SIZE AND SHAPE OF CRYSTAL FACES ON HEATED METAL TIPS

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SIZE AND SHAPE OF CRYSTAL FACES ON HEATED METAL TIPS

Vu Binh, M. Drechsler

To cite this version:

Vu Binh, M. Drechsler. SIZE AND SHAPE OF CRYSTAL FACES ON HEATED METAL TIPS.

Journal de Physique Colloques, 1984, 45 (C9), pp.C9-29-C9-37. �10.1051/jphyscol:1984906�. �jpa-

00224382�

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JOURNAL DE PHYSIQUE

Colloque C9, suppl6ment au n012, Tome 45, d6cembre 1984 page C9-29

S I Z E AND SHAPE OF CRYSTAL FACES ON HEATED METAL T I P S

Vu Thien Binh and M. ~rechsler*

De'partement de Physique des Mate'riaux, associe' au CNRS, Universite' de Lyon I , F-69622 ViZZeurbanne, France

'CRMC~

-

^CNRS, Universite' drAix

-

MarseiZZe, Campus de Lminy, F-13288 MarseiEZe, France

Resume : Nous avons arneliorg la pr6cision du calcul des formes quasi stationnaires des pointes m6talliques chauffees dans le vide pour des angles de cone entre 3 et lo0.

De nouveaux r6sultats sont pr6sentes concernant (1) les courbures locales, (2) les variations des diamgtres des faces, (3) les gllipticites et (4) la taille des faces. Les resultats calcules sont en accord avec les quelques resultats expgrimentaux que nous avons estim6s 2 partir des photos obtenues par differentes methodes microscopiques (FEM, FIM, SEN et TEM).

Abstract : The steadv-state s h a ~ e s of metal tips heated in vacuum are calculated with improved precision for tip cone angles between 3' and 10'. These new data of tip crystals are presented as:

(1) the local curvatures ( 2 ) the variations of the diameters of the faces, (3) the face ellipticities and (4) the face areas.

The calculated data are in agreement with the limited number of data estimated from tip micrographs obtained by different micros- copic methodes (FEM, FIM, SEM and TEM).

Introduction :

The crystal at the end of a metal tip is widely used,for example, for fundamental studies in surGace science. Such a crystal is usuallv pre- pared by heating conical tivs in ultra-high vacuum. The anisotropic crystal shape obtained represents an equilibrium shape (fig. 1) compo- sed of several plane faces and of curved surface regions /1/2/3/4/.

The faces of such a crystal can be observed by using various types of microscopes (field electrorr microscope (FEM) (see f iq. 5)

,

^field

ion microscope (FIM) (see fig. 9 ) , scanning electron microscope (SEMI (see fig. 2) and transmission electron microscope (TEM) /I /I 0 / )

.

Size and shape of the faces of such crystals (formed by surface self- diffusion fluxes) are determined by : (1) anisotropic surface free energies (equilibrium shape) and (2) capillarity. This paper concen- trates on the capillarity effect which has been neqlected so far in FEM and FIM studies. Recent TEM and SEM studies have shown that the capillarity effect cannot be negleted in many cases (fig. 2) /3/4/

and that the variationof the longitudinalfacediameter due to capillarity can be graphicallv calculated /4/. We now have tried (1) to calculate such face changes numerically (instead of graphically) which must lead to a better precision, (2) to calculate not only lonqitu- dinal but also transverse curvatures and face diameters(in the two principal direction) (3) to consider apex distances not only to 60 degree but up to % 80° (4) to calculate such data not only for one example but systematically for tip half cone anales from 3' to

lo0

(5) to calculate not only local curvatures and face diameters but also face ellipticities and face areas. The results presented here briefly nay initiate detailed discussions an6 can be used for the different applications.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984906

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C9-30 30URNAL DE PHYSIQUE

Fig. 1

Model of a tip equilibrium shape (bcc, tantalum)

2 g1 = longitudinal face diameter of ( 1 1 0 1 .

2 gt = transverse face diameter.

Fig.2. Profile of a heated Mi tin crvstal adjusted in the microscope (SEM)

.

The longitudinal diameter ( 2ql) of {Tll), about 7 0 ' awav from the tip apex, is a factor of roughly 1.5 crreater than that of (111).

Micrograph by T.Barsotti, J.M. Bermond and M. Drechsler / 7 / .

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2. Method of Calculation.

The method of calculating the isotropic evolution of a sharp tip towards a steady-state tip shape by capillarity induced surface self-diffusion fluxes has been described basically by NICHOLS and MULLINS / 5 / 6 / and will not to be repeated here. Our calculation is different concerning the following three points :

(1) We try to obtain a better precision.

( 2 ) Not only steady-state tip profiles are calculated and presented

but also the values of local curvatures in the two principal direc-

1 1

tions (

-

and

-

R1 and R2 are the local curvature radii) as well R1 R2, 1 1

as total curvatures

(

E,

⁺R?).The curvatures in planes which passes the tip axis are called'long~tudinal curvatures and those in rectan- gular planes transverse curvatures.

(3) We do not consider only the isotropic tip shapes / 5 / 6 / (which are only rough approximations) but also particularly the shape anisotropy. Calculated are the two principal diameters (longitudinal and transverse, fig. 1) of the lane faces or, more precisely, the variation of these diameters as a function of the angular distance to the tip apex (see fig. 3). This includes a determination of the variation of the face ellipticity (gt/gl) as well as of that of the relative face area (gt.g1)6/(gt.g1) 0 0 as a function of the apex distances.

o r

. . . . . . . . . . .

^l

0 1 2 3 4 5 6 7 8 9 1 C 1 1

T I P A X I S D I S T A N C E X ( ~ I Z ~ W I T )

Fig. 3A. Half steady state profile for different tip half cone angles.

Distances are normalized in tip apex radius units Ro.

Fig. 3B. Longitudinal curvature 1/R1 versus tip apex distance for the different angles.

Fig. 3C. Transverse curvature 1/R2 versus tip apex distance.

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C9-32 JOURNAL DE PHYSIQUE

3. Results

Steady-state tip profiles of improved precision are shown in fig. 3A.

The local longitudinal curvatures 1/R1 as well as the transverse curvatures 1/R2 are shown in fig. 3B and 3C as a function of the tip axis distance (or the angle between an {hkl} face and the tip apex face).

Tips of cone angles between 0" and 3" are not considered, because such cones do not form steady state profiles /5/6/. Fig. 4 shows the variation of the total curvature (l/R1 + 1/R2) which is needed for example, to calculate the changes of the chemical potential along the tip surface due to capillarity or for a better calculation of tip equilibrium shapes / 4 / .

0 1 2 3 4 5 6 7 8 9 1 0 U

T I P A X I S D I S T A N C E X (NW.UIIEDWIT)

Fig. 4 Total local curvature (l/R1 + 1/R2) versus tip apex distance for different tip half cone angles. Tip profiles as in fig. 3A are shown on the bottom.

To facilitate the understanding and use of the results we present a field electron micrograph and a special crystallographic projection for it (fig. 5a and 5b). Fig. 5a shows a wide-angle FEM micrograph of a tungsten tip. The sometimes anisotropic border of a face is determined by other type of investigations /1/2/3/. The question here is for example : which are the real sizes and shapes of (110) and

(011) faces compared to those of (011) (see also fig. 9). Until now it was approximately assumed that such faces have (1) equal diameters,

(2) no face ellipticity (except a possible ellipticity due to the anisotropic values of the surface free energies) and (3) equal face areas. However our calculations show that these assumptions are not correct, which means that the face dimensions change considerably

(fig. 6, 7 and 8). This is illustated in the special crvstallographic projection of fig. 5b. The projection shows approximately the relative real sizes and shapes of the plane faces of the crystal shown in fig. 5a. The precise data are given by fig.6 and 7:One result for example is, that the longitudinal diameter of (011) is not equal but about 2.8 times greater than that of (011). FurtheronL the (011) face must be elliptic (gt/g 0.405) and the area of (01 1 ) should be 3.2 times greater than tkat of (0 1 1 )

.

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Fig. 5a. Field electron micrograph of a tungsten crystal which shows the excep- tional case of two-opposite parallel faces ( ( 0 1 1 ) and

( 0 1 1 ) ) .Micrograph by E.W.

Mcller / 8 / . The observation of more faces than usually is promoted by adsorbed Ba of less than 0 . 1 monolayer.

The tip radius is a 1 3 0 0

A ,

which corresponds roughly to the distance from ( 1 0 1 ) to ( 0 1 1 )

.

^{B y}heating for some minutes ( s 2 8 0 0 K ) the tip equilibrium shape of clean tungsten has been formed. Afterward Ba was adsorbed and spreaded by heating

( a 1 0 0 0 K ) which pratically does not change the crystal shape.

Fig. 5b. An unusual crystal projection. The projection shows the approximate face sizes and shapes of a heated W(or other bcc metals) tip of < 0 1 1 > orientation.

The projection corresponds to fig. 5a. Each of the ellipses shows the calculated size and shape of a face which is free of all distortions. The angular face diameters are assumed t o b e : 2 0 ° for [ O 1 1 1 , 16O for [ 0 0 1 } and 12O for { I 1 2 1 values which are in reasonable agreement with experiments and calculations / I / 2 / 3 / . The tip half cone angle is assumed to be 5'.

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JOURNAL DE PHYSIQUE

18 2'0 3 8 48 5 8 68 78 8 ' ~

dB

A 1 6 U L A R D l S T A R t E F R O n T I P A P E X (nrws)

Y LI 14.-

<

-

a ,

Y

:

^12..

Y

A 1 6 U L A R D I S T A N C E F R O M T I P A P E X (ommrj)

Fig. 6a and 6b : Relative longitudinal (a) and transverse (b) face diameters versus angular distance from the tip apex (angle between face {hkl) and the apex face)

.

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7Q . ib 2b '

3b

^'^4:8^'^5:8^-

6b

^.^$8

.8b

^.^$0

A 1 6 U L A R D l S T A W C E F R O M T I ? A P E X

Fig. 7. Face ellipticity (gt/gl) versus angular distance from the tip apex.

4. Discussion.

4.1 Experimental verification.

The presented calculated results are related to the following known experimental results :

( 1 ) The isotropic morphological evolution of imperfect conical tips

to steady-state shapes is to a large extent confirmed experimentally /6/ : The presented curvature data therefore can be considered as beeing confirmed roughly by ex~erimental data.

(2) The formation of the anisotropic tip equilibrium shape is studied experimentally and is in reasonable agreement with calculated results /I 12/31.

(3) The presented data predict that with the increase of the distance to the tip apex from O 0 to 9 0 ' the face diameters should change in the following manner : an important increase of the longitudal diameter and a smaller increase of the transverse diameter. These pre- dictions are in agreement with SEM microscope observations on Ta and Ni / 3 / 7 / 9 / . In spite of these agreements, it would certainly be of

interest to verify the presented data with better precision and in more detail.

-

TI* WAF (PL I*aE-

Fig. 8. Relative face area versus angular distance from the tip apex.

A N G U L A R D I S T A N C E F R O I I T I P A P E X ( - )

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C9-36 JOURNAL DE PHYSIQUE

Fig. 9. Field ion micrograph of a heated tungsten crystal

(2600' K , 30 minutes, 4.10-'O Torr)

.

Hydrogen'ion Image. Image diameter% 2500 A. Micrograph by A. Miiller and M. Drechsler.

4.2 Real Shape of the Face Borders.

(formed by simultanious action of capillarity and surface energy anisotropy)

.

The basic shape of a face in fig. 1, 5b, 6 and 7 is assumed to be circular. However this assumption is only an approximation because the surface free energy is anisotropic so that a face may not have a circular symmetry / 1 / 3 / . Then the real ellipticity is caused and calculable because of three phenomena : (1) the surface energy anisotropy, (2) the tip surface capillarity and (3) the tip crystal orientation. Real face borders are measurable by means of TEM or SEM. Ellip- tic borders of close packed faces of Ta and Ni are visualized by SEM /9/3/7/. Nevertheless, the problem has not been studied yet, as far as we know, in a precise quantitative manner

4.3 Applications.

When field emitter crystals are used for fundamental studies in physics, surface chemistry, or metallurgy it may be, in general, of some interest to know the size and the shape of the crystal faces involved in such experiments. Examples of special applications are :

( 1 ) An improved construction of tip equilibrium shapes /3/4/.

(2) Improvements in the measurements of anisotropic surface free energies / I O/.

(3) A determination of the capillarity induced variation of the chemical potential of an atom along the tip crystal surface.

(4) In FEM and FIM microscopy an improved analysis of the size and the shape of the imaged faces (fig. 5 and 9) may give some information on the tip cone angle. Moreover, an analysis may enable one to decide if an initially imperfect tip shape has obtained a more or less perfect steady-state shape because of heating.

(5) The knowledge of the relative face diameters (fig.6 and 7) and of the curvatures (fig. 3) may give some information on image distortions in FEM and FIM.

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REFERENCES

1 1 1

^M. Drechsler and J. Nicholas : J.Phys. Chem. Solids

28

( 1 9 6 7 )

2 5 9 7 and 2 6 0 9 .

121 A. MUller, M. Drechsler : Surface Sci.

13

( 1 9 6 9 ) 4 7 1 .

131 M. Drechsler : in "Surface Mobilities of Solid Materials" ed. Vu Thien Binh, Plenum Press, New-York 1 9 8 3 , 4 0 5 - 4 5 7 .

141 M. Drechsler, submitted to Surface Science

151 F.A. Nichols and W.W. Mullins, J. Appl. Physics

36

( 1 9 6 5 ) 1 8 2 6 . 161 VU Thien Binh, H. Roux, R. Piquet, R. Uzan, M. Drechsler ;

Surface Sci.

25

( 1 9 7 1 ) 3 4 8 .

171 R. Barsotti, J.M. Bermond, M. Drechsler, Proc. 2 9 t h Intern. Field

Emission Symp. eds. Andren and Norden, Almquist and Wicksell Intern., Stockholm 1 9 8 2 , 59-66 and unpublished results of these authors.

181 Erwin W. Miiller, Erqebn. d. exakten Naturw. XXVII ( 1 9 5 3 ) 2 9 0 .

191 M. Drechsler, S. Hok : Intern, Field Emission Symp., Oregon,

U.S.A. 1 9 8 1 .

1101 T. Barsotti, J.M. Bermond, M. Drechsler, article accepted by

Surface Science.