Computation of contrasts in atomic resolution electron spectroscopic images of planar defects in crystalline specimens

Ultramicroscopy 81 (2000) 223}233

Computation of contrasts in atomic resolution electron spectroscopic images of planar defects in crystalline specimens Termeh Navidi-Kasmai, Helmut Kohl* Physikalisches Institut and Interdisziplina( res Centrum fu( r Elektronenmikroskopie und Mikroanalyse, Universita( t Mu( nster, Wilhelm-Klemm-Str. 10, D-48149 Mu( nster, Germany Received 29 June 1999; received in revised form 30 November 1999 Dedicated to Professor Harald Rose on the occasion of his 65th birthday

Abstract The image obtained in a conventional transmission electron microscope contains contributions from elastically and from inelastically scattered electrons. The electron spectroscopic imaging mode of an energy-"ltering transmission electron microscope allows us to separate these two di!erent contributions by inserting an energy-selecting slit in the energy-dispersive plane of an imaging energy "lter. Selecting a speci"c energy loss corresponding to the ionization of the inner shell of a particular element one can obtain information on the distribution of the element within the specimen. The contrast is then caused by inelastically scattered electrons. For crystalline specimens, however, the contrast will be in#uenced additionally by the elastic contrast. This elastic contrast arises from electron di!raction and increases with increasing crystal thickness. Therefore the intensity distribution in the image cannot directly be interpreted as an elemental map. For a reliable interpretation of contrast formation in elemental maps it is therefore necessary to compute theoretical energy-loss images for various crystal thicknesses and compare these images with the experimental images. As an example we discuss the in#uence of electron di!raction e!ects on energy-loss images of two crystals with planar defects. Linescans are computed for various thicknesses of these crystals. Our calculations are performed using "rst-order perturbation theory to describe the transitions between the Bloch-wave states of the incident electron. The computed linescans for various crystal thicknesses show clearly that the in#uence of the elastic contrast on an image increases when we investigate thicker specimens. Furthermore, the comparison between elastic and energy-loss images demonstrates the partial preservation of the elastic contrast as a function of thickness. We "nd that for specimens thicker than about one third of the extinction length (here &80}100 As ) it is impossible to interpret an energy-loss image directly as elemental map.  2000 Elsevier Science B.V. All rights reserved. PACS: 61.16.!d; 61.72.Dd; 78.90.#t

1. Introduction In a conventional transmission electron microscope the image is formed by elastically as well as by inelastically scattered electrons. When an imaging energy "lter is used, it is possible to obtain either an elastic * Corresponding author. Tel.: #49-251/83-3-36-40; fax: #49-251/83-3-36-02. E-mail address: [email protected] (H. Kohl) 0304-3991/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 9 9 ) 0 0 1 9 5 - 3

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or an inelastic image depending on the selected energy loss. Choosing the energy loss to be zero (*E"0) an image is formed whose contrast is caused by elastic scattering of the incident electron in the specimen. Such a zero-loss "ltered image contains information about the structure of the object, e.g. surface pro"les, lattice defects and domain boundaries. To obtain analytical information about the local distribution of a particular element in a specimen one can select an energy loss *EO0 which corresponds to an inner-shell ionization energy of this particular element in the specimen. In this way the image is formed exclusively by inelastically scattered electrons which have su!ered this preselected energy loss in the specimen. One thus obtains an elemental map where the intensity is proportional to the local concentration of that element. Hashimoto et al. [1] and Wiedenhorst et al. [2] have presented elemental maps with a high spatial resolution and related the intensity maxima to the atomic positions of this element. For crystalline specimens, the interpretation of such an image is hampered by di!raction e!ects. One then has to distinguish between two di!erent contrast mechanisms. The "rst contrast type is the inelastic contrast which is due the inelastic scattering processes in the crystal. The second one is the so-called `preservation of the elastic contrasta. The elastic contrast is caused by coherent scattering of the primary electrons by the periodic potential of a crystal. The wave function of these electrons inside the crystal is a Bloch wave that exhibits the periodicity of the crystal. One can therefore consider any inelastic scattering process as a transition from one Bloch-wave state into another one because the waves before as well as after the inelastic scattering process exhibit the lattice periodicity. The shape of the Bloch waves is then partly preserved in the inelastic image [3]. In this way, the elastic contrast caused by electron di!raction in the crystal strongly in#uences the image. The importance of the elastic contrast increases with increasing thickness. So it is possible that intensities occur in electron spectroscopic images, which are not caused by the inelastic scattering of the primary electron with the atomic electrons but are solely due to the elastic interactions in the specimen. This fact may lead to a false interpretation of an elemental map, because then intensity maxima may exist which do not correspond to the position of an atom, whose inner shell has been excited. To estimate the in#uence of electron di!raction e!ects on electron spectroscopic images we have therefore performed theoretical computations of the contrast in atomically resolved electron spectroscopic images. For our investigations we have chosen two crystals with planar defects. The "rst one is a SrCuO crystal with  a planar oxygen defect. The second one is an In O -doped ZnO crystal, which shows a planar Indium defect.   The energy-loss images and the elastic images of these two defects are computed for di!erent crystal thicknesses.

2. Theoretical considerations The quantitative treatment of image formation by inelastically scattered electron in electron microscopy has been pioneered 25 years ago by Rose in a famous publication [4,5] dealing with non-crystalline specimens. Recently, he has extended his theory to include the in#uence of multiple elastic scattering on the total inelastic intensity [6]. In this paper we con"ne the discussion to inner-shell excitations. To describe the interactions of the primary electron within the crystal we follow the treatment by Stallknecht and Kohl [7]. We consider a perfect crystal slab with plane and parallel surfaces, which has an in"nite extension in x}y-direction and a thickness d in z-direction. The direction of incidence of the primary electron beam is perpendicular to the crystal surface. The interaction between the incident electron and the crystal is described by the timeindependent SchroK dinger equation [3,8] (H #H #H )"W2"E"W2, (1) # #! ! where E is the kinetic energy of the electron. The operator H describes the interaction between the incident #! electron and the crystal, H is the operator for the propagation of the incident electron in the crystal and # H the crystal Hamiltonian, respectively. !

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The eigenstates "U 2 of the crystal electrons of energy E are solutions of the equation L L H "U 2"E "U 2, n"0,1,2, E "0. (2) ! L L L  The total state vector "W2 in Eq. (1) can be written as sum of product states of the eigenstate "U 2 of the L crystal and the state "u 2 of the primary electron L "W 2" "u 2"U 2. L L L L Inserting Eq. (3) in SchroK dinger equation (1) yields

(3)

(H "U 2#H "U 2#H "U 2!E"U 2)"u 2"0. (4a) # L ! L #! L L L L Multiplying Eq. (4a) by 1U ", the excited electronic state of the crystal, from the left-hand side and K considering the orthogonality relation 1U "U 2"d , we obtain K L KL (H #E !E)"u 2"! 1U "H "U 2"u 2. # K K K #! L L L The expression

(4b)

H " : 1U "H "U 2 (5) KL K #! L is a measure for the excitation probability of the crystal from the state "U 2 into the state "U 2 and L K

k E!E " K (6) K 2m  describes the kinetic energy of the incident electron after the inelastic interaction, where m is the mass and  k the wave number of the primary electron before (m"0) or after an inelastic scattering process. K Inserting Eq. (5) and (6) into Eq. (4b) and using the real-space notation "u 2"u(k , r) we obtain L L ! 

k D! K #H (r) u(k , r)"! H (r)u(k , r). (7) KK K KL L 2m 2m   L$K In real space the matrix element in expression (5) has the form








H (r) " : UH (R)<(r, R)U (R) d,# >, R, KL K L

(8)

where R,(r , r ,2, r # ; R , R ,2, R ! ) denotes the positions of the N electrons and the N nuclei in the   ,   , # ! crystal and r represents the position of the primary electron. The operator <(r, R) describes the Coulomb interaction between the incident electron and the crystal electrons as well as the atomic nuclei. For a su$ciently weak interaction potential the wave functions u(k , r) of the excited states are small. K Using "rst-order perturbation theory, Eq. (7) can then be approximated by








0 if m"0,



k ! D! K #H (r) u(k , r)+ (9) KK K 2m 2m !H (r)u(k , r) if m O0.   K  The homogeneous equation in the "rst line of Eq. (9) as well as the homogeneous part of the second line in Eq. (9) describe the elastic scattering of the incident electron in the crystal before and after the inelastic scattering process occurs. The inhomogenity of the second line in Eq. (9) represents the inelastic scattering process where the crystal is excited from the initial state U into the "nal state U . The solution u (k , r) of  K  K

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the homogeneous Eqs. (9) can be obtained using dynamical theory [9]. We thus obtain a superposition of Bloch waves uH(k , r) [10,11] K u (k , r)" eH(k )uH(k , r)" eH(k ) cH(k ) exp+i(kH#g)r,  K K K K E K K H H E " eH(k ) cH(k ) exp+i(k #cH(k )#ikH(k )#g)r,, (10) K E K K K K H E where the index j represents the Bloch wave number j and the reciprocal lattice vector g denotes the excited Bragg re#ections. The eigenvalues cH, the eigenvectors cH and the excitation coe$cients eH(k ) are obtained E K from the well-known eigenvalue equation for the Bloch waves [9}11]. The set of the absorption coe$cients kH describes the attenuation of the Bloch waves in the crystal. The function u (k , r) is the wave function of   the elastically scattered electrons. To determine the in#uence of inelastic scattering, we have to solve Eq. (9) for mO0, thus including the inhomogenity. Using the Greens function method we obtain for the inelastic wave function u (r) K



u (r) " : u(k , r)" K K

G (r, r) (!H (r)u (k , r)) dr, mO0, K K    where G is the Greens function constructed of Bloch waves [12,13] K 2m ujYH(k, r)uHY(k, r)  G (r, r)"lim dk, K (2p)  j k!k !ig K E where k"(k , k , k ) is the wave vector of the primary electron after an inelastic process. V W KX With the approximations k+k and k +k we "nally obtain the expression X K KX 2m uHYH(k, r)uHY(k, r)  u (r)"! lim dk H (r) eH(k )uH(k , r) dr K K   k!k !ig (2p)  K  E I HY

2m HHYH (k, k )  eH(k )lim K  dk dk dk , ! uHY(k, r)  (2p)  k !k !ig V W X X W V I I I X KX HYH E where the abbreviation







HHYH (k, k ) " : K 

 




uHYH(k, r)H (r)uH(k , r) dr, K 

(11)

(12)



(13)

(14a)

 is the excitation matrix element. Inserting Eq. (8) in Eq. (14a) we obtain for the matrix element



HHYH (k, k ) " : K 

uHYH(k, r)UH (R)<(r, R)U (R)uH(k , r) d,# >,I  d r, K  

(14b) PY 0 which gives the probability amplitude for an inelastic scattering event where the crystal state changes from the initial state U in excited state U while the Bloch wave state uH(k , r) of the incident primary electron  K  changes to the "nal Bloch wave state uHY(k, r). To obtain an expression which describes the inelastic imaging of the whole crystal we choose an exciton-like representation for the excited electronic state U of the crystal, i.e. a linear combination of K localized atomic excitations [14]. We describe the crystal as being composed of planes of atoms perpendicular to the direction of incident electron (z-direction). The state within each plane is then described by two-dimensional exciton with wave vector K"(K , K , 0). In this representation the "nal states of the V W

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crystal can be described by the transferred momentum and therefore we can classify the scattered electrons according to their exit direction for the inelastic imaging. The "nal state of the crystal is then denoted by the wave vector of the exciton K"(K , K , 0), the azimuthal quantum number l, the magnetic quantum V W number m of the excited state of the atom, the wave number of the ejected electron i and the number of the layer. When we also account for the imaging process within the microscope, we obtain for the wave function of the inelastically scattered electron at image plane z"z [7]



2m k  u (x , y )" exp+ik (b#g), exp i K (x #y ) V W KGJYKY_ ) )  K (2p) M 2M f









1 ;lim exp !i [k x #k y ] ;exp+ik d, eH(k ) V W X  M X W V E I I I HHY HHYH (k, k ) K  ;exp+[icHY(k)!kHY(k)]d, cHY(k) exp+ig d, E X k !k !ig  X KX E 1 (k #g ) f (k #g ) f W ; k dk dk dk , V V , W ;exp !i [g x #g y ] ¹ W  K V W X M V k k K K (15)








where x and y represent the image coordinates in x}y-direction, ¹ is the transmission function describing  the e!ect of lens aberrations and the diaphragm in the aperture plane, M the microscope magni"cation, f the focal length and g the reciprocal lattice vector, respectively. To calculate the current density in the image plane we have to compute the modulus square of the wave function and proceed as described in Ref. [15] I (x , y )""u (x , y )". (16) KGJYKY_ )V )W  KGJYKY_ )V )W  To obtain the intensity distribution of the inelastic image one has to sum and to integrate over all "nal states that contribute to an experimental energy-loss image. In addition we have to sum over all non-equivalent positions of the investigated element inside the unit cell and over all crystal planes



I (x , y )" "u (x , y )" dK dK , KG KGJYKY_ )V )W  V W    )V )W *  JY KY K where m denotes the occupied initial state. A detailed treatment is given in Ref. [7]. 

(17)

3. Results and discussion We examined two crystals and calculated the intensity pro"le of the inelastically scattered electrons for various thicknesses of these crystals. For our numerical calculation we used a FORTRAN program named ELSA which is based on Weickenmeiers programs for energy spectroscopic di!raction patterns [16,17] and was extended further by Stallknecht for the computation of the energy-loss images. To obtain the intensity distribution of the inelastically scattered electrons we proceed in three steps: 1. We calculate the eigenvectors and the eigenvalues for the incoming Bloch waves and those leaving the crystal at the exit face. 2. Next the atomic matrix elements are calculated for the transition of the crystal from the initial state into the excited state due to inelastic scattering.

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3. We then compute the energy-loss images for di!erent thicknesses of the examined crystal to study the in#uence of the elastic contrast on the inelastic image For our calculations we neglected multiple inelastic scattering processes and computed background subtracted images, i.e. elemental maps. Due to large computation times we con"ne ourselves to imaging lattice planes and therefore computed linescans. Stallknecht and Kohl [7] have shown in test calculations that a `one-exciton approximationa is su$cient for our computation. The images, calculated for an exciton wave vector K"0 are practically indistinguishable to the results of a full calculation. Because of this fact we have assumed a slit-shaped objective aperture for our calculations in Sections 3.1 and 3.2. We have thus reduced our computation time. The in#uence of the lens aberrations will be presented in Section 3.3. 3.1. Images of a planar oxygen defect in a SrCuO2 crystal The in"nite layer material SrCuO is a superconductor. It contains planar defects for which there exist two  di!erent models. In the "rst model proposed by Zhang et al. [18] an extra layer of Sr}O is inserted in the "nite layer structure of SrCuO and apical oxygen atoms are brought into the in"nite layer. The second  model proposed by Azuma et al. [19] is based on the removal of half of the oxygen atoms in the Cu}O layer and the removal of some cations in the layers adjacent to that Cu}O layer. For these two models we computed the energy-loss images. For the model proposed by Zhang the total concentration of the oxygen atoms in the defect is nearly the same as in the perfect crystal so that the image contrast is almost the same for both regions, contrary to the experimental results obtained by Wiedenhorst et al. [2]. Our calculations have shown that for the Zhang model one never obtains a substantial decrease of the intensity in the oxygen image of the defect. Therefore we con"ne our discussion to the Azuma model and compare our results with the experimental data obtained by Wiedenhorst et al. [2]. Fig. 1 shows a projection of the structure along the [1 0 0] direction. The dimensions of the super cell in the [1 0 0], [0 1 0] and [0 0 1] directions are 7,8, 7,8 and 21 As , respectively. The direction of the incident beam is [1 0 0]. In the perfect crystal the (0 0 1) planes are occupied alternately by CuO and Sr. In the defect layer we  assumed that in the CuO layer 50% of the oxygen positions are occupied. The Sr layers above and below  the defect are also occupied with 50% strontium. For the calculation of Bloch waves we have used a 37 beam case for the (0 0 1)-systematic row. The extinction length is 246 As . We have calculated linescans through oxygen maps using the O}K edge and assuming an energy loss of 550 eV and an acceleration voltage of 300 kV. The intensity is given in arbitrary units and the linescans are normalized to the same current density. Fig. 2 shows the computed linescan for a crystal thickness of 20 As . The image represents the oxygen distribution in the specimen. Each intensity peak corresponds to an oxygen position in the unit cell (Fig. 1). As expected, the intensity of oxygen in the defect layer is lower as compared to the fully occupied Cu}O layers. In this case the image can be interpreted directly as an elemental map. By examining a thicker crystal (for example 100 As ) in Fig. 3, other features appear between the oxygen peaks. One can observe additional peaks at the strontium positions. Comparing Fig. 3 with Fig. 1 we see that this e!ect must be caused by elastic contrast. In Fig. 4 we show a linescan for a crystal thickness of 200 As . The e!ect of the elastic contrast is even more pronounced as compared to Fig. 3. This was expected, because the elastic contrast should increase with increasing crystal thickness. Therefore the energy-loss image will be increasingly in#uenced by the preservation of the elastic contrast when the specimen becomes thicker. Whereas Fig. 2 can readily be interpreted as an elemental map, Figs. 3 and 4 show a intensity distribution which is caused by inelastic contrast due to the ionization of oxygen atom as well by preservation of elastic contrast, caused by elastic scattering of the electron with strontium atoms in unit cell. The comparison with the corresponding elastic images for these three di!erent thicknesses, displayed in Fig. 5, con"rms this assumption.

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Fig. 1. Projection of a SrCuO -crystal in the [1 0 0] direction. 

Fig. 2. Simulated oxygen map of a defect in a SrCuO -crystal  for a thickness of 20 As .

Fig. 3. Simulated oxygen map of a defect in a SrCuO -crystal  for a thickness of 100 As .

Fig. 4. Simulated oxygen map of a defect in a SrCuO -crystal  for a thickness of 200 As .

3.2. In2 O3 -doped ZnO crystal As second example we calculated the energy-loss images of a In O crystal with a planar defect of In. The   projection of this structure in [0 0 1] direction is given in Fig. 6. The (1 0 0) planes are alternately occupied with Zn and O. In the defect area there is In O layer between the ZnO layers. The dimensions of the super   cell are 23,4, 5,629 and 3,25 As in the [1 0 0], [0 1 0] and [0 0 1] direction respectively [20]. For our computations of the In-M energy-loss images we considered a (1 0 0)-systematic row and used a 41 beam  case. The extinction length is 251 As . The direction of the incident beam is [0 0 1]. We assumed an energy loss of 520 eV, which lies above the M edge of indium but still below the O}K edge. The acceleration voltage is  300 kV. The other conditions for the computation are the same as explained in Section 3.1. In Fig. 7 we present the computed energy-loss image of In-M for a crystal thickness of 20 As . All intensity  peaks observed are at the In-positions. For this crystal thickness the image can be directly interpreted as an

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Fig. 5. Simulated elastic images of a SrCuO -crystal for various  thicknesses.

Fig. 6. Projection of a ZnO-crystal with a planar In defect in the [0 0 1] direction.

Fig. 7. Simulated indium map of a defect in a ZnO-crystal for a thickness of 20 As .

Fig. 8. Simulated indium map of a defect in a ZnO-crystal for a thickness of 100 As .

elemental map of indium. Considering a thicker specimen (for example 100 As ) in Fig. 8 one sees two peaks in addition to the intensity peaks at the indium positions. A comparison with Fig. 8 proves that these two peaks appear at the positions of oxygen atoms. This clearly demonstrates that the preservation of elastic contrast in#uences the inelastic image with increasing thickness. At last Fig. 9 presents the energy-loss image of indium for 500 As thickness. The intensities on the positions of oxygen and indium are almost the same. One can hardly distinguish between the intensity peaks which are caused by inelastic contrast at the indium positions and those caused by the preservation of elastic contrast. The computed linescans, Figs. 7}9, clarify that the in#uence of the elastic scattering inside the crystal increases with increasing crystal thickness. As demonstrated in Section 3.1 this e!ect is caused by the preservation of the elastic contrast and its amount depends on specimen thickness. This conjecture is supported by the corresponding elastic images as shown in Fig. 10.

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Fig. 9. Simulated indium map of a defect in a ZnO-crystal for a thickness of 500 As .

231

Fig. 10. Simulated elastic images of a ZnO-crystal for various thicknesses.

3.3. The inyuence of the lens aberrations For the computation of the energy-loss images in Sections 3.1 and 3.2. we followed previous work [7]. We used the `one-exciton-approximationa and neglected the lens aberrations. In the following example we discuss the in#uence of the spherical and chromatic aberration on an energy-loss image and consider a two-dimensional exciton representation for calculating the image. The transmission function describing the e!ects of lens aberrations and the diaphragm in the aperture plane is given by ¹ (x , y ; k )"A(x , y ) exp+!ic(x , y ),    K    

(18)

with the amplitude



1 for x #y )o    A(x , y )"   0 otherwise,

(19)

where o is the radius of the objective aperture, and with the phase shift 










dE x #y C (x #y ) *f  !   . c(x , y )"k Q  #C    2E 4 f 2 f

(20)

Here C and C are the spherical and chromatic aberration coe$cients, f is the focal length, * f is the defocus,   dE is the deviation from the mean energy and E is the acceleration energy of the primary electron. In Eq. (15) the function ¹ represents the according transmission function in the image plane (x , y ), where the  arguments ((k #g ) f/k ) and ((k #g ) f/k ) include the exciton wave vector in x- and y-direction. To V V K W W K determine the in#uence of the lens aberrations we have to multiply the inelastic wave function in Eq. (15) with the transmission function ¹ and integrate over the aperture plane. The microscope parameters used for our  calculations are shown in Table 1. Our results for a crystal thickness of 200 As , displayed in Fig. 11, should be compared to the results in Fig. 4, where the aberrations have been neglected. We "nd that the intensity peaks at the strontium positions outside the defect area have disappeared and that the oxygen peaks have broadered. On the other hand the intensity peaks of strontium in the defect have increased as compared to the oxygen peak so that a distinction

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Table 1 Microscope parameters used for calculating the energy-loss image represented in Fig. 11 a (Collection angle)

9.5 mrad

f *f C  C  dE E

1.7 mm 50 nm 1.2 mm 1.2 mm 12 eV 300 keV

Fig. 11. Simulated oxygen map of a defect in a SrCuO -crystal for a thickness of 200 As including the in#uence of lens aberration e!ects. 

between these three peaks is no longer possible. It can be clearly seen that the lens aberrations lead to a blurring of the image thus decreasing the resolution attainable.

4. Conclusion With modern energy "ltering transmission electron microscopes it is possible to obtain elemental maps with almost atomic resolution which give information about the element distribution in the specimen. For crystalline specimens, however, the interpretation of such images is di$cult. We have discussed the preservation of the elastic contrast which is due to electron di!raction. The linescans for both crystals show that this fact in#uences the inelastic image when one has thicker crystalline specimens. The small crystal thicknesses necessary to avoid this e!ect may not be feasible in practice. Therefore one most be careful with the interpretation of electron spectroscopic images and take the dynamical di!raction e!ects into account for thicker specimens. So far relativistic e!ects have only been considered using the relativistic corrections for the electron mass and wavelenght. As has been shown by Knippelmeyer et al. [21], this is not su$cient when describing images of thin specimens. We are therefore planning to develop a fully relativistic treatment in the near future.

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Acknowledgements We would like to thank Dr. A. Weickenmeier and Dr. P. Stallknecht for providing their programs. Furthermore we thank Dr. B. Freitag (Univ. Bonn) for valuable discussions.

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