Electron holography

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Ultramlcroscopy 47 (1992) 223-230 North-Holland

Electron holography I. Can electron holography reach 0.1 nm resolution? Hannes Llchte

Insttut fur Angewandte Phystk, Unversttat Tubmgen, D-7400 Tubmgen, Germany Received 9 August 1991, at Editorial Office 15 May 1992

The idea of solwng many of the basic problems m electron microscopy by means of electron holography is strmghtforward However, with the progress m experimental reahzatlon of electron holography at high resolution, severe difficulties show up Presently, the major obstacle m reaching 0 1 nm resolution seems to be the imprecise knowledge about the aberratmns of the microscope

I. Introductmn Generally, the electron wave at the exit face of an object Is modulated both m amphtude a and phase ~b In order to determine the structure of the object, which represents a lot of difficulties m itself, amphtude and phase must be known uniquely In electron microscopy, however, there are stdl some severe restrictions which prevent the gain of this knowledge The results of transfer theory, describing the transfer of the object wave into the image plane, show that the Jmage wave is strongly affected by the aberrattons of the objectwe lens, according to the transfer functions cos X and sin X, amphtude and phase of the object are mLxed up m the Jmage plane by means of the wave aberration X This is schematically sketched m fig 1 for a weakly modulated object wave, l e a = 1 and q5 << 27r Since m conventmnal electron mtcroscopy only the intensity and not the phase of the image wave can be recorded, usually about half of the information about the object is lost To avoid this, one can, m principle, by a statable choice of the wave aberration X direct into the ~mage intensity I = A z that part of the object wave, t e amphtude or phase, in which one is interested Ideally, this can

be achieved for the amphtude by X = 0 and for the phase by X - 7r/2 (Zermke phase contrast) Unfortunately, m an electron m~croscope, X strongly depends on the spatial frequency vector with modulus R If rotational symmetry ts assumed and if coma and astigmatism vantsh, then we have to deal with spherical aberratton and defocus, given by C S and Dz, respectively, which give a wave aberration x(R)

= 237"k[0 25

Cs( R//k ) 4-Jr 0 5 D z ( R//A ) 2]

Since C~ Js fLxed for a gwen microscope, defocus is the only free parameter for opt~mtzatlon of the imaging properties Evidently, an electron microscope is a poor wave-optical device

amplitude a

/ / I ~.~ect ! I

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\

phase

wave / / 1" / II \ \ // sin X (R)

/.t"I"

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image wave

/

II phase

Fig 1 Transfer of the object wave rata the image wave b~ objective lens with wave aberration X The Scherzer focu~ optimizes the sm X paths Aberration-free imaging means cos X -= 1 at a vanishing "cross talk" sm X =- 0

0304-3991/92/$05 00 © 1992 - Elsevier Science Pubhshers B V All rights reserved

H Ltchte / Electron holography I

224

The optimum focus for the path connecting object phase and image amplitude, l e to obtain phase contrast, is the famous Scherzer focus which yields a transfer of the broadest possible range of spatial frequencies ranging from Rml n :

0

385 C~ 1/4k3/4

to

Rre ~ = 1 5 c~-l/4k 3/4, within which the modulus of sin ) is larger than 05 Spatial frequencies smaller than Rmm are missing m the image, i e there is no large-area phase contrast Spatial frequencies larger than Rre s are, due to the oscillations of the sin X function, partially transferred with wrong weight and sign, or completely omitted Rre S can be Improved considerably by decreasing the spherical aberration coefficient Cs a n d / o r increasing the wavenumber k In either case, one finds a deterioration of the large-area phase contrast and, m the ~deal case of C s = 0, one would find a highly resolved image without any phase contrast Consequently, a phase contrast technique of high performance is needed Therefore Rose [1] proposed to correct spherical aberration of third order such that in combination with the spherical aberration of fifth order and a special defocus an optimized phase contrast transfer can be achieved at highest resolution Anyway, Scherzer focus helps only m the case of pure weak-phase objects, because amplitude components, if additionally present in the object, contribute to the intensity according to the transfer function cos X Then one faces severe problems with the interpretation of the image, because one cannot dec~de whether the contrast is due to the amplitude or to the phase of the object The situation gets even worse for strongphase objects because then by expansion of the exp(l&) function considerable amphtude components arise at h~gher harmonic spatial frequencies which are transferred by the cos X transfer function Apart from a weak-phase object, a straightforward interpretahon of an electron image is not possible

2. Electron holography Gabor [2] realized that all these problems can be solved by electron holography, l e by recordmg both amplitude A and phase • of the image wave Then by means of a reconstruction procedure, the image wave can be deconvoluted from the aberrations and defocus such that, ideally, a transfer function cos X = 1 and sin X - 0 results In addition, the whole arsenal of wave-optical tools is available, e g for phase contrast generation and for manipulations in real space and in Fourier space Different techniques were elaborated for the realization of electron holography The one that proved most successful is called "image plane off-axis electron holography" developed by Wahl [3], which uses the Mollenstedt electron bIpnsm [4] Beautiful applications at medium resolution were published by the groups headed by Hanszen, Pozzl and Tonomura The reahsatlon of electron holography at high resolution showing the viability of Gabor's idea to correct the aberrations were reported in detail in refs [5,6] Therefore, only the principle is outlined in the following An electron blprism is inserted m between the back focal plane and the intermediate image plane At a positive blprlsm filament voltage, a coherent plane reference wave is superimposed on the image wave A exp0q)) In the resulting "hologram" Ihol(X, y) = 1 + A 2 + 2A c o s ( 2 ~ ' R c x + ¢b), the interference fringes, spacing R c, are modulated in contrast by the amplitude A and in position by the phase @ of the image wave A and @ can be recovered by a reconstruction procedure, that advantageously is performed numerically The hologram xs digitized and fed into the computer The Fourier transformation of the hologram yields S( R ) = 6( R ) + F T ( A 2) zero beam and autocorrelatlon + 6 ( R - R c ) * F T [ A exp(I@)] side-band + 6 ( n + R c ) * F T [ A exp( - lq~)] side-band

H Ltchte / Electron holography I 2 1 Frmge spacing

Hologram Aberrated

! FI

L -1 FT

225

~mage wave

The resolution of holography is limited by the fringe spacing 1 / R c which gives the position of the side-bands in Fourier space The maximum available radius Rmax of the Fourier spectrum of the image wave - hence the resolution found In the reconstructed wave - cannot exceed R c / n , where n is a number between 2 and 3 This was already pointed out by Wahl [3] The finest fringe spacing 1 / R c we realized so far IS 1 / 3 0 nm referred to the object [7], hence the information needed for a resolution of 0 1 nm after reconstruction can be collected In such a hologram

Computer Fourier spectrum wove aberration Correcbon plate

Display Aberrabon-free object wave

Fig 2 ;chematlc representation of holographic correction of aberratmns In the computer a numencal phase plate is generated such that it anmhdates the wave aberration of the objective lens

2 2 Adjustment of the mtcroscope The side-band convoluted around R c 1s cut out and stored as a subtmage, it contains the complex Fourier spectrum of the recorded image wave, which represents the basis for the subsequent procedures To correct for spherical aberration, residual astigmatism, or defocus, a corresponding phase plate exp(Ig(R)) is generated and applied to the image spectrum

When taking a hologram, the microscope should be adjusted such that the maximum amount of Information can be recorded in the hologram First of all, one has to take special care about the effects that cannot be corrected under reconstruction Therefore, one has to take care about axial coma (which cannot yet be corrected) and the incoherent aberrations (which cannot be corrected in principle if the signal is damped below noise level) The chromatic aberration can only be helped by the small energy width which is provided by the indispensable field emission gun, and it could be further improved by means of an energy filter or chromatic correction [8] Chromatic aberration does not seem to prevent one from reaching 0 1 nm resolution with today's electron microscopes equipped with a field emission gun

FT[ A exp(ldP)/exp(ix( R))] which, by inverse Fourier transform, delivers the corrected object wave a exp(lqb) in a field of view given by the width of the hologram This procedure is schematically sketched in fig 2 Experimental results are shown in fig 4 To obtain satisfying results, one has to look closer to the special needs for optimum experimental reahsatlon

2

4

S

a

t 0

1 2

t 4

R(NM-I)

Fig 3 The attenuatlon function due to the condensoraperture changesslgmflcantly w~th focus (0, 100, 200, 300 and 400 nm, from left to right) The optlmum focus is close to 400 nm (U = 300 kV)

226

H Ltchte / Electron holography I

T h e attenuatton function s t e m m i n g from the finite c o n d e n s o r aperture, fortunately, can substantlally be improved by m e a n s of defocus since

tt is given by r exp [ - c o n s t × ( d x / d R )

1 2] ,

Fig 4 Example for holographic correction of aberrations The hologram of carbon fod was taken at some astigmatism In the image wave, a strong phase contrast of the high spatial frequencies is found m the amphtude (A), whereas the phase (B) mainly exhibits the large-area phase contrast After correction of astigmatism, spherical aberration and defocus, the amphtude (E) shows only very httle contrast, whereas the phase now represents nearly the whole object structure (from ref [11])

H Llchte / Electron holography I

to Gausslan focus Fig 3 shows that the condensor aperture does not prevent a resolution of 0 1 nm The optimum focus turns out to provide two addttlonal advantages (l) The corresponding reduction of the width of the point spread function delivered by the objectwe lens in the image plane to a mm~mum value Consequently, more information about the image points m the hologram is avadable for the subsequent reconstruction (n) The reduction of the number of ptxels needed for the numerical reconstruction, again by the factor of 4 compared to Gausslan focus Therefore, the number of plxels avadable in our

It strongly depends on the focus of the microscope Consequently, the focus should be chosen such that the derwatlve of x(R) is minimized over the whole range of spatial frequencies in which one is interested A detailed analysis shows that the optimum focus for taking a hologram is gtven by Dz = - 0

75

227

Cs(Rmax/k) 2,

where R m a x IS the maximum spatial frequency desirable m the reconstructed wave [9] Then the maximum modulus of the derivative of the wave aberration is reduced by a factor of 4, compared

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H Llchte / Electron holography I

228

processing system (4096) is sufficient for the reconstruction at 0 16 nm at 100 kV and at 0 1 nm at 300 kV This point is essential for the intended real-t~me reconstruction using a CCD camera for image pick-up [10], since the number of pLxels of the CCD cameras applicable in electron microscopy is still limited to 1024

For correction, a phase plate ts generated and applied yielding F T [ a exp(,~b)] exp[l(Xm, c --Xnum)] = F T [ a exp(t4))] exP0Xres), where expOxres) means the transfer function resultmg for the complete holographic process Xre,(R) = 27rk[0 25(C .... - C,.°m)(R/k) 4

The Fourier spectrum of the image wave found m the side-band is affected by the wave aberration Xm,c of the microscope F T [ A exp(,@)]

=

by

Xres 1S g i v e n

2 3 Correction of aberrattons

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H Ltchte / Electron holography I

Precision 010

of Cs{mm}

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Fig 7 T h e p r e c i s i o n o f C s n e e d e d f o r 0 1 n m r e s o l u u o n d e c r e a s e s with i n c r e a s i n g 5 z A c c e l e r a t i n g v o l t a g e ( t o p to b o t t o m ) 400, 300, 200, 100 k V

1 a n d a v a m s h m g cross talk sm Xres ~ 0 HOWever, th~s ~s not easy to r e a h z e b e c a u s e severe p r o b l e m s arise with the prec~ston o f the m e a s u r e m e n t o f Csm~ a n d DZm~c A n y correcUon of a b e r r a u o n s ts h m ] t e d by the accuracy w~th which we

Precision

know t h e m T h e q u e s t i o n is, how a c c u r a t e l y do we n e e d to know t h e m 9 L e t ' s a s s u m e that the s p h e r i c a l a b e r r a t i o n coefflc]ent a n d the defocus of the m i c r o s c o p e can be m e a s u r e d within an accuracy of + 5 C a n d

of Cs (mini I I

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8 T h e p r e c i s i o n o f C s n e e d e d to resolve t h e spaUal f r e q u e n c y R f o r d i f f e r e n t 6 z ( t o p to b o t t o m A c c e l e r a t i n g v o l t a g e 300 k V

0,02,04,08

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230

H Ltchte / Electron holography I

_+~z, respectively Then the wave aberration X is determined only within the limits of + gX given by

the lmagmg process by trial and error would be possible even if the accuracy of C~ and D z were initially worse than the limits given above

gX = (dx/dC~) I~Cl + ( d x / d D z ) [gz I Thts gives an estimate of the residual wave aberration Xres Inserting 0 01 mm and 1 nm for 8C and gz, respectively, one can compute the resultmg transfer function (fig 5) Though these accuracies can only be reached by highly sophisticated measurements, they are evidently insufficient to reach 0 1 nm point resolution at 300 kV For an estimation of the maximum acceptable values for gC and gz, the criterton gX N ~-/6 lS used which would guarantee that cos Xres > 0 86 and the undesirable crosstalk sin Xres < 0 5 In the whole range of spatial frequencies It turns out that then both 8C and gz have to be considerably smaller than the values inserted above (fig 6) The possible combinations of different 8C and gz values which fulfill gX N ~ ' / 6 can be found in fig 7 For a 300 kV electron microscope with field emIsston gun, e g the Phllips CM30 F E G which was recently brought on the market, the needed gC value is plotted versus the intended resolutton in fig 8 For example, 8C = 0 003 mm and gz = 0 3 nm would allow a resolution of 0 1 nm The determination of the wave aberration of the electron m~croscope seems to provide a major difficulty for reachmg highest resolution However, this difficulty is not specific for electron holography, it arises for all techniques of electron microscopy, as well Therefore, it would be useful to think about criteria for finding the best focus and the least spherical aberratton under operatlon of the microscope or under the reconstruction procedure then a stepwise tmprovement of

3. Summary For a further reduction of resolution, electron microscopy urgently needs a method for the determination of spherical aberration and defocus with a precision of about 1 txm and 1 nm, respectively

Acknowledgements Invaluable discussions with Dr Frledrlch Lenz and Dr Karl-Heinz Herrmann are gratefully acknowledged Financial support from the Korber Sttftung, the Volkswagen Stiftung and the Deutsche Forschungsgememschaft was mdtspensable for the achieved progress

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