A Reflection Electron Microscopy Investigation of the Divergence of the Mean Correlated Difference of Step Displacements on a Si(111) Vicinal Surface

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A Reflection Electron Microscopy Investigation of the Divergence of the Mean Correlated Difference of Step

Displacements on a Si(111) Vicinal Surface

J. Heyraud, J. Bermond, Claude Alfonso, J. Métois

To cite this version:

J. Heyraud, J. Bermond, Claude Alfonso, J. Métois. A Reflection Electron Microscopy Investigation of the Divergence of the Mean Correlated Difference of Step Displacements on a Si(111) Vicinal Surface.

Journal de Physique I, EDP Sciences, 1995, 5 (4), pp.443-449. �10.1051/jp1:1995137�. �jpa-00247069�

Classification Physics Abstracts 68.30

Short Communication

A Reflection Electron Microscopy Investigation of the Divergence

of the Mean Correlated Diffierence of Step Displace~nents

_{on a}

Si(Ill) Vicinal Surface

J.C.

Heyraud (*),

J.M. Bermond

(**),

C. Alfonso

(*"")

^{and J-J- Métois}

CRMC2-CNRS (****), Campus Luminy, Case 913, 13288 Marseille Cedex 09, France

(Received

21 December 1994, accepted 24

Jauuary1995)

Abstract. A

Si(Ill)

vicinal

(misorientation

_m

0.6)

is studied by _m situ Reflection Electron Microscopy at II 73 K. A statistical study is done of trie distances between pairs ofm~~ neighbours

in a step train. Trie

mean correlated dioEerence of trie step displacements from their _mean positions

G(m)

_=<

(u,

t1,+m)~ _> is determined

as a function of _m. Evidence _is given for the roughness of trie surface. A loganthmic behaviour of

G(m)

_{versus m is} demonstrated

unambiguously _up to _{m =} 7. Quantitative _agreement is found with the theoretical predictions of Villain, Grempel and Lapujoulade. For _more distant pairs of steps _a ^{dioEerent behaviour} _is

demonstrated:

G(m)

_increases faster than

Log(m),

a fact already found by another author.

l, Introduction

The

roughening

transition of _a

crystal

surface has been trie

subject

of _many theoretical and

experimental investigations.

This transition _can be detected

through

the behaviour of trie _mean

correlated

height

diiference

H(r~j

=< (h~

hj)~

>, where _h~

-hj

is the

height

^diiference of the surface

(above

_some reference

plane)

between two

points separated by

_a distance r~j, measured

parallel

_to the surface

plane

and where the brackets denote _an ensemble _average. At T smaller than the

roughening

temperature

(TR), H(r~j)

remains finite _as r~j - cc, whereas it goes ^to

infinity

with r~j ^{at T >} TR. In the latter _case,

theory predicts

a

logarithmic divergence

with _r

il,2]

and references

therein).

Expenmental

evidence for the

loganthmic divergence

was first deduced from He and

X-rays

diffraction expenments on vicinal surfaces [3) ^and references

therein).

It is

only

indirect since

(*) ^{Also Université} de Provence

(Aix-Marseille I),

Marseille, France.

(** Corresponding author. Also Université de Droit, d'Economie et des Sciences d'Aix-Marseille

(Aix-

Marseille

III),

Marseille, France.

(***)Present address: Laboratoire de Métallurgie-URA 443, Université de Droit, d'Economie et des Sciences d'Aix-Marseille

(Aix-Marseille

III), Marseille, France.

(****) Laboratoire _propre du CNRS

(LP7251),

associé _aux Universités d'Aix-Marseille II et III.

@ Les Editions de Physique 1995

444 JOURNAL DE PHYSIQUE I N°4

Experimental

evidence for the

loganthmic divergence

_was first deduced from He and

X-rays

diffraction

experiments

_on vicinal surfaces [3] ^and references

therein).

It is

only

indirect _smce the

only

measurable parameters are the

height

and width of the diffraction

peaks.

In order to check the

roughening

of the surface _it is suflicient to _measure trie normal dis-

placement

_u~ of the

i'~

_{step from its}

average

position

and to

plot

the

quantity G(m)

₌

< (t1~ t1~ +

m)~

> _{as a} function of _m [2]. As shown in

Figure

1, it is

equivalent

to

determining

the correlated

height

diiference H in the direction normal _to the steps since (h~

h~+m)~

₌

sin~

Ù(~t~ t1~+m)~, ^{with the} misorientation

angle

of the vicinal.

Direct observation of the step

position

_is allowed

by

STM.

G(m)

_can then be calculated

from the measurement of the deviations of the steps ^{from their} mean positions. Such _measure-

ments were

attempted by Goldberg

et ai. [4],

and,

_more

recently, by

Masson _[5] and

Hegeman

et ai. [6].

Golberg

et ai.

investigated

vicinal

Si(Ill)

surfaces below the

(7

_{x 7)}

(1

x

1)

surface transition temperature. Their expenments _were made diflicult

by

the

imperfect equilibration

of the surface

(occurrence

of

triple-height steps).

Yet,

by averaging

the data from diiferent

samples,

_a

loganthmic

law _was found. However the authors did not compare the

slope

of their

expenmental straight

hne with the

theoretically expected

value.

Hegeman

et ai. measured

G(m)

_on vicinal

Si(lo0)

surfaces. They ^did not find _a

logarithmic dependence

of

G(m)

_on

m. Instead

they

got a

tendency

to saturation with oscillations

supenmposed.

Masson inves-

tigated

_a

(1,1,11)

_copper surface. Her data _are

compatible

with _a

logarithmic law,

however

600 nm

,

x ,

1-2 1-1 1+1 1+2

, U

Fig. l. Schematic drawing of _a meandenng train of steps and definition of trie parameters used in

text. Top: View normal to the terrace plane. Doted fines: The _mean positions of trie steps, equally

spaced by

il).

They _are trie _zero temperature positions of the steps. Trie _x-axis is parallel to the terrace plane and normal to the step _mean direction. The three "sections" used in the measurements are parallel to the x-axis, _{xi is} trie position of trie i~~ step. Bottom: Cross sectional _view of the vicinal

surface. Trie 0 K positions of trie steps _are drawn _as sobd lines. For clarity, trie displacement of only

one step is shown

(doted

line).

only

for small _m

Ii.e.

^{for the first three} consecutive terraces in _a step

train).

For

larger

m

she found _{a more} complex behaviour which she did not attempt to rationalise. It must be noted that all the above-mentioned expenments were done _on vicinals that had rather

dosely spaced

steps. This _{is necessary} either in diffraction experiments, due to the coherence

length

of the

diifracting beam,

_or in STM experiments where _a suflicient number of

images

must be recorded to extract

statistically significant

information.

Prerequisites

to any verification of the theoretical expectations _seems to be

ii)

the _use of vicinals with

large _step

distances and

(ii)

the measurement of _a suflicient number of steps m the _sonne train.

We checked the behaviour of

G(m) by

_{usmg a}

Si(Ill)

vicinal

displaymg

_{a great} number of rather

loosely spaced

steps. ^The expenments were clone

by

in sitt1Reflection Electron

Microscopy (REM)

at l173

K,

1-e- above the reconstruction temperature of the

(7

_x

7).

This paper reports our

experimental

results and _compare them _to the theoretical

predictions

of

Villain, Grempel

and

Lapujoulade (VGL)

_[2].

2,

Experimental

Results

The

experimental

details have been

explained

elsewhere _[7] and will not be

repeated

here. The

sample

temperature was l173 K. The

region

of the surface which _was observed had a

fairly

uniform train of steps, ^about 30 _nm apart, ^which

corresponds

to a misorientation

angle

(Ù) ^of

roughly

o.6°.

Figure

1 shows _a REM

image

of this

sample.

The _mean step direction is < llo _>

and the steps are

practically parallel

to the electron beam incidence

plane.

Thus the

image

distorsion is

negligible

normal to the step ^direction. Hence, the t1~'s can be

directly

measured.

Figure

1 summarises the definitions.

l' '

' lj

l

~

~.

'-

q ÎÎ

, '1~ ^'

G

Il il ~

~l

~

Îj ~ il

Fig. 2. REM image of trie step ^{train that} was studied. Trie incident electron beam

is perpendicular

to trie 0.25 _pm bar. Since trie _{image is} severely foreshortened

(compare

trie lengths of trie scale bars), trie steps are actually almost parallel to trie electron beam incidence plane.

446 JOURNAL DE PHYSIQUE I N°4

Gim)A~

~

~~4

o

i,i o~

o O.s iog m

Fig. 3.

G(m)

_versus

logio(m)

for groups of 6 steps within the train of16.

(o):

steps 1 to 6.

(x):

steps 6 to il;

(6):

steps 10 ta 16. The sohà fine anà the black àots

(.)

_are extracteà from Figure 4 to show that the slope _is the _same within the uncertainty of _our measurements. For clarity, only _a few

error bars _are àrawn.

Gim)À

.

2,lO~

.

i,i o~

o o,s i o g m

Fig. 4.

G(m)

_versus

logio(m)

for steps 1 to 14. For clarity, only _a few _error bars _are drawn.

We bave studied _a set of16 consecutive steps _in trie train shown in

Figure

2. An

independent

determination of trie autocorrelation function of the step fluctuations showed that the auto- correlation

length along

the step ^direction is

roughly

400 _nm in this train [8].

Consequently

three "sections" of the step train _were

made,

normal to the _average step

direction, roughly

600 _nm apart. ^{Trie distances} between

couples

of steps (z~

i~+m) along

these "sections" _were

Table I. Mean step

distances,

numbers

of

pairs

of first

and most distant

neighbours for

the varions

portions of

the step train that _were studied.

~~~n

ofpairs

Step

labels

distance (/j _{of 1~~} of _mOSt distant

in _nm

neighbours ll~lghb°"~

l to14 29.1 2613 201

to 6 6to II

loto 16 29.0

measured

(see Fig. l). By averaging

_over the _various

pairs

of m~~

neighbours

and _over REM images (1.e. over time [7]) ^the mean distance between

pairs,

then the _mean value of the relevant parameter (u~

u~+m)2

_were determined.

In order _to check the

homogeneity

^{of the} train _we aise determined _< (u~

u~+m)2

> for three sub-sets of 6 steps ^{within the 16} ones studied. The results _are shown in

Figure

3.

The determination of the _errer bars is

explained

below.

Figure

4 shows the

plot

of

G(m)

=

< (u~

u~+m)2

> versus

logio(m)

_up to _{m =} 13 and the relevant _errer bars. In addition, Table I

displays

the numbers of _mth

neighbours

that _were taken into account for

averaging

and the _mean step distances that _were determined. From

Figure

3 and Table I _we conclude that the step ^{train is}

fairly homogeneous

and that _oui results _are

statistically significant.

The _errer bars _were determined _as follows: the

experimentally

measured value U2 of

< (u~

u~+m)2

> is in fact _an estimator of the actual

(unknown)

value

£~

_{[9]. If the distribu-}

tion of the (u~

u~+m)

is Gaussian, then the standard deviation of U2 _is

£~ @fi,

where Nm is the number of

independent

values used to determine U2 [9]. ^{It is reasonable to} assume a Gaussian distribution for the distances between

pairs

of m~~

neighbours

_[7]. Since the _mea-

surements of the distances _were

independent

the

uncertainty

_was ^estimated to be U2

(2 /Nm).

3. Discussion

Figure

4 shows that _a

logarithmic dependence

of G is found _up to _{m =} 7. This

dependence

aise _is found wherever _a subset of 6 steps is selected within the train. Within the _accuracy of

the measurements the

slope

of G _{versus m} is the _same for ail the subsets.

Beyond

_{m =} 8, _a

different

regime

is

obeyed.

In the

logarithmic regime

^the

experimental

value of the

slope

S of G is

Sexp

= 1.Il x102 nm2 if the value of _m is

expressed

in decimal

logarithm.

This _regime should be characteristic of _a

roughened

surface.

This

expectation

cari be checked

by

_{comparing Sexp} to the critical value

SR

₊ 2.302 _x

2(ai)2 _/7r2,

where _ai is the kink

length

normal to the step direction [2]. ^{At T} > TR, S~xp must be

larger

thon

SR. Taking

ai = 0.333 _nm

yields

SR _" 5.18 x 10~2

nm2,

_{hence Sexp »} SR.

Therefore _{we cari} conclude that _our surface _is

rougir.

We shall _see later _on that the measurements _are made within the

validity

of the VGL model.

Then Sexp cari be

compared

to the theoretical value

S =

2.302(ai)~kT/7rzÛ~î~î~

T > TR

(1)

448 JOURNAL DE PHYSIQUE I N°4

1J and _{1J' are} renormalised stiffnesses. Their

expressions

have been obtained

by

Masson [Si for

energetic

step interactions

decaying

_as

1/z~:

~2

1J ⁼

"

[fl +

Ai (2)

~ll

with

i 62

~2

^{4~ A 1/2 2}

~

ajj12 ^{ôçg2 24}

~~~~

_~

~

~

kib2

^~~~

~=o

1J' =

~~~"~)~~~

i

+

(i

⁺

#) ^~j

(4)

41

fl

~

Here

#

_{and A}

are the step stiffness and the step interaction constant,

respectively,

is the

mean

interstep

distance, ajj ^and ai are the kink

lengths parallel

and

perpendicular

to the step direction (ajj = 0.384 nm; ai " 0.333

nm).

In equation

(3),

b2 is the

"step dioEusivity",

1-e- the _{mean square}

displacement

of _a

kink,

which _cari be calculated from _{an atomistic} model of the step

[loi,

^b2 is _a function of _çg, the azimuthal

angle

of the step.

It is reasonable to use

equations (2), (3)

and

(4)

_since the existence of

1/z2

interactions is

experimentally

established [7]. ^{The values of A and}

#

are known for silicon at i173 K [7].

However,

_no

experimental

value exists for A. It _cari be calculated if _a

good

model of the step structure is available. In this respect, Williams et ai. have shown that the orientational

phase diagram

of silicon _cari be well

explained by using

a model of

independent polykinks [loi. Using

this model _we have calculated A. It turns eut that A is much smaller thon

#

_{at l173 K and}

cari be

neglected (A

_m

io~3)).

Trier

equations (2)

and

(3)

show that the

product

_1J1J' does trot

depend

_on

fl

itself.

Instead,

the relevant parameter ^is the

product A#,

which is

directly

obtained from trie measurement of trie terrace width distribution

(Ai

= 4.6 _x 10~~°J2 _at i173

K)

_[7].

Using

the aforementioned numerical

values,

equation

(1) yields

S

= 1.10 _x 102 nm2. _The agreement with Sexp is excellent.

Note that VGL'S

logarithmic

law is _an asymptotic

law, Orly

valid when the parameter p = m[1J/1J']~/~ ^is

larger

than [2].

Using

the calculated values for

1J and 1J', it turns eut that p = 53.7 _m. Thus _{p is}

langer

than for _{any m} and the VGL formalism _cari be used.

Beyond

_m

= 8 the slope of G increases and _no

simple logarithmic

law _can be found. This is _no statistical artefact

(Tab. I)

and trot due _{to an}

inhomogeneity

of the train

(Fig. 3). Therefore,

another

regime

is reached. It looks _as

though

the step ^train now behaved _more like _{a one-} dimensional system. ^It may be _a true

physical effect,

1-e- _a breakdown of the approximations involved in the theoretical treatments. It may as well be _an influence of

,

for

instance,

the finite

length

of the steps. However _we must mention that the auto-correlation function of the steps, normal to the step mean direction (1.e. ^{the function} < u~.u~+m

>)

has aise been measured

as a function of _m, in _an

independent

_mariner, for the _same train [8]. It tutus eut that this auto-correlation _goes to zero for _m > 7. This may ^Dot be coincidental. At the _{moment we} have _no clear-cut

explanation

and leave the question _open. We aise mention that Masson, toc, could Dot fit her data

by

_a

single logarithmic

law

beyond

_{m =} 3 for _a

Cu(1,1,11)

vicinal [5].

4. Conclusion

In this _paper the

logarithmic divergence

of the correlated

height

dioEerence _{over a}

rougir

surface is demonstrated

by meaningful

statistical results. For _a quantitative

comparison

with the VGL

theory

the

Orly

_necessary

quantity (AP

has been measured _on the _same

sample.

The agreement with the

theory

is excellent.

At

large

distances,

however,

the

logarithmic

law breaks down. This second

regime

could Dot be

given

_a ^definite

explanation.

Acknowledgments

We thank Prof. H-J-W- Zandvliet and Dr. P. E.

Hegeman

for

supplying

_us with their

unpub-

lished results. We aise thank Dr. D. Chatain and Prof. R. Kern for _a critical

reading

of the

manuscript.

References

[1] Nozières P., Solids far from Equihbrium, C. Godrèche, Ed.

(Cambriàge

University Press, Cam-

bridge,

1992) _p. 1.

[2] Villain J., Grempel D.R, and Lapujoulade J., J. Phys F. Met. Phys. 15

(1985)

809.

[3] Lapujoulade J., Surf. SC~. Reports 20

(1994)191.

[4] Golberg J-L-, Wang X.-S., Bartelt N-C- and Williams E-D-, Surf. Sct. 249

(1991)

L285 [Si Masson L., Thesis, Paris

(1994),

unpublished.

[6] Hegeman P-E-, Zandvhet H-J-W-, Kip G-A-M-, Kersten B-A- and Poelsema B., to be published.

[7] Alfonso C., Bermond J-M-, Heyraud J-C- and Métois J-J-, Surf. Sci. 262 (1992 371.

[8] Alfonso C., Thesis, Marseille

(1992),

unpublished.

[9] ^Bass J., Eléments de calcul des Probabilités

(Masson,

Paris, 1974) _p. ^201.

[10] ^Williams E-D-, Phaneuf R.J., Wei J., Bartelt N-C- and Einstein T.L., Surf. Sci. 294

(1993)

219;

Surf, Sci, 310

(1994)

451.