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Ultramicroscopy 111 (2011) 912–919 Contents lists available at ScienceDirect Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic Applying an information transmission approach to extract valence electron information from reconstructed exit waves Qiang Xu a,b,n, Henny W. Zandbergen b, Dirk Van Dyck a,c a EMAT, University of Antwerp, 2020 Antwerp Groenenborgerlaan, 171, U316, Belgium National Centre for HREM, Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Lorentzweg 1, Delft, The Netherlands c Vison Vision Lab, University of Antwerp, 2020 Antwerp Groenenborgerlaan, 171, U316, Belgium b a r t i c l e i n f o a b s t r a c t Available online 1 February 2011 The knowledge of the valence electron distribution is essential for understanding the properties of materials. However this information is difficult to obtain from HREM images because it is easily obscured by the large scattering contribution of core electrons and by the strong dynamical scattering process. In order to develop a sensitive method to extract the information of valence electrons, we have used an information transmission approach to describe the electron interaction with the object. The scattered electron wave is decomposed in a set of basic functions, which are the eigen functions of the Hamiltonian of the projected electrostatic object potential. Each basic function behaves as a communication channel that transfers the information of the object with its own transmission characteristic. By properly combining the components of the different channels, it is possible to design a scheme to extract the information of valence electron distribution from a series of exit waves. The method is described theoretically and demonstrated by means of computer simulations. & 2011 Elsevier B.V. All rights reserved. Keywords: Dynamical scattering Electron distribution Information transmission 1. Introduction High resolution electron transmission microscopy (HRTEM) has proven its ability to provide structure analysis down to the atomic scale. But apart from atomic positions, the distribution of electrons, especially the distribution of valence electrons, is also essential for understanding the properties of materials and the experimental measurement of the valence electron distribution would be hence of considerable importance. Whereas electron diffraction has been successfully explored to obtain the averaged valence electron distribution of crystals [1,2], direct imaging of valence electron distributions in real space would give more details about the local structure, for instance, interfaces, nanoparticles, defects, etc. The recent development of aberration correctors improves the resolution of TEM to the sub-angstrom regime, comparable to that of electron diffraction [3–5]. This suggests that the imaging resolution does not put limit any more on the direct imaging of valence electron distribution in real space. However, the task is not easy and has not been possible till now for two main reasons: (1) The images are formed from the scattered incoming electron wave by all charges, nucleus and all electrons (including core electrons and valence electrons). Among the total charge densities, valence electrons contribute only a tiny fraction. (2) The interaction of the incident wave with the total n Corresponding author at: National Centre for HREM, Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Lorentzweg 1, Delft, The Netherlands E-mail address: [email protected] (Q. Xu). 0304-3991/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2011.01.032 charge densities is so strong that the scattering is a dynamical process, making the HRTEM image not intuitive to interpret. In this paper, we will show that it is in principle possible to reconstruct an image with enhanced valence electron distribution information by combining a series of exit waves (that is the high energy complex electron waves at back surface), based on an interpretation of dynamical scattering using an information transmission approach. To facilitate the understanding, the paper is structured as follows: after briefly introducing the information transmission theory, the problem of obtaining the electron distribution from HREM is described as an inverse problem of information transmission, which can be classified into two subinverse problems: (1) retrieval of exit waves from images and (2) retrieval of the object structure from the exit waves. We first reformulate these two inverse problems in a general term of information transmission to show that the success of solving one problem can be applied to the other. Then we will focus on the possible reconstruction of the electron distribution from the exit waves. And finally based on the developed interpretation, we will give one example of a designed scheme which shows the possibility of combining a thickness series of exit waves to reconstruct a rescaled electron density with enhanced contribution of valence electron distribution and test it by image simulations. 2. General information transmission theory for HRTEM The image formation in an electron microscope can be considered as an information transmission process conveying the Q. Xu et al. / Ultramicroscopy 111 (2011) 912–919 913 retrieve the electron distribution of the object, two inverse problems need to be solved successively: (1) restoration of the exit wave from recorded HREM images and (2) restoration of the electron distribution (object structure) from the exit waves. We will firstly investigate proper channel models to understand the whole transmission process. In both transmission sub-processes, the propagation of a fast electron, neglecting back scattering, is accurately described by the high energy equation which is formally equivalent to a time dependent Schrodinger equation, in which the time can be replaced by the distance z along the incident beam direction, as if the electron propagate with a constant speed , _@c , ðR ,tÞ ¼ HcðR ,tÞ i@t ð4Þ with Fig. 1. Illustration of a simple channel model of interpreting of information transmission. ci gives the transmission characteristic properties of the channel i. information of the investigated object to the image. According to information transmission theory, the transmission process is decomposed in different communication channels or simply channels. A channel can refer to either a physical transmission medium such as a wire, the electron optics system of the microscope, etc, or the quantum mechanical models which describe the physical interaction of the electron with the object. Every channel has its own specific transmission characteristics. And since the characteristics of the channels can differ, the combination of the different components at the end of the transmission process may dramatically differ from the input. In order to understand the whole information transmission process, the deformation and the quality of the transmitted information, one has to investigate the transmission characteristics of the different channels. This is illustrated in Fig. 1, which shows a theoretical single source of channel model to be used in the article. As stated above, the information of the source Is can always be decomposed into different components si: X Is ¼ si ð1Þ i , _ DeVðR ,tÞ 2m ð5Þ , where H is taken in the x–y plane perpendicular to the z axis. R denots a two dimensional vector in the x–y plane; D is the Laplacian operator in the x–y plane; e is the charge of one electron , and VðR ,tÞ is the potential acted on incident electron at time t. The propagation in vacuum through the magnetic lenses and the scattering in the crystal are described by two different Hamiltonians. 3.1. Channel model I (from exit waves to images) Between the exit face of the object and the image plane, the electron wave senses the effect of the magnetic field of the imaging lenses. Since the magnetic field does not change the potential energy of the electron (ignoring spin effects), the corresponding eigen states of the Hamiltonian are plane waves, representing only the kinetic energy. As shown by Scherzer, the magnetic field of the objective lens affects the phases of the different plane wave component. The information transmission from exit wave to image wave is described in the reciprocal space by the following formulas X Cex ¼ fg , ð6Þ g Each of them transfers through a different channel. During the transmission, they are modulated by the corresponding channels so that the information received by the destination Id is X Id ¼ wi si ð2Þ i and wi ¼ fi ðp1 ,p2 ,:::Þ H¼ ð3Þ where wi gives the transmission characteristic properties of the channel i, called its transmission rate; p are those parameters that influenced the transmission properties. This information transmission model is generally applied in quantum mechanics. In this paper, we will use it to describe the information transmission in TEM. 3. Information transmission in TEM The information transmission process in TEM is accomplished in two steps: firstly, the electron interacts with the specimen through dynamical scattering, resulting in a complex electron wave at the exit surface of the sample (simply called exit wave). Secondly, the exit wave is transferred through the electron optics system of the microscope to the image plane where it is recorded. In both steps, the information is distorted. Therefore, in order to X Cim ¼ wg fg , ð7Þ g 3 wg 6expðiðpDf lg 2 þ 12pCs l g 4 þ:::ÞÞ ð8Þ where wg describes the transmission characteristic of the channel, which is well known as the phase transfer function or (for weak objects) the contrast transfer function (CTF) [6]; Cs is the spherical aberration of the objective lens, (other instrumental parameters Cc might be also important, but are ignored here for simplicity), Df is the defocus value, describing the imaging condition; l is the wavelength of the incident beam, g denotes the spatial frequency of the plane wave. According to formulas (6)–(8), the information transmission process from exit wave to image wave can be described by decomposing the signal into different communication channels of plane waves, which are the eigen states of the Hamiltonian representing the kinematical energy. The electron optic system thus behaves as a spatial frequency filter, which modulates the phases of the transmitted plane waves according to the spatial frequencies. Note that the retrieval of the source information (that is the exit wave in this step) has been successfully solved based on the interpretation of image formation as an information transmission. One can use appropriate combinations of the content of the channels to reconstruct the exit wave by varying the 914 Q. Xu et al. / Ultramicroscopy 111 (2011) 912–919 defocus or the wavelength (.e.g. focal series reconstruction) [5,7–10]. 3.2. Channel model II (from object to exit waves) Inside the sample, the electron senses the crystal potential of the object, and thus carries this information of the object to exit wave. This process can be also described in the high energy Eqs. (4)–(5) which can be regarded as a time-dependent Schrödinger equation in which the time is replaced by the depth z (sample thickness) using t ¼ mz=_k and the crystal potential is simplified , as an averaged projected potential UðR Þ along the z direction with Z , , 1 2m jd VðR ,zÞdz ð9Þ UðR Þ ¼ 2 d _ ðj1Þd where d is the thickness of one repeated unit along the z direction; Thus, the Eq. (5) is simplified as , , @cðR ,zÞ i ¼ HcðR ,zÞ @z 4pk ð10Þ where H is H¼ _ DeUðRÞ 2m ð11Þ , The solution of cðR ,zÞ denotes the high energy electron wave at exit surface after penetrating a sample of thickness z. Inspired, by the usual quantum mechanics approach, we expand cðR ,zÞ using the complete set of eigen functions of the Hamiltonian (11) [11]. X ! cðR ,zÞ ¼ wn ðzÞfn ð R Þ , ð12Þ n with En z wn ðzÞ ¼ Cn exp ip El ð13Þ the energy and wavelength of the where E, and l are, respectively, , incident electrons; fn ðR Þ, and En are, respectively, the eigenfunction and the corresponding eigenvalue of the Hamiltonian, which can be obtained by solving the eigen equation , , Hfn ðR Þ ¼ En fn ðR Þ the eigen states to the exit wave. Unlike the thickness dependency is addressed in the normal understanding of dynamical scattering, we would like to point out here the importance of the En dependence of dynamical scattering, especially on imaging the electron distribution. HREM imaging is usually taken,along a low order zone axis orientation. The crystal potential UðR Þ can therefore be considered as a two-dimensional assembly of potential wells, corresponding to the different projected atom columns , UðR Þ ¼ X , , UðR R i Þ ð16Þ i , The eigen-state function fn ðR Þ behave like molecular-orbitals in two-dimensions. The deepest states are highly localized, bound to the core of projected atom columns, and provide the atomic information (the nucleus). They are similar to the 1s states of atoms, but only in two dimension projection plane; whereas the higher states are delocalized among neighboring atom columns, and more sensitive to the valence electrons. In (14) and (15), we only need to consider those bound states (En o 0), since the unbound states with En Z0 are not standing waves, similar to plane waves and provide only nearly constant background and gives no effective information of the object. When the sample is thin , (z 5 El=pjEn j for all En), the transmission rate of the channel fn ðR Þ (see (13)) can be written as wn i pz El Cn En ð17Þ One can see that all the states are transferred in the same phase, but only the deepest states with large 9En9 have a high transmission rate (see Fig. 2). Thus, the exit wave highlights only the nucleus information, for instance the atom positions or atomic types. In contrast, the high energy states, associated with the valence electron distribution, do not effectively contribute to the exit wave because of their small 9En9. When the sample is thick, the transmission rate wn depends nonlinearly on the energy (see Fig. 2), the information of some particular valence states can ð14Þ with H is given in (11). The coefficients Cn will be determined from the boundary condition , Note that the eigen functions fn ðR Þ are only related to the , projected potential UðR Þ, not dependent to the thickness. The total ! density rð R Þ of all the eigen states of the projected potential X X ! ! ! rð R Þ ¼ rn ð R Þ ¼ 9fn ð R Þ92 ð15Þ n n provides the projected electron distribution of the object, and thus is an intrinsic property of the object. In contrast, the exit , wave cðR ,zÞ is strongly dependent on the sample thickness z, due to the dynamical scattering. The thickness dependency of the different channels is explicitly given by (12) and (13). Comparing (12) and (13) to (7) and (.8), one can notice the similarity between the formulas used to describe the two different information transmission steps. This similarity suggests that the eigen state fn of the Hamiltonian can be regarded as a communication channel that transfers the information of the object to the exit wave. The transmission characteristic of each channel fn is given in (13), which can be considered as a ‘‘transfer function’’ for the dynamical scattering. The channels are labeled by n, the quantum number of the electron state or labeled by En, and the eigen energy of the state n. Thus, the dynamical scattering can be then understood as an Eigen state filter, modulating the contribution of Fig. 2. Plot of imaginary part of wn as the function of En. The electron eigen functions act as channels transferring the information of the objet to the exit waves. The exit wave conserves the information of all the occupied two dimensional electron states, but modulated by Eigen energy dependent transmission rate wn. When the specimen is thin, only the deepest states (core state) have enough transmission rates. As a result, the exit wave of thin specimen highlights the information of the core state, indicating the atom positions and atomic type; however, it damps the information of high energy states, thus hardly showing valence electron distribution (see Fig. 4(a) and (g)). When sample is thick, the transmission rate is nonlinearly dependent on the energy, some states are highlighted and some not. The exit wave can be hardly interpreted without known sample thickness. Q. Xu et al. / Ultramicroscopy 111 (2011) 912–919 be more effectively transmitted into the exit wave than those core states. However, without knowing the sample thickness, it is impossible to determine which states are enhanced and how much they are enhanced. The information of valence electron is scrambled and needs to be restored. 3.3. Channel filter of eigen states M X N X win fn ð19Þ n¼1 Bringing (19) into (18), the channel filter is then described as M X N X xin win fn ð20Þ ð25Þ ð26Þ Once each eigen-state has been reconstructed using the channel filter given in (20), the total electron distribution can be then calculated from (15) 3.4. A simple filter to enhance the valence electron distribution The channel filter method, although theoretically possible, is not exactly needed for the retrieval of the valence electron distribution. One of the main reasons is that the retrieval of the electron distribution only 2 requires obtaining the sum of all density functions of fn , instead of the details of each Eigen wave function fn . Other methods can be designed by following the idea of channel filtering. Here we propose a simpler method as an example. One possible filter can be designed by calculating the amplitude square of the standard deviation of a series of exit waves with different sample thicknesses (named as STD method), mathematically written as 2 M 1 X 2 Istd ¼ 9stdðcz1 , cz2 , cz3 ,. . ., czM Þ9 ¼ ðczi cÞ ð27Þ M i¼1 M 1 X c M i ¼ 1 zi x2n ð28Þ where czi denotes the exit wave of the thickness zi and M denotes the total number of the exit waves used for the calculation. For a large number of randomly chosen data sets, Istd will converge to its expectation value Iev, R 2 zu zl ðcðzÞcðzÞÞdz 2 ð29Þ Istd Iev ¼ 9stdðcðzÞÞ9 ¼ R zu zl dz with cðzÞ N X wn fn , ð30Þ n¼1 Eq. (20) can also be expressed in a simple matrix calculus. Let W, Xn, and Yn denote an M*N matrix, an M order vector and an N order vector, respectively, with the elements listed as follows: 2 1 3 w1 w21 wM 1 6 1 7 h i 6 w2 w22 wM 2 7 7, W ¼ win ð21Þ ¼6 6 ^ & ^ 7 MN 4 ^ 5 1 2 M wN wN wN Yn ¼ d1n ð24Þ Xn ¼ W 1 Yn i¼1n¼1 i¼j Thus, the vector of weighting factors Xn is given by c¼ the ith exit wave in the series; M denotes the total where number of the exit waves and xin is the weighting factor of the ith exit wave that is used to create the filter. The weighting factor xin can be derived as follows: From (12), one can write Ciex as h 1 Xn ¼ xn ia j 1 Then the desired xin for n state channel filter can be obtained by solving the linear equation ð18Þ Ciex is fn ¼ 0 with xin Ciex i¼1 Ciex ¼ dij ¼ WXn ¼ Yn As stated above, the dynamical scattering modulates the information of all electron states according to their energy levels. The way of modulation can be described in the formulas similar to that used for interpreting image formations. Thus, the restoration of the exit wave from HREM images and the restoration of the electron density from the exit waves are actually similar problems. The success of solving one of them can be then exampled to the other. One possible way would be to reconstruct each electron state by designing a channel filter. In the information transmission interpretation of the electron dynamical scattering, each electron state behaves as a single channel, transferring into the exit waves with its own predicated character. Thus, in principle one can construct an image representing only the component of one specifically chosen channel by combining a number of exit waves obtained at different modulation conditions with properly chosen weighting factors. These factors can be even complex numbers. Similar example can be found in the retrieval of the exit wave from HREM images taken at different imaging conditions (varying focus or wavelength). The thickness of the object is one of the parameters that influences the characteristic of the channels and free to change. Therefore, the channel filter of an eigen state can be constructed by combining a thickness series of exit waves. This filtering process can be mathematically described as follows. We want to reconstruct the content of the channel n in a linear combination fn ¼ where ( 915 xM n xin d2n dNn T iT , ð22Þ En z wn ¼ Cn exp ip , El cðzÞ ¼ Z zu cðzÞdz ð32Þ zl where zl and zu are the lower and upper integral limit of the thickness, set by the minimum and maximum thicknesses of the series, respectively. Note that wn behaves as a sine wave, following the orthogonality property R z2 ð23Þ ð31Þ lim ðz1 z2 Þ-1 z1 z i expðikn z þ yÞ9z21 expðikn z þ yÞdz ¼0 ¼ lim Rz ðz1 z2 Þ-1 kn ðz2 z1 Þ 0 dz ð33Þ 916 Q. Xu et al. / Ultramicroscopy 111 (2011) 912–919 Bringing (30), (31), (34) and (33) into (29), one can easily get Istd N X 2 9Cn 9 9fn 9 2 ð34Þ n¼1 for zu zl Z 2p kmin ð35Þ with kmin ¼ pEN ð36Þ El where EN denotes the highest eigen energy of the bound state need to be investigated. Thus, Eq. (34) provides the physical meaning of the STD image. Comparing (34) to (15), one can see that the image Istd resembles the total projected electron distribution and can be interpreted as one kind of weighted projected electron distribution with the weighting factor given by jCn j2 for the density of the n state. In another way, by taking the Istd as the final received information ! and taking the projected electron distribution rð R Þ as the input information, one can also build an information transmission channel model to interpret the STD image: the total electron ! distribution rð R Þ is composed of the electron density of all the 2 eigen states 9fn 9 ; each of them transfers into STD images through the channel n with the transmission characteristic 2 described by 9Cn 9 . 2 Note that two properties of the transmission rate 9Cn 9 provide STD images suitable to investigate the valence electron distribu2 tions of samples. One is that 9Cn 9 is only determined by the incident boundary condition, and is not dependent on the sample thickness. Thus the ideal STD image would be free of the influence of the thickness. Secondly, for the condition of plane wave 2 incidence, it has been shown that 9Cn 9 is inversely proportional to the corresponding absolute eigen energy [12], 1 2 9Cn 9 p ð37Þ En This energy dependence is very important for imaging valence electrons: valence electrons constitute only a small fraction of the electrons, and they are delocalized between atoms and distributed in a large space. Thus, the density of valence electron is much smaller than that of the core electrons. An image presenting ! an exact electron density map rð R Þ will only indicate the information of core electron, not give a clear view of the valence electrons, especially for heavy atoms. Thus, the enhancing of the transmission rate of the valence electron information, like what achieved by the standard deviation method, is essential for viewing valence electron distributions. This feature will be further demonstrated in the simulation part. A typical TEM sample is usually wedge shaped, providing a certain thickness variation. For a crystal material, the reconstructed exit waves contain information from an area of hundreds of unit cells. The sub-image with the size of one unit cell corresponds to the exit wave of a certain thickness. One may easily get a series of exit waves with different thickness for the application of the standard deviation method. When investigating the local electronic structure, for instance the electron distribution of near an interface or defects, one may not easily obtain such a thickness series of the exit waves. In this case, one need vary the wavelength of the incident beam to get the series of exit waves with proper modulations, which requires a further research. 2 Note that for different incidence conditions 9Cn 9 will have different energy dependencies. This provides us the flexibility of designing various transmission filters to image the desired Fig. 3. Comparison plot of the transmission rate of electron distributions as the function of eigen energy. The transmission rate of the STD image shows the feature that the electron distributions of the higher energy states have relatively the higher transmission rate. This helps STD image to highlight the valence electron distributions (see Fig. 4(p) and (v)), comparing to the image with the undistorted transmission rate (See Fig. 4(q) and (w)), which gives exact total projected electron distribution map. If the transmission rate behalves close to a delta function, the resulted image will show the electron distribution at one energy level only. The transmission rate of the image obtained at kinematical scattering condition (thin sample) is also included, which determines that the image take at this condition cannot be used to visualize the valence electron distributions. electron density by combining a series of STD images taken with different incidence conditions. For instance, if the electron density of one particular state needs to be highlighted (for instance: O 2p), one can design a channel filter, that only allows the electron density of a particular energy state to pass, as described in the Eqs. (20)–(26). For this purpose, the characteristic transmission rate should approximate a delta function (see Fig. 3). Although the reconstructed images generally provide only the two dimensional projected information, one could apply a tomography technique to obtain the complete three dimensional information, since the reconstructed images are ideally not dependent on the sample thickness. We will not further elaborate all these possibilities in the current paper and leave them in the future work. 4. Simulations In order to test if the STD method follows the design or not, an image simulation has been carried on two structure models of SrTiO3 with different electron distribution configurations. In the first configuration all atoms are regarded as isolated neutral atoms, ignoring valence electron redistribution induced by bonding. In the second configuration bonding effects are taken into account using a multi-pole model to describe the electron transfer and redistribution [13,14]. Thus, these two structure models are mostly same, except slight difference of the valence electron distribution. The exit waves, projected potential and STD images are simulated for these two configurations along [1 0 0] direction and respectively shown in Fig. 4. In Fig. 4, one can hardly see the contrast difference from the kinematical images of the two structure models, for instance, the exit waves acquired at thin sample thickness, Fig. 4(a) and (g) or the projected potential images, Fig. 4(f) and (l). This is expected from Eq. (17) as the kinematical images highlight only the core electron information and the two structure models have nearly same core electron distributions. From the exit waves of the thick samples, the contrast difference between the images of the two structure models can be visualized due to the dynamical Q. Xu et al. / Ultramicroscopy 111 (2011) 912–919 917 Fig. 4. Comparison of Exit Waves, Projected Potential and STD images for two different electron distribution models of SrTiO3: (a)–(e) shows the exit waves of the isolated neutral atom model at the thickness 5, 50, 150, 250, 350 Å along [1 0 0], respectively, and (f) shows the projected potential of the neutral model. A square frame outline the unit cell and the atomic types column are only indicated at its projected position in (a) for simplicity. (g)–(k) shows the exit waves of the multipole model at the thickness 5, 50, 150, 250, 350 Å, respectively, and (l) gives the projected potential of the multipole model. (m)–(o) show the color map of STD images of the neutral model constructed from different ways and (m) is build from a thickness series containing 500 simulated exit waves with thickness range from 1 to 500 Å (step size 1 Å) based on the Eq. (27), representing the ideally expected STD image based on (29); (n) and (p) are constructed from 50 exit waves with the thickness randomly chosen in the range of 1–500 Å, representing the image experimentally available. The STD image (m) is also shown in contour map (p), compared with the contour map (q) of the total electron density and a rescaled total electron density (r), in which the core electron density is rescaled into logarithm scale and the valence electron density keeps normal linear scale. Accordingly the STD images of multipole model are shown in (s)–(x), created using the same way to that of the neutral atom model. (colar images are used for better viewing the details. They are originally all 8 bit gray images, with the gray level translated into the corresponding color (rainbow color map) by using in Digitalmicrograph. The gamma is always set as 0.67 for better viewing the details.).(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) scattering, for instance, Fig. 4(d) and (j); Fig. 4(e) and (k), etc. However, the contrast difference varies with the sample thickness and hardly interpretable. Among these images, the STD images can provide easily interpretable maps of the two different valence electron distributions. For the neutral atom structure model, one can see that all the atoms in the STD image (Fig. 4(m)–(p)) exhibits the nearly spherical symmetry, which is the feature of isolated neutral atoms; whereas the STD image for the multi-pole model (Fig. 4(s)–(v)) indicates more asymmetry feature caused by the bonding effects; for instance the asymmetry effect caused by the dipole of the p orbital for O and the quadrupole pole of the d orbital for Ti atom column can be seen. Note that comparing the intensity level at the center positions of the three atom columns (Fig. 4(m) and (s)), one can see that in the STD images the center peak of the oxygen column is as high as the center peak of the Sr column or the center peak of Ti+ O column; whereas in the projected potential images the center peak of the oxygen is much weaker than that of the others. This also implies that STD images enhance the ‘‘weak’’ information of the light atom signal. The enhancement of oxygen center peak can be also understood from Eqs. (34) and (37), which provide the physical meaning of STD image and its energy related transmission characteristic. To briefly explain here, the center peak of each atom, can be described by the 1s state density function of the corresponding atom column, but modified by multiplying a weighting factor (note this 1s state function used here is a two-dimensional eigen function of the projected potential, close to the Bessel function, not exactly same to the conventional 1s state of H atom). The weighting factor is related to the eigen energy of the corresponding 1s state and influences the contrast contribution of the 1s state to the image. Without considering the weighting factor, the 1s state density function of a heavier atom column is sharper and higher than the 1s state function of a lighter atom column. One could expect a higher central peak for heavier atom columns. However, in the STD image the weighting factor is inversely proportional to the absolute value of the eigen energy of 1s states. It gives a larger weighting factor for the 1s state of the lighter atom column. By taking this effect into account, the final peak height of light atom columns in STD is adjusted to the same level of heavy atom columns. Note that the STD images created from the same structure model resemble each other; despite they are calculated from different thickness series of exit waves (see Fig. 4(m)–(o) for the neutral model and Fig. 4(s)–(u) for the multipole model). The corresponding contour maps are so similar that only one of them is given for each model in the Fig. 4. One can compare the STD images to the corresponding total electron density images (Fig. 4(q) and (w)). From the total electron density images, it is hard to distinguish the two different valence electron configurations, even though three times more contour lines are used in Fig. 4(q) and (w). The similarity of the total electron density images is caused by that the contrast of the image is dominated by the core electron density and the core electron density has nearly no difference in the two structure models. Therefore, the enhancement of high energy states information is crucial for STD images for visualizing the information of valence electrons. There are many ways to enhance the information of the valence electrons, for instance, Fig. 4(r) and (x) shows the contour maps of the rescaled total electron density images of the corresponding structure models. Each rescaled electron density map is created from summing up an image representing the logarithm of the core electron density rc and an image representing the valence electron density rv of the corresponding structure 918 Q. Xu et al. / Ultramicroscopy 111 (2011) 912–919 model ¼ tot Irescaled ¼ Iðlogðrc ÞÞ þ Iðrv Þ ð38Þ Obviously, the way of the rescaling in (38) improves the contrast contribution of valence electrons. Thus, in Fig. 4(r) and (x), one can see the difference of the two valence electron distribution configurations. Note that they resemble to the corresponding contour maps of the STD images Fig. 4(p) and (v), though not exactly same because of the different imaging enhancement mechanics. From the above comparison, one may see that STD provides a way of rescaling the density of electrons for a better visualization of valence electron information. This rescaling is necessary but not unique. To reach the convergence, STD requires a thickness series with a large thickness range (in our simulation, thickness range around 40 nm has been tested). This might hurdle its practical application since the exit wave of thick sample ( 420 nm) can be hardly reconstructed so far. However, this is the current limitation of exit wave reconstruction. Holography and phase plate methods are developing. Furthermore other better rescaling method may be designed, requiring less number of exit waves and less thickness ranges. Towards this direction, more efforts and investigation are required. 5. Discussion From the high energy equation, we have built up an information transmission approach to interpret the dynamical scattering and proposed a scheme to retrieve the information about valence electron distribution. In order to make our approach physically easy accessible, we have only considered the elastic dynamical scattering and ignored absorption effects. However, all these type of simplifications will not influence the major validity of the information transmission approach, since decomposing the exit waves into a series of eigen functions of the Hamiltonian is always allowed by the general quantum mechanics and furthermore, the set of eigen functions of the Hamiltonian always represent the intrinsic properties of the object, not thickness dependent. It is possible to include other factors, such as inelastic scattering. This lead to more complicated mathematics, and more complicated way to design the filters without losing the essence. For instance, one can consider those inelastically scattered electrons as a kind of pseudo absorption effects [15,16], thus the real potential U in the Hamiltonian in (11) is replaced by a complex potential Uu. In the first order approximation, one can write Uu ¼ bU ð39Þ with b ¼ 1 þ ia ð40Þ where the imaginary part a describes the absorption factor and a 51. It can be proved that Enu ¼ bEn , ð41Þ fnu ðRuÞ ¼ fn ðRÞ ð42Þ with R Ru ¼ pffiffiffi b ð43Þ The corresponding exit wave cex u ðzÞ, including absorption, can then be written as X bEn z cex u ðzÞ ¼ Cn exp ip fnu El n X En z aEn z Cn exp ip fnu exp p El El n Since a 51, the exit wave can be simplified as X En z paz X cex u ðzÞ ¼ Cn exp ip fnu þ Cn En fnu E El n l n X En z paz ¼ Cn exp ip fnu þ Uu E El l n ð44Þ ð45Þ where we have applied the boundary condition at the condition of a plane wave incidence [12]. X Cn En fnu ¼ Uu ð46Þ n From (45), one can similarly obtain the expectation value of the corresponding standard deviation image by taking use of the orthogonality of sine wave: R 2 zu zl ðcuðzÞcuðzÞÞdz 2 Istd u Iev ¼ 9stdðcex u ðzÞÞ9 ¼ R zu zl dz X 1 paðzu zl ÞUu 2 2 2 9Cn 9 9fnu 9 þ ð47Þ 12 El n where zl and zu are the lower and upper integral limit of the thickness, set by the minimum and maximum thicknesses of the series. From (47), one can see that absorption will cause an extra correction term in the standard deviation image. But the term is in the second order of absorption factor a(a 5 1), thus, can be ignored in the first order approximation. 6. Summary We have introduced an information transmission approach to describe the image formation in an electron microscope, both for the lenses and for the dynamical scattering in the object. Based on this interpretation, there is no inherent difference between the interaction of high energy electron with the specimen and the interaction of the electrons with the electron optics. Both of them can be considered as information transmission process, through the communication channels, which are the eigen functions of the Hamiltonian. The description of the information transmission in terms of the decomposition in different information channels, each with its own characteristic, also enable to design a particular filter to enhance the content of a particular channel. Based on this interpretation, we have extended the idea of retrieving the exit wave from HREM images to retrieving the object from the exit wave. By applying a so called ‘‘standard deviation method’’, the distribution of valence electrons, though scrambled in the strong dynamical scattering, can be enhanced in the image reconstructed from a thickness series of exit waves. This approach is not limited to the application of imaging the electron distribution. It can be also extended to obtain other desirable properties of the object as well as to design various suitable methods for acquiring them. Acknowledgement The authors are grateful for the financial support by FOMprogram 08IP05 in Netherlands and FWO-project G.0188.08 in Belgium. References [1] J.M. 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