffraction, together

with trie

optical

reflection of _a Laser

beam,

bas been used to

identify

trie orientation of trie terraces. Trie _accuracy of this measurement is not better than few

degrees (< 5°).

This _is

essentially

due to trie small _size of trie terraces.

Trie fracture of _a silicon

single crystal,

mduced

by

trie method described in

Figure 1, gives

use to trie surface

morphology

shown _m

Figure

2

[14,15].

In trie mirror

region,

_no

significant

contrast is

observed,

either

by Optical Microscopy (x1000),

_or

by

SEM

(even

at trie

largest

magnifications

x5000

x10000). Optical Microscopy

at small

magnification

shows that trie mirror

region

of trie

sample presented

in

Figure 2,

^is curved in trie

propagation direction,

1-e-, trie fracture does not follow _a

cleavage plane

in trie _mirror

region. During

trie crack

propagation,

trie initial _mirror

region disappears.

Instead _terraces and

steps

are

simultaneously

nucleated

along

trie crack front. This transition _is observed in

Figures

2a and 2b.

Perpendicular

to trie

propagation

direction

(1.e., along

trie crack

front),

trie

morphology

of

trie _mirror region

investigated by

_means of trie mechanical

profilometer

is shown _m

Figure

3a.

Figure

3b shows _a

profile along

trie crack front

acquired

_using trie _same

technique

_m trie

river

pattem.

Within trie precision of trie

profilometer,

trie terraces are

parallel

and present a

roughness

much smaller than trie

typical

_step

height

between two terraces.

Optical Microscopy

(6)

f

z

'%

ai

0 20 40 60 80 100

Y axis pm

f

,9 g

'~ b)

0 20 40 60 80 100

Y axis _pm

Fig.

3. Profiles measured

by

_a mechanical

profilometer

in trie direction

parallel

ta trie Y.axis.

a)

m trie _mirror

region,

but very close ta trie frontier between trie mirror and the _river pattern

regions.

Most of trie

profile

is characteristic of the mirror

region,

however _some terraces have

already

been created.

b)

in the river pattem

region.

The terraces have been made

parallel

to Y.axis _on _purpose

by

orientation of trie

sample.

with

polarized light

also shows _a

unique

orientation of trie terraces and _no clear variation of

contrast within _one terrace, which is consistent with _a limited

roughness.

Trie combination

of trie Laue diffraction and of trie reflection of _a Laser beam _on trie surface of this

sarnple

indicates that these terraces exhibit _a

(111)

orientation. In contrast, a

step

^between two terraces is

clearly

not flat but _can be _a rather

heavily

^distorted ^surface. ^This _can ^be _seen _on

Figure

2a.

As trie crack front _moves away from trie

origin

of trie steps, trie

steps gradually coalesce,

which induces trie formation of trie river

pattern.

At _any

point

of trie

pattern,

trie crack front is made of alternated

steps

^and terraces.

Indeed, by changing

trie scale in

Figure 3b,

it is

possible

to bave _a

description

of trie crack front _at different

stages

^{of trie} river

pattern.

^This is _an

experimental

evidence for trie scale invariance of trie crack front. This

pattern

^invariance

on scales

varying

from 1 to 100 /Jm is also

supported by

trie

comparison

^of

Figures

2a and 2b.

Appearance

of _new

steps

^and terraces is not observed

during

trie crack

propagation

alter trie initial creation of steps at trie interface with trie mirror region. Also trie crack front _appears

as a staircase in trie

presentation

of

Figure

3: ail terraces are

parallel

to trie Y-axis and ail steps are

up-steps Ii.e.,

with _a

positive slope).

However

during

trie

propagation

of trie crack

m trie

X-direction,

these

steps

_con

turn,

either to trie

left,

_or to trie

right,

_as _seen in

Figure

2.

Therefore, although

trie crack front is

clearly asymmetric Ii.e.,

_no

down-step),

this

asymmetry

is not a

property

of trie whole river pattern.

(7)

68ù JOURNAL DE

PHYSIQUE

I N°6

Y

X ~~~

L~ L~~i

~~

~ii ~i _~i+1

Fig.

4. Schematic definition of Li, trie width of _a terrace and of K, trie position of _a step.

The

steps

^do ^not

propagate

as

straight

lines. To trie

contrary,

a step curves m trie direction of trie step ^{it is}

going

to

join.

One

quite systematically

observes trie increase of this curvature

as trie intersection is

getting

closer. Furthermore when _a small terrace is located between _two

larger

terraces, trie _two

steps

_on trie side of trie small _terrace

usually join

and trie small _terrace

disappears.

These two

points provide

_some evidence that trie

propagating steps

^do not meet at random but that there exists _an effective interaction between the

steps

which _seems to be

determined

by

trie size of trie

neighboring

terraces. This statement _can be formalized

by

trie

following equation:

~~

à _~LÎ

~ ~

+

LÎÎÎ

^~~~

where

Lz

and

Lz+i

are trie terrace widths _on sides of _a step. Trie

position

of the step

along

trie crack front is

given by l~ (see Fig. 4).

C is _a

coupling

constant and X is trie _axis

along

the direction of crack

propagation. High steps

close to the end of the crack

propagation

_m

Figure

2a do not appear

strongly

curved

although they

_can be

separated by large

terraces of

various _sizes.

Therefore,

that

dl~/dX

_con be

directly proportional

to trie difference of terrace

widths,

_appears rather

unlikely.

Trie denominator

(Lz

+

Lz+i)

is introduced in

equation (1)

to

prevent

an

extremely _strong

interaction between steps

separated by large neighbonng

terraces of different sizes

[20].

After measurement of trie step

positions

at one

stage

of trie _river

pattern, equation (1)

enables _one to

numerically

calculate trie number and the

position

of

steps

at any further

stage.

Figure

5

presents

the result of trie numerical simulation based _on

equation (1).

The initial

positions

of

steps

^have ^been ^measured

along

the black line

parallel

to trie Y.axis _m

Figure

2a.

Straight

steps ^bave ^been

imposed

at both

edges (black

lines

parallel

to trie X.axis _m

Fig. 2a).

It is then

possible

to

directly

_compare trie simulation

(Fig. 5)

with the

expenmental

results

(Fig. 2a).

The

path

of each step ^is

only approximately

descnbed. However the

hierarchy

of the steps ⁱⁿ ^the

region analyzed

is

reproduced.

Therefore

Figure

5 shows that trie main features

of _river

patterns

^observed ⁱⁿ ^trie ^brittle ^fracture _can be

reproduced by

this

simple algonthm

although

this

description

based _on

equation (1)

is

only empirical.

This also indicates that _a

precisely

defined initial distribution of _terrace widths _can

completely

determine trie

subsequent

evolution of this

pattern.

A river

pattern

can be further charactenzed

by

trie

change

of trie

density

of steps as trie crack

propagates.

This _is trie result

presented

in

Figure

6. Trie step

density

bas been measured _on SEM

pictures

^with

magnification

_rangmg from 300 to 16000.

Together

with trie direct observation of _a

pattern (Figs.

2a and

2b), Figure

6 shows that trie

density

of steps can

continuously

(8)

Fig.

5. Calculation of the step pattern

produced dunng

the

propagation

of the crack. This _numer- ical simulation is based _on

equation (1) (see text).

This

figure

should be

directly compared

to the _area

defined

by

black fines _m

Figure

2a. The initial

positions

of steps used in the calculation have been measured in

Figure

2a

along

the black fine

parallel

to the Y.axis.

o o

E_

OE U

à

~

(

à

_0.1

Éi

o.oi

o.i i io ioo iooo

x(pm)

Fig.

6. Variations of the step

density along

the direction of the fracture propagation

(X-axis).

The

origin

of the

curve

(X

=

ù)

is at the interface between the _mirror and the

stepped regions.

This

density

of steps has been extracted from SEM

pictures

with diflerent

magnifications (x30ù, Xsùù,

X3ùùù, X5ù0ù,

X16ùoù).

The fuit fine

(-)

is calculated using

N(x)

=

A/~.

decrease

by

two orders of

magnitude during propagation

^of trie crack. As trie

pattern develops

away from the _mirror _region

(X

> 10

~tm),

the

density

of

steps

appears m

Figure

6 to be

descnbed

by

_a _power law with _an

exponent

^close to -1. However measurements _on different

samples

^mdicate ^that other values of this

exponent

_can ^be ^found. ^Further

experiments

^with im-

proved

control of trie crack

growth parameters

_are needed to

clearly

establish what determines trie interaction between

steps

and this exponent

[21].

JOLRNAL DEPHYSIQUE -T. 5,N°6,JUNE 1995 26

(9)

682 JOURNAL DE

PHYSIQUE

I N°6

In

conclusion,

_a detailed

investigation

of trie river

patterns

observed in trie brittle fracture of silicon shows that

they

_can be

analyzed

_as trie

propagation

of

interacting

interfaces.

Although

its

origin

is not

experimentally identified,

this effective interaction favors the

large

distances between the steps ^and ^thus ^leads to _a

coarsening

of the terrace widths.

By

_usmg the

simplest

interaction that _one _can deduce from the

experimental results,

the main features of _river

patterns produced by

brittle fractures of silicon _can be

numerically reproduced.

Trie calculated

pattern

_appears

completely

determined

by

trie interaction between

steps

^and

by

trie initial

terrace width distribution.

We bave not

directly

addressed trie transition between trie mirror and trie

stepped regions.

A relevant

experimental description

ofthis transition in silicon

certainly

needs to show whether _or not _a

plastic

deformation is present. ^Also ^the charactenzation of the surface

morphology

close to this transition calls for further

experimental

work with better

spatial

resolution.

Interesting aspects

^of ^these ^river

patterns

_may be: 1)

they

_are

examples

of surface

morphologies produced by

brittle fracture which _can be

quantitatively analyzed by

_means of direct _space

techniques

like Atomic Force

Microscopy, ii) previous

works _on brittle fracture

[6,17,22] _suggest

that it may be

possible

to induce and to control the _appearance of steps. ^To some extent, this _means that these patterns can be _seen _as _a

quantitative

tool to

study

the

propagation

of brittle cracks.

Although

the

present

^work ^remains

prelimmary

in view of trie

dilliculty

of this

problem,

it is

an

atteiupt

ⁱⁿ ^this ^direction.

Acknowledgments

I

gratefully.

thank trie Alexander _von Humboldt Foundation for _a Research

Fellowship.

The

support

^and ^trie comments of G. Comsa bave been of

great help during

this ~vork. I have benefited from the technical

help

of U.

Linke,

B. Schumacher and S. Nitsche and from stimu-

lating

discussions with R.

I<ern,

U.

Linke,

Y.

Brechet,

A.

Pimpinelh

and D.E. Wolf. Detailed

analyses

of trie surface

morphology

with _an Atomic Force

Microscope

_are in progress thanks to J. Dernen.

References

iii

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Statistical Models for trie Fracture of Disordered Media, H-J- Herrmann and S.

Roux,

Eds.

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[7] ^Rice R-W-,

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(10)

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3592.

[12] ^Michot

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[13] ^Brede

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70

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[14] ^Tokumoto H.,

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Technol. 89

(1991)

695 and references therein.

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_pp. 321-326.

[19] ^In ^references

[14,15],

it is noticed that the _mirror surface is associated with trie part of the

sample initially

in extension and that the part of the

sample,

which

usually

exhibits the _river pattern,

is at the

begmning

_m compression. This _is

expenmentally

correct.

However,

ta our

knowledge,

it has net been shown how this remark coula be relevant ta describe the transition between the two

regimes.

[20]

Changes

_m step

heigths

_as the crack propagates should

certainly

_appear in

equation (1).

However

no direct measurement of the step

heigths

_is here

presented.

The necessary introduction of

(Li

₊

Li+i)

in equation

(1)

could appear as a way to represent

changes

_in the step

heigths

and the

influence of the step

heigth

_on the step

propagation.

[21] The variation ofthe

density ofsteps

_m simulated patterns ^based _on

equation (1)

_is also described

by

_a transient

regime

close to trie origm of trie pattern followed

by

_a well defined _power law with

an exponent

equal

to -1.

[22] ^Yuse ^A. ^and ^Sano M., Nature 362

(1993)

329.