The resolution of the low energy electron reflection microscope

Ultramicroscopy 17 (1985) 51-56 North-Holland, Amsterdam

51

T H E R E S O L U T I O N OF T H E LOW E N E R G Y E L E C T R O N R E F L E C T I O N M I C R O S C O P E E. B A U E R Physikalisches Institut, Teehnisehe Universiti~t Clausthal, D- 3392 Clausthal- Zellerfel~ Fe~ Rep. of Germany

Received 17 August 1984

The resolution attainable by imaging a surface with reflected electrons with energies from zero to several hundred eV in an immersion lens is discussed. It is shown that resolutions in the nm range can be obtained.

1. Introduction The direct imaging of surfaces was one of the first achievements of electron microscopy in its infancy, both by emission microscopy [1] and high energy reflection microscopy [2]. However, only in recent years have the two methods become useful techniques for the study of surfaces [3-5]. This p a p e r describes the fundamentals of another powerful surface imaging technique, low energy electron reflection microscopy (LEERM), which was invented more than 20 years ago [6] but put into operation successfully only recently [7]. In this technique the surface to be imaged is the cathode of an immersion lens and is illuminated by a nearly parallel beam of slow electrons at near n o r m a l incidence. The immersion lens forms in its focal plane the low energy electron diffraction ( L E E D ) pattern of the surface from which the central region can be selected by an aperture and used to form an image of the surface by standard electron optical methods. The instrument can also be operated in various emission modes so that a wide range of surface characterization techniques can be combined [7]. The stimulus for developing the technique was provided by the revival of LEED by G e r m e r et al. [8] and the desire to use the diffracted electrons for imaging in a m a n n e r similar to transmission diffraction electron microscopy. The obstacle was the general opinion - based on the work of Recknagel [9] - that the resolution of an immersion lens is

limited by the chromatic aberration of the accelerating field to 8c = V/Uo (in units of the field length L; E = e V f f i energy of the emitted electrons, U0 = accelerating potential). This leads to a resolution in the 100 nm range for L E E R M , which w o u l d hardly justify the effort to develop this technique. The first step was, therefore, to reexa m i n e the optical properties of the homogeneous accelerating field from the p o i n t of view of L E E R M and to calculate the optical properties of a real immersion lens in o r d e r to obtain quantitative resolution data. This was done 20 years ago and never published - except for abstracts [10,11] and is the subject of sections 2 and 3. The optical properties of the other imaging elements are known from the literature and will not be discussed here. The results of these calculations encouraged the construction of a prototype instrument [12]. The further development and present state of the instrument is discussed in ref. [7].

2. The optical properties of the homogeneous acceleration field for electrons with arbitrary energy An immersion lens may, in principle, be divided into an acceleration part and an imaging part and, in a first step, the optical properties of the two p a r t s may be treated separately. The solution of the equation of m o t i o n of an electron in a h o m o geneous electric field is a textbook task which

0304-3991/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

52

E. Bauer / Resolution of LEERM

yields the trajectory

The tangents R t ( Z ) for arbitrary a and p intersect this p l a n e at R I ( Z 0) = 2 0 sin o/

Here distances and energies have already been expressed in reduced units: R =r/L, Z = z / L and P = V/Uo, where r,z d e n o t e the position of the electron, L is the length of the field, Uo the acceleration potential, eV= E the inital energy and a the emission angle of the electron (see fig. 1). The electron is leaving the field in the direction of the t a n g e n t to the trajectory at z = L, i.e. Z = 1, w h i c h is given by

Rt(Z)

= 20 sin

a

[ (Z +1)/20 +c°s2a (1/0 + cos2ot

] cos a .

- 1) + × [ (°°/°)(lv/T-T~-- °.° + 1cos~

-

[ ( 1

+

20 c o s 2 a - cos a

~ + cos2a

)]

]

and thus p r o d u c e the disc of confusion in the " G a u s s i a n image p l a n e " caused by the " a b e r r a t i o n s " of the field. If cos a is expressed by sin a and eq. (5) expanded in powers of sin a then one o b t a i n s to the o r d e r sinaa:

Rt(Z°)=sina2o{-~w [~((-W-oo-1)+l]-I} 1

+ sin3a 20

1

1

2 FVw q w~/-w

The t a n g e n t intersects the Z axis ( R t ( Z ) -- 0) at =

-cos

(5)

(2)

zl

a2Osc

(3)

w h i c h is the location of the virtual i m a g e for an emission angle a. The magnification is M = 1 bec a u s e all r values are equivalent. If the electrons have an energy distribution from E0 = eV0 to E = eV, i.e., from P0 to 0, then the " G a u s s i a n i m a g e p l a n f f ' for energy Eo, defined by a ~ 0, is located at

where the abbreviation w = 1 / 0 + 1 was introduced. The term linear in sin a may be written as

.r om

)]

1/ kTo 1

Rlt(Z°)=sinazP[---P V ~ - 1-~wo

.

(7) This term is zero for O -- 0o, w - Wo, i.e. for zero energy spread of the electrons and, therefore, it represents the chromatic aberration

[0(

0+1-1)

r,R -o0 -Z,7

U=O z=L Z=I

U=Uo

Fig. 1. Optical parameters of homogeneous acceleration field. For explanation see text.

+ 1 - 1

.

(8)

The third-order term in sin a is the first term which is not vanishing when O = 00. Therefore, it represents the spherical aberration for 0 = 0o:

3 0 (1 Rt(Za)=sin3a2p 2 (

~1 +-2-ww1 ) ' 1 ) 2.

8= = sin3a p 1 l/-1/p+ 1

(9a) (9b)

E. Bauer / Resolution of LEERM

If the aberrations are defined in the usual way as the coefficients of the expansion of R t ( Z 0) in powers of a, then the same result is obtained because the contribution of the a3-term from sin a = a - a 3 / 6 in eq. (8) to eq. (9) is zero for p = P0. The maximum emission angle o/A which is accepted by the imaging lens following the accelera t i o n field is determined by the exit aperture at the end of the field, except at very low energies when a is not limited b y the aperture (a = 90°). If R A = r A / L is the aperture radius, then eq. (1) yields for Z - - 1 the limiting angle

s i n ot A -

RA I / 1 + 2p + ~4p2 + 4 p - R2A 2Vr~ V 1 + RE

(10)

Up to this point all expressions except the expansion eq. (6) are valid for all a, p and R A. They are unhandy, however, and will now be replaced by approximations valid for emission microscopy (p0 = 0, p - P0 = c --, 0) and low energy reflection microscopy (c = p - P0 << P << 1). In the first case all electrons pass the aperture so that sin a = 1; in the second case sin aA ---- RA//2V/p. Then, eq. (8) yields with p = Po + c

{

-2c

8o

=

-

2¢;

for emission microscopy (RA >__ 2 f p ) ,

(lla)

8s=

t

1___1__R3

8¢7 A

energy distribution but is of the o r d e r ~ - are intimately related and 8c is solely determined by c. In reflection microscopy energy spread c and energy p are independent quantities, c being determined by the energy spread of the field emission current from the illumination electron gun and p( >> ~) by the voltage difference between gun and specimen. Furthermore, 8c is limited by RA, so that the Chromatic abberration is reduced for two reasons: RA << 1, c < V/-p. Depending upon the relative magnitude of RA and p, the spherical aberration may be larger or smaller in reflection microscopy than in emission microscopy. For A = 8 s / 8c = R 2 / 4 c > 1 the spherical aberration is dominating, for A < 1 the chromatic aberration. Inasmuch as 8s - 1 / P ~ 0 - ~oo, the resolution is improved by decreasing U0 when A > 1 but by increasing U0 - i.e., the field strength at fixed L - when A < 1 because 8c - E / 1 / ~ - - 1 / ~ o . The rule known from emission microscopy (Be - c , 8s - p , i.e. both 8 ' s - l / U 0 ) that the field strength should be increased to increase resolution is, therefore, not valid for reflection microscopy. In o r d e r to obtain optimum resolution at a given energy E = e V = e # Uo and energy distribution A E = e c Uo the aperture has to be optimized taking into account diffraction at the aperture using the radius of the diffraction disc of confusion (13)

8 D = 1 . 2 • / R A,

for reflection microscopy

where )~ is also expressed in units of L: (R A < 21/p- ).

(11b)

For the spherical abberation one obtains to the same approximation ( 1 / p >> 1) from eq. (9) with sin aA = R A / 2 f ~P

53

150

10 - ' °

Uo + V

L

(12a)

for reflection microscopy (12b)

~ 150 10 -1° ~

Uo

L

( L in m )

The condition d S / d RA = 0 with 6 = ~/8~z + 8~ + * ~

for emission microscopy (RA_> 2 1 : ) ,

)" =

(14)

leads to (15)

< 2¢7). Eqs. (11) and (12) clearly show the difference between emission and reflection microscopy. In emission microscopy energy spread c and mean energy - which depends on the details of the

Because of V<< U0 ( h - 1 / ~ o ) , R~pt depends upon V only via p. This dependence is shown in fig. 2a for U0 = 2 5 kV, c = 1 × 1 0 -5 and L = 3 mm [13]. Fig. 2b shows the resolution obtained with the optimum aperture by inserting eq. (15) in

54

E. Bauer / Resolution of L E E R M

20

!

15 A

RADIUS

E

-=_1o @.

U. = 25 kV AV : 0.25 V L

100

I

l

I

RESOLUTION LIHIT

0.25 V 3 mm

A

= 3ram

I

0

I

b

I

I

200 300 Vo(VOLT) ,

1

I

td)O -

500

0

I

100

I

I

200 300 Vo (VOLT)

400

500

Fig. 2. (a) Optimum aperture rA and (b) resolution 8 obtained with optimum aperture for U0 = 25 kV, c = 1 x 10 -5 and L = 3 mm as a function of inital energy Eo = e Vo [13].

'

15

1

10

'

'

t.l. =

2SkY

\

,,v: o.s v," /

v

2.sv

:

I

' I /

ISt~

/ / -

l

\~0

6

~/ / ~ s

I

I

\

uo = 25 kV v =250V

\

~v -- 0.2s v L =

E

~5

2

0

5

10 rA (IJm)

15

20 ---

25

3mm

g:,o

0

5 rA

10 15 (pro)

20

25

---

Fig. 3. Contributions of spherical aberration 8s, chromatic aberration 8c and diffraction ~D to the resolution 8 as a function of aperture radius rA for (a) p = 1 0 -4 and (b) p = 1 0 -2. U0 = 25 kV, e =1 × 1 0 -5, L = 3 mm [13].

eq. (14) [13]. The deterioration of the resolution with deviations from the o p t i m u m aperture is illustrated for the same parameters for two limiting cases (p = 10 -4 -~ E = 2.5 eV and p = 10 -2 ~ E = 250 eV) in figs. 3a and 3b, respectively [13]. It is seen that in the LEED r a n g e the accelerating field allows a resolution 2-3 nm, much better than expected on the basis of the considerations valid for emission microscopy [9].

3. The optical properties of a real electrostatic immersion lens for electrons with arbitrary energy It r e m a i n s to be seen now to what extent the conclusions d r a w n in section 2 are modified by the deviations from the homogeneous field w h i c h occur in any real i m m e r s i o n lens. For this p u r p o s e a p r o v e n electrostatic i m m e r s i o n lens [14] was cho-

sen for two reasons: (i) at the time of the lens design [11] U H V technology was not developed e n o u g h to make the incorporation of a magnetic lens into the U H V specimen c h a m b e r feasible; (ii) a well-tested c o m p u t e r p r o g r a m for the calculation of the potential distribution and of the trajectories in electrostatic lenses was available [15]. A l t h o u g h only a resolution of 70 nm had been achieved with the chosen lens in secondary electron emission microscopy, its theoretical resolution was calcul a t e d to be 8 = 8.5 nm for U0 = 40 kV and p = 5 × 10 -5 when the o p t i m u m aperture rA = 7 /~m is used [14]. If this resolution were largely determined by the aberrations in the emission mode then a resolution in the low nm r a n g e could be expected on the basis of the considerations of section 2 for the reflection mode. The procedure for the field calculation is described elsewhere [15]; the trajectories were

55

E. Bauer / Resolution of L E E R M

calculated by solving the equations of m o t i o n by the modified E u l e r (predictor-corrector) technique. The location of the principal and focal p l a n e s were o b t a i n e d to 3 - 4 significant figures, the spherical aberration constants C3, Cs to 2-3 significant figures. Fig. 4 shows the result Of a typical calculation for fixed initial energy E0 = eVo, specimen and center electrode potentials U0, U¢ and distance zo between these two electrodes. T w e n t y two trajectories are calculated, of w h i c h only thirteen are plotted. Trajectories 1 and 3 which start at r = 0.5 mm with a = 0% and 45 °, respectively, are calculated in the paraxial approximation of the field, and determine i m a g e position zB, magnification M and the location of the focal p l a n e ZF and of the principal p l a n e zp with ze - zF = f . C u r v e 2 differs from curve 1 in energy by AE = e A V = + 0 . 2 5 eV and is used to o b t a i n the chromatic aberration. Curves 1 5 - 2 2 , which are e n h a n c e d by a factor of 50 in the r-direction and which cover the r a n g e a = 0.02 rad to 0.16 rad, serve to calculate the spherical aberration constants (73, Cs. Curves 4-14 w h i c h cover the a - r a n g e from 5 0 - 3 0° in 5 ° steps, from 30 ° to 80 ° in 10 ° steps, give the

CATHODE (OBJECT:

CENTER ELECTRODE

/I Jl 41 Jl il •'1 A 4 4 /11 /I fl /I /I 44 /[

,

]q~d~,l[ I

' ~

I ~

:

l :

1 [

I [ I I

~ 1 I

I 1

[ I I I I I

[

i.~ll

ll~'.l II/ll l~ll YJ~ll YYll IF~II

I

I I |

I

I I I I

L

I I I I II I I

10

I I I I l l l l I ] [

~I~ ~'~,,i._J /I ,4

I

1

L J . ~ = . . ~ J . ~ . .Jf "- f

i

/1

II tl

III III

/I /I /I

II II II

I11 I II I II

I

I

I I I I I

I

U

I

I

I

I

l

I

I

I

I

I

i.J4_

I I i I I i I l l l l l l l I I I I I I I I I I I I I I - - i I~t~l,.kJ.I

I I I

II 1

II II

II II

I

II

II II II

II II II

I

I I

I I

I

'l

~ll'l I q l P l

L L L I

I I I Pr llil I I P l l ~ , i . ~ l I I I'~I P f " l ' l l l I l l I I I I I

i

I I

k " k ~ b 4 - - l - g . 4 - t - - ~ T W LJ - 4 - i . . . . . . . . . .

A

~FI /

I I II'Y I [ I Ig-~ ! IIUgl i.~' I ILgJl V~g.I I IX I I

I I li [ ~

1

I

i [ IJ~

! g l l l ~ ,

; i [ [ I J. !

I I t

[ ~J?'l

i / d / # f J F l . ~ ~ W / ~ ' A I i/.xx~Jr, I 8 X I A ' I I IB'd'/~[ I I IM'iXII I I , / A " d I ] I I"~r~fll I I I I K ~ r d I I f I IlgXII I I 1 I f ~ r ~ l I ] I I IT [ [ I I [I Y I-I I I ] [ I I [ l

J

, b A

ANODE

g / / / M

distortions of the diffraction p a t t e r n (intersections with the focal p l a n e at ZF)- Uo was fixed in all calculations at 25 kV, Uc was varied from - 1 kV to + 1 kV, z0 from 0.6 mm to 1.2 mm. For every U~, z0 pair V was varied initially in large steps in o r d e r to determine the V r a n g e in which real images with high magnification were formed and subsequently in this r a n g e in small steps. The results of these calculations which are imp o r t a n t for the operation of the lens are compiled in ref. [13]. Here only the resolution data will be presented. Fig. 5 shows a typical example of how chromatic aberration 8c, spherical aberration 8s and diffraction 8d contribute to the total resolution 8. T h e s e data may be compared directly with those of fig. 2b which correspond to nearly the same electron energy (Vo = 250 V versus 245 V). It is evident that both 8¢ and 8, are much larger in the i m m e r s i o n lens than in the homogeneous field so that the aperture has to be reduced by a factor of two and the resolution is by a factor of 2 worse. How this situation changes with object position z0 and electron energy Eo -- eV0 is illustrated in figs. 6a and 6b. Here the center electrode potential U¢ is chosen in such a m a n n e r for each z0 and V0 that a focussed i m a g e is o b t a i n e d at an i m a g e distance of z0 = 300 ram. The o p t i m u m aperture radius increases as expected with increasing z0 but the resolution o b t a i n e d with the o p t i m u m aperture is not very sensitive to z0, except at s m a l l z0 and V0. Fig. 7 compares the curves for z0 = 0.8 mm with

I I I I I I I I I I I I ]l II

ft

\<-

//u(=o.skv y v : v.sv

\ ,o

~

2

~V = 0.25 V

/ / ' - - . . . . . L o - - lmm-

Jl -

Fig. 4. Typical computer plot of the trajectories in the electrostatic immersion lens. For explanation see text.

0

5

10 15 20 rA ( p r o ) -

25

Fig. 5. Contributions of 8s, 8c and 8 D tO the resolution 0 of the electrostatic immersion lens. U0 = 2 5 kV, Uc = 0.5 kV, Vo = 245 V, AV = 0.25 V, z0 = 1.0 mm [13].

E. Bauer / Resolution of L E E R M

56

o 'OPT, OM A ERTU E

8

j

.O~~-j I

100

0

I

I

I

I

200

300

4.00 -

500

Vo (VOLT)

I

I

I

I

RESOLUTION LIMIT

\^.

I

"B~ - ~ T E R : OBJECT ] POSITIONI ,.6/ / U0=25 kV, AV=0.25 V -I

I

b

PARAMETER: OBJECT /

POSITION_]

.~.b

1

--6

Uo= 25 kV

4 I

100

0

I

I

I

I

200

300

t~00

500

V. (VOLT)

=

Fig. 6. (a) Optimum aperture rA and (b) resolution 8 (obtained with the optimum aperture) of the electrostatic immersion lens for U0 = 25 kV, ( = 1 x 10 -s as a function of initial energy Eo = e Vo for various object positions [13].

20

I

IISf

b

HOMOSENEOUS FIELD

/.

OPTIMUM APERTURE

A

lo

I 6

@ I

0

I

I

100 200 300 V, (VOLT)

.

I

I

N LENS

HOMOSENEOUS FIELD

"'" IMMERSION LENS

¢-"~ 5

~

I

RESOLUTION LIMIT

I

z,O0 =

I

500

0

100

I

I

I

200

30O

tOO

V.(VOLT)

500

-

Fig. 7. Comparison of homogeneous acceleration field and electrostatic immersion lens [13].

t h o s e for the h o m o g e n e o u s acceleration field of fig. 2~ It is e v i d e n t that the i m m e r s i o n lens requires a much s m a l l e r a p e r t u r e than the h o m o g e n e o u s field and that its r e s o l u t i o n is by a f a c t o r of 2 p o o r e r . This s h o w s that the r e s o l u t i o n is l i m i t e d not by the accelerating field but by the electros t a t i c lens. R e p l a c i n g it by a m a g n e t i c lens with s m a l l e r a b e r r a t i o n s should, therefore, a l l o w one to a p p r o a c h the l i m i t set by the h o m o g e n e o u s accelerating rid&

4. Conclusions L o w e n e r g y e l e c t r o n reflection m i c r o s c o p y ( L E E R M ) is, in principle, c a p a b l e of p r o d u c i n g a r e s o l u t i o n of 2 - 3 nm, d e p e n d i n g upon e l e c t r o n energy, p r o v i d e d that the a b e r r a t i o n s of the i m a g ing lens do not d e t e r i o r a t e the o p t i c a l p r o p e r t i e s of the accelerating field. T h u s , L E E R M is c o m p a r a ble in resolution with high e n e r g y reflection electron m i c r o s c o p y and s u p e r i o r to it b e c a u s e of its c o m p a t i b i l i t y with v a r i o u s e m i s s i o n m i c r o s c o p y m o d e s and with L E E D .

References [1] E. B~che and H. Johannson, Naturwissenschaften 20 (1932) 353; Ann. Physik 78 (1932) 17; Physik. Z. 33 (1932) 898. [2] E. Ruska, Z. Physik 83 (1933) 492. [3] N. Osakabe, Y. Tanishiro, K. Yagi and G. Honjo, Surface Sci. 97 (1980) 393. [4] T. Hsu and J.M. Cowley, Ultramicroscopy 11 (1983) 167, 239. [5] H. Bethge, Th. Krajewski and O. Lichtenberger, Ultramicroscopy 17 (1985) 21. [6] E. Bauer, in: Prec. 5th Intern. Congr. on Electron Microscopy (Academic Press, New York, 1962) p. D-11. [7] W. Telieps and E. Bauer, Ultramicroscopy 17 (1985) 57. [8] L.H. Germer and C.D. Hartman, J. Appl. Phys. 31 (1960) [9] [10] [11] [12]

2085. A. Recknagel. Z. Physik 117 (1941) 689; 120 (1943) 331. E. Bauer, J. Appl. Phys. 35 (1964) 3079. D.R. Cruise and E. Bauer, J. Appl. Phys. 35 (1964) 3080. G. Turner and E. Bauer, J. Appl. Phys. 35 (1964) 3080;

see also E. Bauer, in: Adsorption et Croissance Cristalline, Colloq. Intern, CNRS No. 152 (CNRS, Paris, 1965) p. 21; E. Bauer and W. Dietzel, in: Moderne Verfahren der Oberfl~chenanalyse, Dechema Monographie, No. 78 (Verlag Chemie, Weinhein, 1975) p. 23. [13] W. Telieps, PhD Thesis, Clausthal (1983). [14] G. Bartz, in: Prec. Intern. Kongr. Elektronenmikroskopie, Berlin, 1958, Vol. 1, p. 201; G. Bartz, Optik 17 (1960) 135. [15] D.R. Cruise, J. Appl. Phys. 34 (1963) 3477.