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PROFILES OF TILTED TIP CRYSTALS
D. Drechsler, D. Laporte
To cite this version:
D. Drechsler, D. Laporte. PROFILES OF TILTED TIP CRYSTALS. Journal de Physique Colloques,
1986, 47 (C2), pp.C2-443-C2-449. �10.1051/jphyscol:1986268�. �jpa-00225702�
JOURNAL DE PHYSIQUE
Colloque C2, supplément au n°3. Tome 47, mars 1986 page C2-443
PROFILES OF TILTED TIP CRYSTALS
M. DRECHSLER and D. LAPORTE
CRMC2-CNRS, Vnlverslte d'.Six-Marseille III, Campus de Luminy, F-13288 Marseille Cedex 2, France
Résumé - Pour obtenir des informations plus précises sur l'évolution
géométrique des pointes cristallines visualisées en microscopie électronique on a introduit une technique d'analyse des pointes inclinées. Pour l'analyse des images des pointes inclinées on a calculé le profil des pointes en fonc- tion de l'angle d'inclinaison et sur ces pointes on a déterminé la position et la forme des faces.
Abstract - To get more precise informations on the geometrical evolution of heated tip crystals visualized by electron microscopy a tilt analysis technique is introduced. To analyse micrographs of tilted tip crystals (Ni, Ta, W) the profile of such crystals is calculated for different tilt and tip cone angles. The position and shape of faces on such tips is determined.
I - INTRODUCTION
The study of field emitter crystals by transmission or scanning electron microscopy (TEM, SEM) was so far limited to non-tilted tips (see /1/2/3/4/5/) because (I) usually micrographs of tilted tips had been not made, (2) the existing microscope tilt devices or goniometers offer only limited analysis possibilities and (3) no calculation of the profile of tilted tips exist so far. Nevertheless, an extension of such studies to tilted tips may be of interest in order to obtain more complete and more precise information on the geometry of tip crystals. So we have tried to calculate the profile of tilted tips and present the result in this paper. Our study is limited to heated tips or steady-state tips /6/2/7/, formed by surface self-dif- fusion under the action of capillarity. Such tips are in fact most interesting in regard to experiments, theory and applications.
II - EXPERIMENTS
The tips are usually made from a crystal or wire by electrolytic etching (see / 8 / ) . It is of interest and possible to etch conical tips of definite cone angle 19/.
The tip is usually mounted on a heating loop and then introduced in ultrahigh vacuum (UHV) in a way that temperature and heating time can be regulated (see / 1 0 / U / ) . The heat treatment yields (1) mostly an adsorption free surface and (2) a tip blunting by surface self-diffusion under the action of capillarity forces /6/1/7/. Surface cleanliness or adsorption as well as blunting are controllable by field electron microscopy. Tip cone angles are by preference greater than 3°, which is necessary to form steady-state shapes /6/7/. On steady-state shape crystals angles and ratios of distances are independent of the crystal size (tip radius). After such an UHV experiment the vacuum system is opened and the tip is transported and mounted on an object holder of an electron microscope (TEM, SEM). In former studies the tip axis was more or less perpendicular to the microscope axis. Now we have considered tips which have been tilted either by commercial or by laboratory build tilt devices or goniometers. Typical micrographs are presented in the second part of this paper.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986268
JOURNAL
DE
PHYSIQUEF i g . I
P r o j e c t i o n ( p r o f i l e ) c o n s t r u c t i o n of a t i l t e d t i p
I11 - DETERMINATION OF TILTED TIP PROFILES
The s t e a d y - s t a t e shape i s an approximation i n which shape a n i s o t r o p i e s a r e n o t y e t considered (they w i l l b e considered i n s e c t i o n V ) . So t h e s t e a d y - s t a t e shape h a s a symmetry of r e v o l u t i o n and i t s s u r f a c e can be r e p r e s e n t e d by a s e r i e s of p a r a l l e l c i r c l e s . I f t h e t i p a x i s i s more o r l e s s p e r p e n d i c u l a r t o t h e a x i s of t h e microscope (TEM, SEM) t h e s u r f a c e c i r c l e s appear a s s t r a i g h t l i n e s . ( f i g . I and 2 ) . However, when t h e t i p a x i s i s t i l t e d by an a n g l e B ( f i g . I ) t h e s e r i e s of c i r c l e s appears a s a s e r i e s of p a r a l l e l e l l i p s e s of c o n s t a n t e l l i p t i c i t y (a/b = c o n s t . ) . The t i l t e d t i p p r o f i l e i s obtained when t h e r o t a t e d z e r o t i l t p r o f i l e i s p r o j e c t e d t o a plane which passes t h e X-axis ( f i g . l ) p e r p e n d i c u l a r t o t h e drawing p l a n e . It i s p o s s i b l e t o reduce t h e threedimensional t o a twodimensional problem by imaginating t h a t t h e p r o j e c t i o n plane i s turned by 90" around t h e X-axis. Then t h e drawing p l a n e becomes a l s o p r o j e c t i o n p l a n e . OA1 i s t h e p r o j e c t i o n of t h e t i p a x i s d i s t a n c e OA (OA1= OA.
cos B). The g r e a t a x i s of each e l l i p s e i s equal t o t h e corresponding c i r c l e r a d i u s r,. The v a l u e s of r, a r e n u m e r i c a l l y c a l c u l a t e d a s a f u n c t i o n of t h e t i p apex d i s - tance z and the t i p cone angle a / 7 / . Each small e l l i p s e a x i s can be e a s i l y construc- ted o r c a l c u l a t e d ( A { B l = AB s i n B ) . Thus t h e e l l i p s e around A1 c a n b e c o n s t r u c t e d a s t h e p r o j e c t i o n of t h e t i p s u r f a c e c i r c l e of around A and r a d i u s r,. Any o t h e r p r o j e c t i o n e l l i p s e can b e determined i n t h e same manner. F i n a l l y , t h e p r o f i l e of t h e p r o j e c t i o n of t h e t i l t e d t i p i s obtained by c o n s t r u c t i o n of t h e envelope of a l l e l l i p s e s ( f i g
.
2 ) .I n o r d e r t o draw t h e t i l t e d t i p s by a computer d i r e c t e d g r a p h i c r e c o r d e r i t i s n e c e s s a r y t o f i n d t h e corresponding e l l i p s e e q u a t i o n . The i m p l i c i t e q u a t i o n of cen- t r a l e l l i p s e s i n xy-coordinates i s
(1) x 2 / a 2
+
y2/b2 = Iwhere a and b a r e t h e l e n g t h s of t h e e l l i p s e s h a l f axes i n X and y d i r e c t i o n s . I n t h e p r e s e n t c a s e i t has t o b e considered t h a t ( f i g . 1) :
(1) X and y have t o b e r e p l a c e d by X and Y ; ( 2 ) t h e c e n t e r of t h e e l l i p s e s i s d i s - placed i n X d i r e c t i o n o r X h a s t o be replaced by X
-
c ; (3) c i s given byc = z cos 6; ( 4 ) b = r z ; ( 5 ) a = r, s i n B . Then t h e e l l i p s e e q u a t i o n becomes :
(2) 2 2 2
(X
-
z cos B) / ( r Z s i n B)2 + Y / r Z = II n p r a c t i c e t h e e l l i p s e s a r e drawn by an automatic p l o t t e r (Benson 1 1 0 2 ) .
F i g . 2
-
C a l c u l a t e d p r o j e c t i o n s of a t i l t e d t i p c r y s t a l v a l i d f o r a n y t i p r a d i u s v a l u e( s t e a d y - s t a t e s h a p e ) . Example of cone a n g l e cr = 4"
( 2 a ) I n i t i a l t i p of d e f i n i t e c o n e a n g l e a s p r o d u c i b l e b y e l e c t r o l y t i c e t c h i n g / 9 / (2b) T i p of ( 2 a ) a f t e r s u r f a c e s e l f - d i f f u s i o n b y c a p i l l a r i t y f o r c e s . A s t e a d y - s t a t e
s h a p e of r e v o l u t i o n symmetry i s formed / 6 / 1 / 2 / 7 / . The t i p i s b l u n t e d a s w e l l a s r e d u c e d in l e n g t h . C i r c u l a r c r o s s s e c t i o n s a p p e a r a s s t r a i g h t l i n e s . The s h a p e a n i s o t r o p y c a n b e c o n s i d e r e d s u p p l e m e n t a r y ( s e e f i g . 6 and 7 and / 3 / / 5 / . ( 2 c ) T i p ( 2 b ) a f t e r 20" t i l t . The c i r c l e s a p p e a r a s e l l i p s e s . The t i p p r o f i l e ( a s
v i s u a l i z e d by e l e c t r o n m i c r o s c o p y ) i s g i v e n b y t h e e n v e l o p e of t h e e l l i p s e s (2d) T i p a s i n ( 2 c ) b u t 40" t i l t a n g l e . T i p apex and p r o f i l e apex a r e n o t i d e n t i c a l ( 2 e ) T i p a s i n ( 2 c ) b u t 60" t i l t a n g l e . The a p p a r e n t t i p l e n g t h a s w e l l a s t h e
a p p a r e n t t i p neck c u r v a t u r e r a d i u s a r e d e c r e a s e d . The a p p a r e n t cone a n g l e i s i n c r e a s e d
( 2 f ) T i p a s i n ( 2 c ) b u t 80" t i l t a n g l e . The e l l i p s e s a r e n e a r l y c i r c l e s . The appa- r e n t neck p r o f i l e h a s o b t a i n e d a b r e a k . The a p p a r e n t n e c k t o p r o f i l e - a p e x d i s -
t a n c e i s r e d u c e d .
JOURNAL DE PHYSIQUE
A s r e s u l t a t y p i c a l s e r i e s of t i l t e d t i p s i s shown i n f i g . 2a t o 2 f . Q u a l i t a t i v e c h a r a c t e r i s t i c s of t i l t e d t i p s a r e i n d i c a t e d i n t h e c a p t i o n s . Q u a n t i t a t i v e charac- t e r i s t i c s a r e p r e s e n t e d i n t h e f o l l o w i n g s e c t i o n .
IV - CHARACTERISTICS AND ANALYSIS OF TILTED T I P PRDFILES
The f o l l o w i n g d e s c r i p t i o n i s based on t h e v e r i f i e d h y p o t h e s i s t h a t a t i p v i s u a l i z e d i n a micrograph (TEM, SEM) h a s a s t e a d y - s t a t e shape. When a t i p ( i n i t i a l l y p a r a l l e l t o t h e image p l a n e ) i s t i l t e d by an a n g l e B (measurable on t h e t i l t d e v i c e ) t h e a n g l e between t h e t i p and t h e image p l a n e i s u s u a l l y n o t B . The measurable a n g l e i s only an apparent t i l t a n g l e (Ba) and n o t t h e r e a l t i l t a n g l e Br ( a n g l e between t i p a x i s and micrograph p l a n e ) . So t h e q u e s t i o n a r i s e s : How can a r e a l t i l t a n g l e be determined. By t r y i n g t o s o l v e t h e problem one h a s t o b e aware a l s o t h a t t h e t i p co- ne a n g l e measurable on a t i l t e d t i p micrograph i s only an apparent and not t h e r e a l cone a n g l e .
To s o l v e both problems i t i s necessary t o search and determine geometrical t i p p r o f i l e q u a n t i t i e s a s a f u n c t i o n of r e a l t i l t and cone a n g l e s . The f i r s t measurable q u a n t i t y we t r y t o u s e f o r t h i s purpose i s t h e a n g l e y between t h e t i p a x i s and t h e t i p p r o f i l e t a n g e n t a t t h e r e v e r s a l p o i n t a s shown i n f i g . 3 . The second q u a n t i t y proposed i s t h e r a t i o of t h e neck r a d i u s R, t o t h e apex r a d i u s Ra ( s e e fig. 4) The d a t a of f i g . 3 and 4 a r e determined from p r o j e c t i o n s a s i n f i g . 2.
l I
M M 3 0 1 0 5 U 6 0 N r n
Tilt aqk
F i g . 3
-
The a n g l e (y) between t h e t i p a x i s and t h e p r o f i l e t a n g e n t a t t h e r e v e r s a l p o i n t v e r s u s t h e t i l t a n g l e f o r cone a n g l e s of 3O,5',7Oand 9 " . A t t h e r i g h t s i d e of t h e
broken l i n e t h e neck p r o f i l e s have a break and t h e r e y i s t h e
tangent-axis a n g l e a t t h e b r e a k
Tilt mgle
P
F i g . 4
-
The r a t i o neck r a d i u s (Rn) t o apex r a d i u s (Ra) v e r s u s t h e t i l t a n g l e B f o r cone a n g l e s of 3O,5",7O and 9 O Real t i l t and cone a n g l e s c a n now b e determined from a t i l t e d t i p micrograph i n t h e following way :( 1 ) y j s measured on t h e micrograph and y i e l d p r o v i s i o n a l l y a rough v a l u e of a r u s i n g f i g . 3 . (2) Rn/Ra i s measured on t h e micrograph and y i e l d s w i t h t h e rough v a l u e of a, and f i g . 4 a rough v a l u e of B,. ( 3 ) The rough v a l u e of Br e n a b l e s t o o b t a i n a b e t t e r ar v a l u e u s i n g f i g . 3 . Then t h e b e t t e r a, v a l u e i s used t o o b t a i n a b e t t e r &-value.
A s a t e s t we u s e h e r e t h e micrograph of a t i l t e d n i c k e l t i p ( f i g . 5 a ) . The r e s u l t of t h e a n a l y s i s i s approximately ar = 6.8' and B, = 38". F i g . 5c shows f o r compa- r i s o n a c a l c u l a t e d p r o f i l e having n e a r l y t h e same a n g l e s . Reasonable agreement i s o b t a i n e d .
I n conclusion : The r e a l t i l t and t h e r e a l cone a n g l e s of a s t e a d y - s t a t e t i p a r e roughly determinable by an a n a l y s i s of only one micrograph p r o f i l e .
Fig. - 5
-
Shape of a t i l t e d t i p :(5a) Transmission e l e c t r o n micrograph of a til- t e d n i c k e l t i p of a d s o r p t i o n f r e e s u r f a c e a f t e r h e a t i n g (1700 K ) . Micrograph by T . B a r s o t t i , J.M. Bergond and M. Drechsler
181.
The e f f e c t i v e t i l t and cone a n g l e s a r e roughly
5a determined by f i g . . - 3 and 4
(8=38',
~ ~ ' 6 . 8 ' ) .Tip r a d i u s 0.5 m .
(5b) P r o f i l e a s i n f i g . 2 b u t 7 O cone a n g l e and 0' t i l t angle
t
5b (5c) P r o f i l e a s i n (5b) b u t 40' t i l t a n g l e
F i g . 6b
-
F i e l d i o n micrograph of a n annealed t u n g s t e n t i p corresponding t o6a. Micrograph by A . Mhller and M . D r e c h s l e r , s e e / l 1 /
Fig. 6a
-
T i p p r o j e c t i o n i n c l u d i n g a cons i d e r a t i o n of t h e approximate posi-t i o n and s i z e of t h e p l a n e f a c e s of t h e e q u i l i b r i u m shape without a d s o r t i o n
110131.
Example of t u n g s t e n of701
11- o r i e n t a t i o n , 6O cone a n g l e and 60" t i l t a n g l eF i g . 6c - F i e l d e l e c t r o n micrograph of a t u n g s t e n t i p a s i n (6b)
~ 2 - 4 4 8 JOURNAL DE PHYSIQUE
V - POSITION AND SHAPE OF PLANE FACES ON TILTED TIP MICROGRAPHS
The d e t e r m i n a t i o n of t h e a p p a r e n t p o s i t i o n and s h a p e of f a c e s on t i l t e d t i p micro- g r a p h s (TEM, SEM) i s f a c i l i t a t e d by t h e f a c t t h a t t h e t i p s u r f a c e i s d e s c r i b e d a s a s e r i e s of e l l i p s e s . The h e a t e d t i p s a r e n o t a b s o l u t e s t e a d y - s t a t e s h a p e s . R a t h e r s u p e r p o s e d i s t h e c r y s t a l e q u i l i b r i u m shape 1 3 1 . As t i p c r y s t a l s a r e used t o s t u d y phenomena on d e f i n i t e f a c e s of m i c r o - c r y s t a l s i t i s i m p o r t a n t t o t r y t o d e t e r m i n e t h e p o s i t i o n and t h e s h a p e of f a c e s on t i l t e d t i p m i c r o g r a p h s . The e q u i l i b r i u m s h a p e of a c l e a n m e t a l c r y s t a l i s composed of c u r v e d r e g i o n s and p l a n e f a c e s ({1101,{@021 and { l 1 2 1 f o r a c r y s t a l a s t u n g s t e n ) . For s i m p l i f i c a t i o n t h e curved r e g i o n s c a n b e assumed t o b e s p h e r i c a l
1 3 1 ,
which i s a p p r o x i m a t e l y t r u e f o r t h e p a r t of t h e t i p s u r f a c e v i s i b l e on f i e l d e m i s s i o n m i c r o g r a p h s . I n t h i s s u r f a c e r e g i o n a p l a n e f a c e h a s a p p r o x i m a t e l y a c i r c u l a r b o r d e r which s h o u l d a p p e a r i n a t i l t e d t i p m i c r o g r a p h a s a n e l l i p s e . The d i a m e t e r of a p l a n e f a c e i s a p p r o x i m a t e l y c a l c u l a b l e / 3 / 1 0 / . Then by u s i n g g e o m e t r i c a l t r a n s f o r m a t i o n s which a r e n o t d e s c r i b e d i n d e t a i l h e r e , t h e p o s i - t i o n , t h e s i z e and t h e e l l i p t i c i t y of t h e p l a n e f a c e s a p p e a r i n g on a t i l t e d t i p p r o j e c t i o n c a n b e c a l c u l a t e d .F i g . 7a
-
T i l t e d T a n t a l t i p a f t e r h e a t i n g (2000 K, %20 h o u r s ) . Scanning e l e c t r o n m i c r o g r a p h by S . Hok and M. D r e c h s l e r 1121.Only 3 o r 4 of t h e { l 10) f a c e s a r e j u s t v i s i b l e a s e l l i p s e s . The c u r v e d neck l i n e r e p r e s e n t s a s m a l l g r a i n boundary g r o o v e
( b r o k e n l i n e i n f i g . 7b) which i s c l e a r l y v i s i b l e on t h e m i c r o g r a p h of t h e u n t i l t e d t i p .
F i g . 7b - P r o j e c t i o n a s i n f i g . 2 b u t cone a n g l e 7" and t i l t a n g l e 70" c o r r e s - ponding a p p r o x i m a t e l y t o f i g . 7 a . I n d i - c a t e d a r e a d d i t i o n a l l y t h e i n d i c e s o f the { 110)-faces and t h e [ I l l]-zone l i n e s c o r r e s p o n d i n g t o t h e t a n t a l c r y s t a l o r i e n t a t i o n and t h e r o u g h l y c a l c u l a t e d e q u i l i b r i u m s h a p e s i z e of t h e s e f a c e s / l O /
As a n example f i g . 6 shows t h e p l a n e f a c e s o f a h e a t e d and t i l t e d c l e a n t u n g s t e n t i p c r y s t a l t o g e t h e r w i t h t h e c o r r e s p o n d i n g f i e l d i o n and e l e c t r o n m i c r o g r a p h . Another example shows f i g . 7. T i l t and cone a n g l e and t h e p r o j e c t i o n of t h e t a n t a l u m t i p a r e d e t e r m i n e d u s i n g t h e few v i s i b l e {110} f a c e s i n f i g . 7a. The p o s i t i o n of t h e f a c e c e n t e r s i s d e t e r m i n e d b y geometry u s i n g c r y s t a l l o g r a p h i c d a t a . The f a c e d i a m e t e r s a r e g i v e n b y c a l c u l a t i o n / 1 0 / 3 / . F i n a l l y a s a c u r i o s i t y f i g . 8 shows a low m a g n i f i - c a t i o n image of a t i l t e d t i p .
I n c o n c l u s i o n : P r o j e c t i o n s of t i l t e d t i p s c a n b e c a l c u l a t e d i n c l u d i n g t h e p o s i t i o n and s h a p e o f c r y s t a l f a c e s a p p e a r i n g i n t i l t e d t i p m i c r o g r a p h s .
F i g . 8
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T i l t e d (-80') a n n e a l e d t a n t a l t i p . Scanning e l e c t r o n m i c r o g r a p h b y S. Hok and M. D r e c h s l e r / 1 2 / . The s t e a d y - s t a t e s h a p e a p p e a r s i n t h e c e n t e r ( a r r o w ) .The combined u s e of f i e l d e m i s s i o n microscopy and e l e c t r o n m i c r o s c o p y (TEM, SEM) i s i m p o r t a n t b e c a u s e i t combines t h e p o s s i b i l i t y of u l t r a h i g h vacuum e x p e r i m e n t s on m i c r o - c r y s t a l s under d e f i n e d c o n d i t i o n s w i t h t h e p o s s i b i l i t y of a p r e c i s e measure- ment o f g e o m e t r i c a l s u r f a c e d a t a i n c l u d i n g i t s e v o l u t i o n . So t h e d e s c r i b e d i n t r o - d u c t i o n of t i p t i l t i n g and a n a l y s i s may c o n t r i b u t e t o more and a more p r e c i s e u s e of t h i s u n i q u e combined t e c h n i q u e i n domains a s : P r e p a r a t i o n and a n a l y s i s of t i p c r y s t a l s f o r s t u d i e s o n s u r f a c e m a t t e r f l u x e s , c r y s t a l growth, c r y s t a l d e f e c t s , a d s o r p t i o n , s h a p e e q u i l i b r i a , g r a i n b o u n d a r i e s , f i e l d s t r e n g t h d e t e r m i n a t i o n and t h e f a b r i c a t i o n of h i g h q u a l i t y f i e l d e m i t t e r s .
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(1965) 1826 171 Vu T h i e n Binh, D r e c h s l e r M . , J . d e Physique 45 (1984) C o l l . C9, p . 29-37 ( P r o c . I n t e r n . F i e l d E m i s s i o n Symp., P a r i s , 1 9 8 4 r1 8 1 Miiller Erwin W., and Tsong, T.T., F i e l d I o n Microscopy, E l s e v i e r , New York, 1969 / 9 / Vu T h i e n Binh, P i q u e t , A., Uzan, R . , D r e c h s l e r , M., Revue d e Physique Appliquge 5 , 1970, 645
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(1969) 471-490/ 1 2 / Hok, S., D r e c h s l e r , M., P r o c . I n t e r n . F i e l d E m i s s i o n Symp., P o r t l a n d , Oregon 1981, p . 110-113. Also Hok, S . , ThSse d e 3e c y c l e , U n i v e r s i t G d ' A i x - M a r s e i l l e 111, 1981.
